May 26, 2026
In this paper, we consider the problem of simultaneous testing of multivariate normal means under arbitrary covariance dependence. Specifically, let \(\boldsymbol{X}\sim N_n(\boldsymbol{\theta},\boldsymbol{\Sigma})\), where \(\boldsymbol{\theta}\in\mathbb{R}^n\) is unknown and \(\boldsymbol{\Sigma}\) is a known positive definite covariance matrix. The objective is to test \(H_{0i}:\theta_i=0\) against \(H_{Ai}:\theta_i\neq 0\), simultaneously for \(i=1,\ldots,n\). We establish a general admissibility theorem for a broad class of monotone residual-based step-down multiple testing procedures which iteratively rank the active hypotheses using statistics obtained through locally adaptive strictly increasing transformations of suitably standardized residual statistics arising from conditional normal distributions. Our main result shows that every such procedure is admissible with respect to a vector-valued loss function whose components are the usual individual \(0\)–\(1\) testing losses. The proof relies on a delicate geometric analysis of the induced acceptance regions together with structural invariance properties of the adaptive stagewise rejection indices. The theorem substantially extends the admissibility theory developed for the maximum residual down procedure of [1] and reveals that admissibility under dependence is fundamentally driven by the monotone ordering structure induced by the residual statistics rather than by the precise functional form of the testing rule itself.
Keywords: admissibility; arbitrary dependence; decision theory; multiple testing; residual-based step-down procedures; residual statistics; vector loss.
MSC 2020 subject classifications: Primary 62C15, 62F15; secondary 62C25, 62H15.
Multiple hypothesis testing has emerged as one of the central themes in modern statistical inference, particularly in the analysis of high-dimensional datasets arising in genomics, bioinformatics, brain imaging, astronomy, economics, finance, medicine, and related scientific disciplines. In many contemporary applications, a very large number of hypotheses must be tested simultaneously in order to identify a relatively small number of significant signals hidden among a vast collection of null effects. Such large-scale inferential problems require statistical procedures that effectively distinguish true signals from noise while maintaining adequate control over erroneous decisions at a global level.
The need for simultaneous inference naturally introduces multiplicity issues, since procedures designed for individual testing problems may accumulate errors rapidly when applied repeatedly across a large collection of hypotheses. Consequently, a vast literature has developed on multiple testing procedures controlling global error criteria such as the family-wise error rate (FWER) and the false discovery rate (FDR); see, for example, [2], [3], [4], [5], [6], [7], [8], [9], and the references therein. While classical FWER-controlling procedures often become excessively conservative in large-scale settings, FDR-based methodologies typically provide a more flexible trade-off between false discoveries and false non-discoveries, and have therefore become central to modern large-scale multiple testing.
A substantial portion of the theoretical development in this area initially relied on the assumption that the underlying test statistics are independent. However, in many scientific applications, the test statistics exhibit significant dependence arising from spatial, temporal, biological, or network-driven interactions. Ignoring such dependence may severely distort the behavior of standard multiple testing procedures. Several authors have demonstrated that, under strong dependence, traditional \(p\)-value based stepwise procedures may exhibit highly unstable behavior, leading to excessive variability in the numbers of false discoveries and false non-discoveries; see, for instance, [10], [11], [12], [13], and [14]. Moreover, [15] emphasized that failure to incorporate dependence may produce substantially misleading inferences, particularly in highly correlated settings. On the other hand, dependence may also provide additional structural information that can potentially be exploited to improve inferential efficiency. In this sense, dependence may become a “blessing” rather than a “curse”; see [16], [17], and [18].
These considerations have motivated a growing literature on multiple testing procedures that explicitly account for dependence structures among the test statistics; see, for instance, [4], [19], [20], [21], [8], [22], [9], and the references therein. Nevertheless, despite substantial progress on controlling global error criteria such as FWER and FDR under various forms of dependence, comparatively less attention has been devoted to the decision-theoretic validity of such procedures. As emphasized by [23], [24], and [25], the study of admissibility and related optimality properties is important not only for comparing competing procedures, but also for understanding the deeper structural principles governing multiple testing rules viewed as compound decision problems.
This issue becomes particularly significant under dependence. While several commonly used procedures continue to possess desirable global type I error controlling properties in dependent settings, a number of authors have shown that many standard \(p\)-value based stepwise procedures become inadmissible under natural vector-valued loss functions when the test statistics are correlated; see [26], [27], [28], and [29]. Inadmissibility indicates the existence of alternative procedures which perform at least as well under every parameter configuration and strictly better under some configurations. Thus, admissibility provides a fundamental decision theoretic validation that a procedure cannot be uniformly improved upon.
A major development in this direction was the Maximum Residual Down (MRD) procedure proposed by [1], which explicitly incorporates the dependence structure through adaptively formed residual statistics arising from conditional normal distributions. The MRD procedure was shown to possess an important admissibility property under a natural vector-valued loss formulation, thereby providing one of the very few known examples of a nontrivial admissible stepwise multiple testing procedure under arbitrary covariance dependence.
However, the admissibility proof for the MRD procedure is closely tied to the specific structure of the residual statistics employed there, and the underlying structural principle responsible for admissibility remains somewhat implicit. In particular, because the residual statistics are constructed through adaptive stagewise index selections depending on the observed data vector itself, the associated acceptance regions possess highly nontrivial data-dependent geometry. As a consequence, it is not immediately clear whether admissibility is a special feature of the precise MRD construction itself, or whether it reflects a deeper structural phenomenon associated with covariance-adapted residual geometry and more general locally adaptive residual-based step-down mechanisms.
The residual statistics underlying the MRD procedure naturally induce a stagewise ordering of the active hypotheses according to the strength of the corresponding residual evidence against the null hypotheses. It is therefore natural to ask whether admissibility fundamentally depends on the precise algebraic form of the MRD statistics themselves, or rather on the monotone ordering structure generated by these covariance-adjusted residual quantities. Since strictly increasing transformations preserve the local ordering of residual evidence, they provide a natural framework for investigating the structural stability of admissibility under more general adaptive residual-based scoring mechanisms.
Motivated by these considerations, the present paper investigates admissibility properties of a broad class of residual-based step-down multiple testing procedures under arbitrary dependence structures within a multivariate Gaussian modeling framework. The present paper identifies and formalizes a fundamental monotone ordering structure underlying admissibility in this setting and develops a rigorous invariance-based framework for analyzing the adaptive geometry induced by the step-down mechanism. Unlike classical p-value based procedures which often treat dependence primarily as a nuisance requiring correction, the residual-based framework underlying the present work actively exploits the covariance structure through adaptive conditional residualization at each stage of the procedure. The resulting residual statistics dynamically incorporate information from previously eliminated coordinates and therefore evolve in a manner intrinsically tied to the underlying dependence geometry. In this sense, the proposed framework may be viewed not merely as dependence-robust, but fundamentally dependence-adaptive. Our analysis reveals a remarkable invariance phenomenon along suitable admissibility directions: despite the highly adaptive and combinatorial nature of the procedure, the stagewise elimination sequence among the competing coordinates remains invariant locally along the direction of admissibility. This invariance structure yields a tractable geometric characterization of the induced acceptance regions. In particular, we show that a remarkably broad class of highly heterogeneous and locally adaptive residual scoring mechanisms based on suitably standardized residuals arising from conditional normal distributions gives rise to admissible step-down multiple testing procedures under arbitrary covariance dependence.
Although each generalized residual score is obtained through a strictly increasing transformation of the corresponding MRD residual magnitude, the locally adaptive nature of the transformations allows the induced stagewise rejection ordering itself to differ substantially from that of the original MRD procedure. The result substantially extends the existing admissibility theory for dependent multiple testing procedures and provides a unified decision-theoretic justification for a broad class of adaptive residual-based step-down testing rules, including the MRD procedure as an important special case. Importantly, the generalized procedures considered here need not preserve either the adaptive stagewise rejection ordering or the threshold structure of the original MRD procedure. Thus, admissibility persists far beyond exact structural equivalence with the MRD rule itself. The nontriviality of the present framework arises fundamentally from the adaptive recursive structure of the residual-based step-down mechanism itself, since even globally monotone residual transformations may generate substantially different future residual systems and rejection paths through changes in the adaptive conditioning histories. This reveals that admissibility under dependence is fundamentally driven not by the precise algebraic form of the testing rule, but rather by the underlying covariance-adapted residual geometry and the associated monotone ordering structure. It is worth emphasizing that the present framework is entirely non-asymptotic and does not rely on any sparsity assumptions, weak dependence conditions, or large-sample approximations. The admissibility results established here hold for arbitrary finite dimensions and general positive definite covariance structures.
The remainder of the paper is organized as follows. In Section 2, we formulate the multiple testing problem, introduce the vector loss framework, and describe the class of monotone residual-based step-down procedures considered in this paper. Section 3 develops the main admissibility theory together with the underlying structural results. Concluding remarks and several directions for future research are presented in Section 4. The proofs of several key auxiliary results instrumental in establishing the main theorem are deferred to the Appendix.
In this paper, we consider the problem of simultaneous testing for means of a set of jointly normal variables. Towards that, let us assume that we observe a random vector \(\boldsymbol{X}=(X_1,\dots,X_n)\) (obtained through some suitable transformation, if necessary) such that \(\boldsymbol{X} \sim N_{n}(\boldsymbol{\theta},\boldsymbol{\Sigma})\) where \(\boldsymbol{\theta}=(\theta_1,\dots,\theta_n)\) is the vector of unknown means and \(\boldsymbol{\Sigma}=((\sigma_{ij}))\) is an \(n \times n\) known positive definite matrix with an arbitrary covariance structure. We are interested in testing simultaneously \[\label{TESTING95PROBLEM} H_{0i}:\theta_{i}=0againstH_{Ai}:\theta_{i}\neq 0,fori=1,\dots,n.\tag{1}\]
Note that since \(\boldsymbol{\Sigma}\) is known, without loss of generality, one may assume \(\boldsymbol{\Sigma}\) to be the correlation matrix of the \(X_i\)’s so that \(X_i \sim N(\theta_{i},1)\) for each \(i=1,\dots,n\). This is so since if \[\begin{align} \label{DIAG95MATRIX} \boldsymbol{D} &=& \begin{pmatrix} \sigma_{11} & 0 & 0 & \dots & 0 \\ 0 & \sigma_{22} & 0 & \dots & 0 \\ \vdots \\ 0 & 0 & 0 & \dots & \sigma_{nn}\end{pmatrix}, \end{align}\tag{2}\] then letting \(\boldsymbol{U}=\boldsymbol{D}^{-1/2}\boldsymbol{X}\) we have \(\boldsymbol{U}\sim N_{n}(\boldsymbol{\mu}, \boldsymbol{\Lambda})\), where \(\boldsymbol{\mu} = \boldsymbol{D}^{-1/2} \boldsymbol{\theta}\) and \(\boldsymbol{\Lambda}=\boldsymbol{D}^{-1/2}\boldsymbol{\Sigma} \boldsymbol{D}^{-1/2}\) is simply the correlation matrix of \(\boldsymbol{X}\). Therefore, testing \(H^{'}_{0i}:\mu_i=0 against H^{'}_{Ai}:\mu_i\neq 0\) simultaneously for \(i=1,\dots,n\), is equivalent to the original testing problem (1 ).
We now introduce the Maximum Residual Down (MRD) procedure proposed by [1], which serves as the canonical residual-based step-down procedure under arbitrary covariance dependence. For that, we adopt here similar convention of notations used in [1].
Let \(\boldsymbol{X}^{(i_{1},\dots,i_{t})}\) be an \((n-t)\times1\) vector consisting of those components of \(\boldsymbol{X}=(X_1,\dots,X_n)\) with \((X_{i_{1}},\dots,X_{i_{t}})\) left out. Suppose \(\boldsymbol{\Sigma}_{(i_{1},\dots,i_{t})}\) is the \((n-t)\times(n-t)\) sub-matrix obtained after eliminating the \(i_{1},\dots,i_{t}\)-th rows and the corresponding columns of \(\boldsymbol{\Sigma}\). Let \(\boldsymbol{\sigma}_{(j)}^{(i_{1},\dots,i_{t})}\) be the \((n-t-1)\times1\) vector obtained by eliminating the \(i_{1},\dots,i_{t}\)-th and \(j\)-th elements of the \(j\)-th column vector of \(\boldsymbol{\Sigma}\). Define \[\begin{align} \sigma_{j\cdot(i_{1},\dots,i_{t})} &=& \sigma_{jj} - {\boldsymbol{\sigma}_{(j)}^{(i_{1},\dots,i_{t})}}^{T}{\boldsymbol{\Sigma}^{-1}_{(i_1,\dots,i_t,j)}}{\boldsymbol{\sigma}_{(j)}^{(i_{1},\dots,i_{t})}}. \nonumber \end{align}\]
The MRD procedure is based on a collection of adaptively formed residual statistics defined as \[\begin{align} \label{MRD95STATISTICS} U_{tj}^{(i_1,\dots,i_{t-1})}(\boldsymbol{X}) &=& \frac{X_j - {\boldsymbol{\sigma}_{(j)}^{(i_1,\dots,i_{t-1})}}^{T}{\boldsymbol{\Sigma}^{-1}_{(i_1,\dots,i_{t-1},j)}}{\boldsymbol{X}^{(i_1,\dots,i_{t-1},j)}}}{\sigma^{\frac{1}{2}}_{j\cdot(i_1,\dots,i_{t-1})}},\nonumber \end{align}\tag{3}\] for \(t,j=1,\dots,n\), \(1\leqslant i_{1}\neq\dots\neq i_{t-1}\leqslant n\) and \(i_{l}\neq j\) for all \(l=1,\dots,t-1\).
For \(1 \leqslant t \leqslant n\), we define the index \(\widetilde{j}_t(\boldsymbol{X})\) as \[\begin{align} \label{MRD95INDEX} \widetilde{j}_t(\boldsymbol{X})&=\mathop{\mathrm{argmax}}_{j \in \{1,\dots,n\}\setminus\{\widetilde{j}_1(\boldsymbol{X}),\dots,\widetilde{j}_{t-1}(\boldsymbol{X})\}} |U^{(\widetilde{j}_1(\boldsymbol{X}),\dots,\widetilde{j}_{t-1}(\boldsymbol{X}))}_{tj}(\boldsymbol{X})|. \end{align}\tag{4}\]
Given a set of positive constants \(C_1 \geqslant C_2 \geqslant \dots \geqslant C_n\), the MRD method works in a step-down manner as follows:
At stage 1, consider the statistics \(|U_{1j}(\boldsymbol{X})|\), where \(j \in \{1,\dots,n\}\). If \(|U_{1\widetilde{j}_1(\boldsymbol{X})}(\boldsymbol{X})| \leqslant C_1,\) stop and accept all \(H_{0i}\)’s. Otherwise reject \(H_{0\widetilde{j}_1(\boldsymbol{X})}\) and continue to stage 2.
At stage 2, consider the statistics \(|U^{(\widetilde{j}_1(\boldsymbol{X}))}_{2\widetilde{j}_1(\boldsymbol{X})}(\boldsymbol{X})|\), where \(j\in\{1,\dots,n \}\setminus\{\widetilde{j}_1(\boldsymbol{X})\}\). If \(|U^{(\widetilde{j}_1(\boldsymbol{X}))}_{2\widetilde{j}_2(\boldsymbol{X})}(\boldsymbol{X})| \leqslant C_2,\) stop and accept all the remaining \(H_{0i}\)’s. Otherwise, reject \(H_{0\widetilde{j}_2(\boldsymbol{X})}\) and continue to stage 3.
In general, at stage \(t\), consider the statistics \(|U^{(\widetilde{j}_1(\boldsymbol{X}),\dots,\widetilde{j}_{t-1}(\boldsymbol{X}))}_{tj}(\boldsymbol{X})|\), where \(j\in\{1,\dots,n\}\setminus\{\widetilde{j}_1(\boldsymbol{X}),\dots,\widetilde{j}_{t-1}(\boldsymbol{X})\}\). If \(|U^{(\widetilde{j}_1(\boldsymbol{X}),\dots,\widetilde{j}_{t-1}(\boldsymbol{X}))}_{t\widetilde{j}_t(\boldsymbol{X})}(\boldsymbol{X})| \leqslant C_t\), stop and accept all the remaining \(H_{0i}\)’s. Otherwise, reject \(H_{0\widetilde{j}_t(\boldsymbol{X})}\) and move to stage \((t+1)\).
We continue in this fashion until an acceptance occurs or there are no more null hypotheses to be tested, in which case we must stop.
The primary objective of the present paper is to identify the broader structural principle underlying admissibility of residual-based step-down procedures under dependence. Observe that the MRD procedure ranks the active hypotheses at each stage according to the magnitudes of the corresponding residual statistics \[\left| U_{tj}^{(i_1,\dots,i_{t-1})}(\boldsymbol{X}) \right|.\] This observation naturally leads to consideration of a more general class of residual-based step-down procedures obtained through monotone transformations of these absolute residuals.
The monotonicity assumption is natural from both statistical and decision-theoretic perspectives. Intuitively, larger values of \[\left| U_{tj}^{(i_1,\dots,i_{t-1})}(\boldsymbol{X}) \right|\] correspond to stronger residual evidence against the associated null hypothesis, and hence any reasonable residual-based testing mechanism should respond monotonically to such evidence. Importantly, the admissibility theory developed in this paper does not rely on any globally fixed residual scoring rule. Rather, the main result shows that admissibility is fundamentally driven by the underlying covariance-adapted residual geometry together with the local monotone manner in which evidence accumulates through the residual statistics.
Let \[h_{tj}^{(i_1,\dots,i_{t-1})}:[0,\infty)\to\mathbb{R}\] be any strictly increasing function, where the functional form may locally depend on the stage \(t\), the coordinate \(j\), and the previously eliminated indices \((i_1,\dots,i_{t-1})\). For each stage \(t\) and each active index \(j\), define the generalized residual statistic \[\begin{align} S_{tj}^{(i_1,\dots,i_{t-1})}(\boldsymbol{X}) = h_{tj}^{(i_1,\dots,i_{t-1})} \left( \left| U_{tj}^{(i_1,\dots,i_{t-1})}(\boldsymbol{X}) \right| \right), \label{GENERAL95STATISTIC95DEFN} \end{align}\tag{5}\] for \(t,j=1,\dots,n,\) \(1\leqslant i_{1}\neq\dots\neq i_{t-1}\leqslant n\) and \(i_{l} \neq j\) for all \(l=1,\dots,t-1\).
Importantly, the transformation functions are allowed to vary locally with the stage, the coordinate under consideration, and the previously eliminated indices. Thus, the proposed framework permits highly heterogeneous and adaptive residual scoring mechanisms rather than a single global monotone transformation applied uniformly across all stages and coordinates. In particular, the framework permits stage-dependent, coordinate-dependent, and history-dependent residual scoring mechanisms within a unified admissible step-down formulation.
For \(1\leqslant t\leqslant n\), define the indices \[\begin{align} \label{GENERAL95INDEX} j_t(\boldsymbol{X}) = \mathop{\mathrm{argmax}}_{ j\in\{1,\dots,n\}\setminus \{j_1(\boldsymbol{X}),\dots,j_{t-1}(\boldsymbol{X})\} } S_{tj}^{(j_1(\boldsymbol{X}),\dots,j_{t-1}(\boldsymbol{X}))} (\boldsymbol{X}). \end{align}\tag{6}\] Since the residual statistics involved are continuously distributed under the assumed Gaussian model, ties occur with probability zero and may therefore be ignored throughout the theoretical development. An arbitrary measurable tie-breaking convention may be adopted on the null-probability tie set without affecting admissibility.
Given constants \[C^{G}_1\geqslant C^{G}_2\geqslant \cdots \geqslant C^{G}_n>0,\] the corresponding monotone residual-based step-down procedure operates as follows. At stage \(t\), reject the null hypothesis \(H_{0j_t(\boldsymbol{X})}\) if \[S_{tj_t(\boldsymbol{X})}^{ (j_1(\boldsymbol{X}),\dots,j_{t-1}(\boldsymbol{X})) } (\boldsymbol{X}) > C^{G}_t.\] Otherwise, stop and accept all remaining null hypotheses. The procedure continues until an acceptance occurs or all null hypotheses have been rejected.
The MRD procedure is recovered as the special case corresponding to the identity transformations \[h_{tj}^{(i_1,\dots,i_{t-1})}(x)=x\] for all admissible choices of \(t,j,i_1,\dots,i_{t-1}\).
Remark 1. Note that the indices \(j_t(\boldsymbol{X})\) and \(\widetilde{j}_{t}(\boldsymbol{X})\) defined in 6 and 4 , respectively, need not coincide in general. Thus, the generalized monotone residual-based procedure considered here is not simply a reparameterization of the MRD procedure, but may induce a substantially different stagewise ordering of the active hypotheses. Even globally common monotone residual transformations need not preserve the future adaptive residual systems, since differences in the stagewise threshold structure may recursively alter the adaptive conditioning histories. This makes the admissibility analysis considerably more delicate, since the proof cannot rely on an exact correspondence between the stagewise rejection sequences generated by the two procedures.
Remark 2. The class of procedures introduced above preserves the fundamental monotone relationship between the magnitudes of the residual statistics and the corresponding stagewise ordering of the active hypotheses. The main result of the next section shows that this local monotone structural property is sufficient to guarantee admissibility under the vector loss formulation considered in this paper.
In this section, we establish the main admissibility result of the paper and provide a decision theoretic justification for the class of monotone residual-based step-down procedures introduced in the previous section. The central question is whether admissibility persists under such highly general locally adaptive residual transformations despite the complicated data-dependent geometry induced by the adaptive stagewise ordering mechanism.
Recall that any multiple testing procedure \(\Phi(\boldsymbol{x})=(\phi_1(\boldsymbol{x}),\dots,\phi_n(\boldsymbol{x}))\) induces an individual test function \(\phi_j(\boldsymbol{x})\) for testing \(H_{0j}\) against \(H_{Aj}\), where \(\phi_j(\boldsymbol{x})\) denotes the probability of rejecting the \(j\)-th null hypothesis when the observation \(\boldsymbol{X}=\boldsymbol{x}\) is realized. We consider the standard \(0-1\) loss function corresponding to \(\phi_j\), given by \[\label{INDIVIDUAL95LOSS} L_j\big(\phi_j(\boldsymbol{X}),\boldsymbol{\theta}\big) = I\{\theta_j=0\}\phi_j(\boldsymbol{X}) + I\{\theta_j\neq0\} \big(1-\phi_j(\boldsymbol{X})\big),\tag{7}\] while the corresponding risk function is given by \[\label{INDIVIDUAL95RISK} R_j\big(\phi_j,\boldsymbol{\theta}\big) = I\{\theta_j=0\} E_{\boldsymbol{\theta}:\theta_j=0}\big(\phi_j(\boldsymbol{X})\big) + I\{\theta_j\neq0\} E_{\boldsymbol{\theta}:\theta_j\neq0} \big(1-\phi_j(\boldsymbol{X})\big). \nonumber\tag{8}\]
We consider the overall loss function for the multiple testing procedure \(\Phi(\boldsymbol{X})\) to be the vector loss \[\label{VECTOR95LOSS} L\big(\Phi(\boldsymbol{X}),\boldsymbol{\theta}\big) = \big( L_1\big(\phi_1(\boldsymbol{X}),\boldsymbol{\theta}\big), \dots, L_n\big(\phi_n(\boldsymbol{X}),\boldsymbol{\theta}\big) \big),\tag{9}\] with corresponding vector risk function \[\label{VECTOR95RISK} R\big(\Phi,\boldsymbol{\theta}\big) = \big( R_1\big(\phi_1,\boldsymbol{\theta}\big), \dots, R_n\big(\phi_n,\boldsymbol{\theta}\big) \big). \nonumber\tag{10}\]
A multiple testing procedure \(\Phi(\boldsymbol{X})\) is said to be inadmissible with respect to the vector loss function (9 ) if there exists another multiple testing procedure \(\Phi^{*}(\boldsymbol{X})\) such that \(R_j\big(\phi^{*}_j,\boldsymbol{\theta}\big) \leq R_j\big(\phi_j,\boldsymbol{\theta}\big)\) for all \(j=1,\dots,n\) and all \(\boldsymbol{\theta}\in\mathbb{R}^n\), with strict inequality holding for at least one \(j\) and some \(\boldsymbol{\theta}\in\mathbb{R}^n\). A multiple testing procedure is said to be admissible if it is not inadmissible in the aforesaid sense. It is natural that a multiple testing procedure which is inadmissible with respect to the vector loss function (9 ) also becomes inadmissible whenever the loss is a non-decreasing function of the numbers of type I and type II errors. In this context, it is worth recalling that [26], [27], [28] and [29] showed that in many common applications involving dependent test statistics, typical \(p\)-value based stepwise testing procedures, including the celebrated BH method, are inadmissible with respect to the vector loss function (9 ). Consequently, such procedures also become inadmissible whenever the risk is a non-decreasing function of the expected numbers of type I and type II errors. This reveals an undesirable feature of many traditional stepwise multiple testing procedures under dependence.
We now establish that the class of monotone residual-based step-down procedures introduced in the previous section does not suffer from this deficiency. It is easy to see using standard contrapositive arguments that a multiple testing procedure is admissible with respect to the vector loss function (9 ) if and only if each induced individual testing procedure is admissible with respect to the standard \(0-1\) loss (7 ). Therefore, in order to establish admissibility of the proposed methodology, it suffices to show that the induced test for testing \(H_{01}\) against \(H_{A1}\) is admissible under the assumed setup. As in [1], our proof relies on a classical result due to [30], which provides a necessary and sufficient condition for admissibility of a test of \(H_{01}\) versus \(H_{A1}\) when the joint distribution of \(\boldsymbol{X}\) belongs to an exponential family.
We emphasize that admissibility of the generalized monotone residual-based step-down procedures considered here does not follow as an immediate consequence of the admissibility of the MRD procedure established in [1]. Indeed, for each stage \(t\), the statistics \(S_{tj}^{(j_1(\boldsymbol{x}),\dots,j_{t-1}(\boldsymbol{x}))}(\boldsymbol{x})\) depend on the adaptive stagewise index sequence \(j_1(\boldsymbol{x}),\dots,j_{t-1}(\boldsymbol{x})\), each of which itself depends on the observed data vector \(\boldsymbol{x}\). Consequently, the quantities \(\sigma_{j\cdot(j_1(\boldsymbol{x}),\dots,j_{t-1}(\boldsymbol{x}))}\) and the associated acceptance regions possess highly nontrivial data-dependent geometry. Moreover, as noted earlier, the stagewise rejection indices generated by the generalized monotone residual-based procedure need not coincide with those of the MRD procedure. Even under globally common monotone residual transformations, differences in the stagewise threshold structure may recursively alter the adaptive conditioning sets, thereby inducing substantially different future residual systems and rejection paths. This makes the admissibility analysis considerably more delicate, since the proof cannot rely on an exact correspondence between the stagewise rejection sequences generated by the two procedures. Instead, a more careful structural analysis of the induced acceptance regions becomes necessary. In the process, we establish a broader structural admissibility result showing that, for the multiple testing problem (1 ), any step-down multiple testing procedure based on statistics \(S_{tj}\), where each \(S_{tj}\) is obtained through a locally adaptive strictly increasing transformation of the absolute value of the corresponding MRD statistic \(U_{tj}\), is admissible with respect to the vector loss function (9 ). To the best of our knowledge, this general structural result appears to be new in the literature and substantially extends part of the admissibility theory developed in [1].
Let \(\phi_j(\boldsymbol{x})\) denote the test function induced by the generalized monotone residual-based step-down procedure for testing \(H_{0j}\) versus \(H_{Aj}\) when the observation \(\boldsymbol{X}=\boldsymbol{x}\) is realized. The following lemma due to [30] provides a necessary and sufficient condition for admissibility of a testing procedure for testing \(H_{01}\) versus \(H_{A1}\) when \(\boldsymbol{\Sigma}\) is known.
Let \[\boldsymbol{Y} = \boldsymbol{\Sigma}^{-1}\boldsymbol{X}.\]
Lemma 1. A necessary and sufficient condition for a test \(\phi(\boldsymbol{y})\) of \(H_{01}\) versus \(H_{A1}\) to be admissible is that, for almost every fixed \(y_2, \dots,y_n\), the acceptance region of the test is an interval in \(y_1\).
Proof. See [30]. ◻
Lemma 1 reduces the admissibility problem to a geometric analysis of the induced acceptance region along suitable one-dimensional perturbation directions. The remainder of the argument therefore focuses on understanding how the adaptive residual-based step-down mechanism evolves along such perturbation paths.
Note that, for each fixed \((y_2,\dots,y_n)\), to study the test function \(\phi(\boldsymbol{y})=\phi_1(\boldsymbol{x})\) as \(y_1\) varies, it would be enough to consider sample points of the form \(\boldsymbol{x}+r\boldsymbol{g}\), where \(\boldsymbol{g}\) is the first column of \(\boldsymbol{\Sigma}\) and \(r\) varies. This is true, since \(\boldsymbol{y}\) is a function of \(\boldsymbol{x}\), and so \(\boldsymbol{y}\) evaluated at \(\boldsymbol{x}+r\boldsymbol{g}\) is \[\begin{align} \boldsymbol{\Sigma}^{-1}(\boldsymbol{x}+r\boldsymbol{g}) &=& \boldsymbol{y}+(r,0,\dots,0) = (y_1+r,y_2,\dots,y_n). \nonumber \end{align}\] Thus, varying \(r\) changes only the first coordinate of \(\boldsymbol{y}\) while keeping the remaining coordinates fixed. Lemma 1 therefore reduces the admissibility problem to showing that, along each such line, the acceptance region of the induced test for \(H_{01}\) is an interval.
Lemma 2. The functions \(U_{tj}\) as given in (3 ) have the following properties.
For \(t \in \{1,\dots,n\}\) and for \(j_1,\dots,j_{t-1} \in \{2,\dots,n\}\) with \(j_i \neq j_{k}\) for \(i \neq k\), \[\begin{align} U^{(j_1,\dots,j_{t-1})}_{t1}(\boldsymbol{x} + r\boldsymbol{g}) &=& U^{(j_1,\dots,j_{t-1})}_{t1}(\boldsymbol{x}) + r\sigma^{\frac{1}{2}}_{1\cdot( j_1,\dots,j_{t-1})}. \nonumber \end{align}\] Moreover, for \(t \in \{1,\dots,n\}\) and for \(j \in \{2,\dots,n\} \setminus \{j_1,\dots,j_{t-1}\},\) \[\begin{align} U^{(j_1,\dots,j_{t-1})}_{tj}(\boldsymbol{x} + r\boldsymbol{g}) &=& U^{(j_1,\dots,j_{t-1})}_{tj}(\boldsymbol{x}). \nonumber \end{align}\]
Proof. See the proof of Lemma 3.2 of [1]. ◻
Lemma 2 shows that, along the direction \(\boldsymbol{g}\), the residual statistic corresponding to the first coordinate changes linearly in \(r\), whereas the residual statistics corresponding to the remaining active coordinates remain invariant. This separation property is the basic geometric mechanism behind the admissibility argument.
Remark 3. Since \(\sigma_{1\cdot(j_1,\dots,j_{t-1})}>0\), it follows from Lemma 2 that for each fixed \(\boldsymbol{x}\in\mathbb{R}^n\) and given any \((t-1)\) many indices \((j_1,\dots,j_{t-1})\), \(U^{(j_1,\dots,j_{t-1})}_{t1}(\boldsymbol{x}+r\boldsymbol{g})\) is strictly increasing in \(r\). Consequently, \(|U^{(j_1,\dots,j_{t-1})}_{t1}(\boldsymbol{x}+r\boldsymbol{g})|\) initially decreases and then increases as \(r\) increases. Therefore, when \(|U^{(j_1,\dots,j_{t-1})}_{t1}(\boldsymbol{x}+r\boldsymbol{g})|\) decreases in \(r\), \(U^{(j_1,\dots,j_{t-1})}_{t1}(\boldsymbol{x}+r\boldsymbol{g})\) is negative, while when \(|U^{(j_1,\dots,j_{t-1})}_{t1}(\boldsymbol{x}+r\boldsymbol{g})|\) increases in \(r\), \(U^{(j_1,\dots,j_{t-1})}_{t1}(\boldsymbol{x}+r\boldsymbol{g})\) is positive. This observation is crucial for deriving the interval structure of the induced acceptance region.
Corollary 1. For any \(r \in \mathbb{R}\), we have \[\begin{align} U_{1j}(\boldsymbol{x} + r\boldsymbol{g}) = U_{1j}(\boldsymbol{x})for allj \in \{2,\dots,n\}, \nonumber \end{align}\] which, in turn, implies the following: \[\begin{align} S_{1j}(\boldsymbol{x} + r\boldsymbol{g}) = S_{1j}(\boldsymbol{x})for allj \in \{2,\dots,n\}. \nonumber \end{align}\]
In other words, Corollary 1 shows that perturbations along the admissibility direction \(g\) leave all residual scores corresponding to the competing coordinates unchanged, while only the residual statistic associated with the first coordinate evolves along the \(g\)-direction. Consequently, the induced geometry along the admissibility direction is governed by a single evolving residual component against a collection of geometrically invariant competing residual scores.
In Lemma 3.2 of [1], the term \(\sigma^{1/2}_{1\cdot(j_1,\dots,j_{t-1})}\) was omitted, most likely due to a typographical oversight, and its role was not explicitly analyzed in the subsequent theoretical argument. In the present setting, this term requires careful attention since \(\sigma_{1\cdot(j_1(\boldsymbol{x}),\dots,j_{t-1}(\boldsymbol{x}))}\) depends on the adaptive stagewise indices \(j_1(\boldsymbol{x}),\dots,j_{t-1}(\boldsymbol{x})\), each of which is itself a function of the observed data vector \(\boldsymbol{x}\). A major difficulty here is that both the residual statistics and the associated conditional variance structure evolve adaptively with the previously rejected indices \(j_1(\boldsymbol{x}),\dots,j_{t-1}(\boldsymbol{x})\). Consequently, the local geometry of the step-down procedure is itself data-dependent and potentially unstable under perturbations. It is therefore necessary to understand how these adaptive indices behave as \(\boldsymbol{x}\) varies along the admissibility direction \(\boldsymbol{g}\). The following lemma shows that, remarkably, perturbations along the special direction \(\boldsymbol{g}\) preserve the common portion of the stage-wise elimination sequence prior to rejection of \(H_{01}\). This local invariance principle partially freezes the adaptive combinatorial structure of the procedure and initiates the reduction of the analysis to a one-dimensional monotone evolution along the \(g\)-trajectory.
Suppose \(\phi_1 (\boldsymbol{x}^{*})=0\) when \(\boldsymbol{x}^{*}\) is observed, that is, \(\boldsymbol{x}^{*}\) is an acceptance point of \(H_{01}\). Then the testing procedure must stop before \(H_{01}\) gets rejected. Suppose the testing procedure stops at some stage \(t\) without rejecting \(H_{01}\). Let \(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}\) be a point of rejection of \(H_{01}\), that is, \(\phi_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})=1\), with \(r_{0}\neq 0\). Let the testing procedure reject \(H_{01}\) at some stage \(t_0\) when \(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}\) is observed. The next lemma gives an important identity between the set of indices \(j_l(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})\) and \(j_l(\boldsymbol{x}^{*})\), for \(1\leqslant l\leqslant \min\{t_0,t\}-1\), which shows that these indices remain invariant whenever \(\min\{t,t_0\}>1\).
Lemma 3. Under the conditions \(\phi_1 (\boldsymbol{x}^{*}) =0\) and \(\phi_1 (\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}) = 1\), with \(r_{0}\neq 0\), the following holds when \(t >1\) and \(t_0 >1\): \[j_l(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})= j_l(\boldsymbol{x}^{*})for alll=1,\dots,\min\{t_0,t\} -1, \nonumber\] where \(t_0\) and \(t\) are defined as before.
Proof. See Appendix. ◻
Lemma 3 shows that, along the admissibility direction \(\boldsymbol{g}\), the adaptive stage-wise elimination sequence among the competing coordinates remains invariant up to stage \(\min\{t,t_0\}-1\). In other words, perturbations along the \(\boldsymbol{g}\)-direction affect only the residual statistic associated with the first coordinate, while the relative ordering among the remaining active coordinates remains unchanged throughout the common portion of the adaptive rejection path. Consequently, the otherwise highly adaptive multistage testing procedure reduces locally to a partially frozen one-dimensional monotone evolution along the \(\boldsymbol{g}\)-trajectory. This local invariance property forms the key structural mechanism underlying the admissibility argument developed below.
Lemma 3, coupled with Corollary 1, leads to the following important result on the relation between \(t_0\) and \(t\) defined before.
Lemma 4. Under the conditions \(\phi_1 (\boldsymbol{x}^{*}) =0\) and \(\phi_1 (\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}) = 1\), with \(r_{0}\neq 0\), the monotone residual-based step-down procedure must reject \(H_{01}\) within \(t\) steps when \(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}\) is observed, that is, \(1\leqslant t_0 \leqslant t,\) where \(t_0\) and \(t\) are defined as before.
Proof. See Appendix. ◻
Combining Lemma 3 and Lemma 4, the following invariance principle becomes immediate. This invariance principle freezes the adaptive combinatorial structure of the procedure throughout all stages preceding rejection of \(H_{01}\) and reduces the analysis locally to a one-dimensional monotone evolution along the \(\boldsymbol{g}\)-trajectory.
Corollary 2. Under the framework of Lemma 4, we have \[j_l(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})= j_l(\boldsymbol{x}^{*})for alll=1,\dots,t_0-1. \nonumber\]
Lemma 4 establishes that, once a perturbation along the admissibility direction \(g\) causes the testing procedure to reject \(H_{01}\), the rejection cannot occur strictly later than the stage at which the original procedure stopped without rejecting \(H_{01}\). Thus, movement along the \(g\)-direction cannot delay the rejection of \(H_{01}\). This property plays a crucial role in controlling the evolution of the adaptive rejection path and prevents the acceptance region from developing disconnected geometric structures along the admissibility direction.
Lemma 5. Suppose that for some \(\boldsymbol{x}^{*}\) and \(r_0 > 0,\) \(\phi_1(\boldsymbol{x}^{*})=0\) and \(\phi_1(\boldsymbol{x}^{*}+r_0\boldsymbol{g})=1.\) Then \(\phi_1(\boldsymbol{x}^{*}+r\boldsymbol{g})=1\) for all \(r > r_0.\)
Proof. See Appendix. ◻
Lemma 5 establishes a monotonicity property of the induced decision rule along the admissibility direction \(g\). Specifically, once a perturbation \(\boldsymbol{x}^{*} + r_0\boldsymbol{g}\) becomes a rejection point for \(H_{01}\), every further perturbation in the same direction must also remain a rejection point. Consequently, the acceptance region along the \(\boldsymbol{g}\)-direction must necessarily have an interval structure. This monotone geometric behavior is precisely the key condition required by the Matthes–Truax characterization of admissibility.
Using Lemma 1 and Lemma 5 above, it follows that for testing \(H_{01} { vs } H_{A1}\), the individual decision \(\phi_1(\boldsymbol{X})\) induced by the monotone residual-based step-down procedure is admissible with respect to the standard \(0-1\) loss (7 ). The proof that the other tests induced by the procedure for the remaining individual testing problems are admissible follows analogously. Since admissibility of each individual induced decision implies admissibility of the corresponding multiple testing procedure under the vector loss (9 ), we obtain the desired admissibility property in the following theorem.
Theorem 1.
Suppose \(\boldsymbol{X}\sim N_{n}(\boldsymbol{\theta},\boldsymbol{\Sigma})\), where \(\boldsymbol{\theta}\in\mathbb{R}^{n}\) is unknown, but fixed and \(\boldsymbol{\Sigma}\) is an \(n \times n\) arbitrary but known positive definite covariance matrix. Then, for the two sided multiple testing problem (1 ), any step-down multiple testing procedure based on statistics \(S_{tj}\)’s, where each \(S_{tj}\) is obtained through a locally adaptive strictly increasing transformation of the absolute value of the corresponding MRD statistic \(U_{tj}\), is admissible with respect to the vector loss function 9 .
Theorem 1 identifies a broad structural mechanism underlying admissibility, showing that the admissibility property is fundamentally driven by the local monotone ordering structure induced by the MRD residual statistics rather than by the precise functional form of the residual scoring rule itself. In particular, the admissibility property of the MRD procedure due to [1] follows immediately by taking \(S_{tj}=|U_{tj}|\).
Importantly, the admissibility proof does not require the generalized, covariance-adaptive monotone residual-based procedure to preserve either the stagewise rejection ordering or the threshold structure of the original MRD procedure. Moreover, the theorem does not require the generalized residual statistics to arise from a single globally defined transformation of the MRD statistics. The admissibility result continues to hold even when the residual scoring functions vary locally across stages, coordinates, and adaptive rejection histories. Consequently, Theorem 1 substantially enlarges the currently known class of admissible covariance-adaptive residual-based procedures.
From a geometric viewpoint, the residual-based construction underlying the MRD procedure may be interpreted as a sequential covariance-adapted orthogonalization mechanism. Along the admissibility directions arising from the Matthes–Truax characterization [30], the residual corresponding to the coordinate under consideration evolves monotonically while the competing residual statistics remain invariant. This effectively reduces the high-dimensional adaptive step-down problem locally to a one-dimensional ordered geometric structure and induces an interval-type acceptance geometry which ultimately drives the admissibility property. In this sense, Theorem 1 reveals that admissibility under arbitrary covariance dependence is fundamentally a structural consequence of the covariance-adapted residual geometry underlying the MRD construction.
Remark 4. The Bayesian Step-Down (BSD) procedure proposed by [31] also fits naturally within the framework of Theorem 1. The BSD statistics arise naturally from stagewise posterior probability based Bayesian residual scoring mechanisms under suitable prior specifications and may be expressed as locally adaptive strictly increasing transformations of the absolute values of the corresponding MRD residual statistics. Consequently, admissibility of the BSD procedure with respect to the vector loss formulation follows immediately from Theorem 1.
In this paper, we established a general admissibility result for a broad class of monotone residual-based step-down multiple testing procedures under arbitrary covariance dependence within a multivariate Gaussian framework. The main result shows that admissibility with respect to the vector loss function 9 is fundamentally a structural consequence of the covariance-adapted residual geometry underlying the MRD construction rather than a consequence of any particular residual scoring rule. In particular, admissibility persists across a remarkably broad class of highly heterogeneous and locally adaptive residual-based procedures whose stagewise scoring mechanisms may vary across coordinates, stages, and adaptive rejection histories. Consequently, the admissibility property extends far beyond the original MRD procedure introduced by [1] and remains valid for a substantially larger class of generalized residual-based procedures.
A noteworthy aspect of the present analysis is that the generalized procedures considered here may induce adaptive stagewise rejection sequences that differ substantially from those generated by the original MRD procedure. Consequently, the admissibility proof cannot rely on any direct stagewise correspondence argument, but instead requires a careful geometric analysis of the induced acceptance regions together with structural invariance properties of the adaptive stagewise indices. From a methodological standpoint, this substantially enlarges the class of covariance-adaptive admissible multiple testing procedures and permits considerable flexibility in constructing dependence-aware residual-based step-down procedures beyond the original MRD formulation.
The proof of Theorem 1 further demonstrates that the admissibility phenomenon is fundamentally geometric in nature. From a structural viewpoint, Theorem 1 reveals that admissibility is remarkably stable under broad classes of locally adaptive monotone perturbations of the underlying covariance-adjusted residual geometry. Along the admissibility direction arising from the Matthes–Truax characterization, the covariance-adjusted residual coordinate corresponding to the hypothesis under consideration evolves monotonically while the competing residual statistics remain invariant. This effectively reduces the high-dimensional adaptive step-down problem to a locally ordered one-dimensional geometric structure. The resulting interval-type acceptance geometry ultimately drives the admissibility property. From this perspective, the present work suggests that admissibility under dependence may often be governed less by the precise algebraic form of a multiple testing rule and more by deeper monotone geometric structures induced through covariance-adjusted residual representations.
The present work naturally raises several challenging open problems. One major difficulty arises when the vector loss formulation is replaced by additive loss functions involving the total numbers of type I and type II errors. Under vector loss, admissibility reduces to admissibility of the induced individual tests, thereby permitting the use of geometric characterizations such as the Matthes–Truax criterion [30]. In contrast, additive losses introduce highly nontrivial global interactions among the component decisions, and admissibility can no longer be analyzed through coordinatewise arguments. As a consequence, the structure of admissible multiple testing procedures under additive losses appears to be fundamentally more complicated, particularly under dependence.
Indeed, even under relatively simple dependence structures, a satisfactory structural characterization of admissible procedures under additive losses remains largely unknown. The difficulty becomes considerably more pronounced under arbitrary covariance dependence, where the adaptive geometry of the rejection regions may become extremely complicated. We hope that the structural ideas developed in the present paper may provide some useful insight toward future investigations in this direction.
The present work establishes a broad sufficient condition for admissibility under arbitrary covariance dependence. Whether related monotone residual ordering structures are also necessary for admissibility of covariance-adaptive step-down procedures remains an interesting open question.
Several additional extensions also appear worthy of investigation. These include admissibility theory for one-sided multiple testing problems, problems involving unknown covariance structures, and asymptotic high-dimensional settings in which the covariance matrix itself must be estimated from the data. Another important direction would be to investigate the interaction between admissibility and modern global error criteria such as the false discovery rate under general dependence structures.
The present proof fundamentally exploits the Gaussian conditional residual representation together with the Matthes–Truax characterization for exponential families. It would therefore be interesting to investigate whether the local monotone residual geometry underlying the present admissibility theory extends beyond the Gaussian setting to broader dependence structures and exponential family models. The adaptive residual statistics considered here evolve sequentially through conditioning on previously eliminated coordinates, thereby inducing a locally evolving covariance-adjusted geometric structure along the admissibility direction. This suggests intriguing connections between admissibility under dependence and broader geometric structures arising in adaptive high-dimensional inference problems. Another intriguing direction is to study whether covariance-adaptive residual-based step-down procedures can simultaneously achieve admissibility together with meaningful false discovery rate control under general dependence structures. Establishing rigorous theoretical guarantees for such phenomena remains an interesting open problem for future research.
In this appendix, we provide the proofs of Lemmas 3– 5 which were instrumental in proving Theorem 1 of this paper.
Proof of Lemma 3
Proof. We shall prove the result using the method of induction.
First, we establish the result when \(l=1\).
Observe that since both \(t>1\) and \(t_0>1\), one must have \[j_1(\boldsymbol{x}^{*}+r_0\boldsymbol{g})\neq 1 \quad \text{and} \quad j_1(\boldsymbol{x}^{*})\neq 1.\]
Then, using Corollary 1, we obtain \[\begin{align} \label{LEM95INDEX95INVARIANCE95PARTIAL95EQ1} j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}) &=& \mathop{\mathrm{argmax}}_{j \in \{2,\dots,n\}} S_{1j}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})\nonumber\\ &=& \mathop{\mathrm{argmax}}_{j \in \{2,\dots,n\}} S_{1j}(\boldsymbol{x}^{*})\nonumber\\ &=& j_1(\boldsymbol{x}^{*}). \end{align}\tag{11}\]
Hence, the result is true for \(l=1\).
Now suppose that, for some \(2\leq l\leq \min\{t_0,t\}-1\), \[\label{LEM95INDEX95INVARIANCE95PARTIAL95INDUCTION} j_k(\boldsymbol{x}^{*}+r_0\boldsymbol{g})=j_k(\boldsymbol{x}^{*}), \;\textrm{ for all } k=1,\ldots,l-1.\tag{12}\]
By the induction hypothesis 12 , the active coordinate sets and the corresponding conditioning index sets coincide at \(\boldsymbol{x}^{*}+r_0\boldsymbol{g}\) and \(\boldsymbol{x}^{*}\) up to stage \(l-1\), which, for \(l=2\), follows directly from 11 .
Our aim is now to show that \[j_l(\boldsymbol{x}^{*}+r_0\boldsymbol{g}) = j_l(\boldsymbol{x}^{*}), \textrm{ for all } l=2,\dots, \min\{t_0,t\}-1.\]
Now using Lemma 2, it follows that for all \(j\in\{2,\dots,n\}\setminus \{j_1(\boldsymbol{x}^{*}+r_0\boldsymbol{g}),\dots, j_{l-1}(\boldsymbol{x}^{*}+r_0\boldsymbol{g})\},\) \[U_{lj}^{(j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots,j_{l-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}))}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})=U_{lj}^{(j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots,j_{l-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}))}(\boldsymbol{x}^{*}).\notag\]
Therefore, we obtain for all \(j\in\{2,\dots,n\} \setminus \{j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots,j_{l-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})\},\) \[\label{LEM95INDEX95INVARIANCE95PARTIAL95EQ3} S_{lj}^{(j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots,j_{l-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}))}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})=S_{lj}^{(j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots,j_{l-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}))}(\boldsymbol{x}^{*}).\tag{13}\]
Again using Lemma 2 coupled with the induction hypothesis 12 , we have \[\begin{align} U^{(j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots,j_{l-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}))}_{l1}(\boldsymbol{x}^{*} + r\boldsymbol{g}) &=& U^{(j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots,j_{l-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}))}_{l1}(\boldsymbol{x}^{*})\notag\\ && \quad + r\sigma^{\frac{1}{2}}_{1\cdot( j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots,j_{l-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}))}\nonumber\\ &=& U^{(j_1(\boldsymbol{x}^{*}),\dots,j_{l-1}(\boldsymbol{x}^{*}))}_{l1}(\boldsymbol{x}^{*}) + r\sigma^{\frac{1}{2}}_{1\cdot( j_1(\boldsymbol{x}^{*}),\dots,j_{l-1}(\boldsymbol{x}^{*}))},\nonumber \end{align}\] which shows that, for each \(1<l<\min\{t_0,t\}\), the perturbation affects only the statistic corresponding to coordinate \(1\).
Since \(H_{01}\) is rejected only at stage \(t_0\) when \(\boldsymbol{x}^{*}+r_0\boldsymbol{g}\) is observed, for every \(1 <l<\min\{t_0,t\}\), the statistic \[S_{l1}^{(j_1(\boldsymbol{x}^{*}+r_0\boldsymbol{g}),\dots, j_{l-1}(\boldsymbol{x}^{*}+r_0\boldsymbol{g}))} (\boldsymbol{x}^{*}+r_0\boldsymbol{g})\] cannot attain the maximum among the active statistics at stage \(l\). Consequently, the maximizing index \(j_{l}(\boldsymbol{x}^{*}+r_0\boldsymbol{g})\) at stage \(l<t_0\) when \(\boldsymbol{x}^{*}+r_0\boldsymbol{g}\) is observed must belong to the competing set \[\{2,\dots,n\}\setminus \{j_1(\boldsymbol{x}^{*}+r_0\boldsymbol{g}),\dots, j_{l-1}(\boldsymbol{x}^{*}+r_0\boldsymbol{g})\},\] whose associated statistics remain invariant under the perturbation.
Hence, using 11 , 13 , and the induction hypothesis 12 , we obtain \[\begin{align} j_l(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}) &=& \mathop{\mathrm{argmax}}_{j \in \{2,\dots,n\}\setminus\{j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}), \dots, j_{l-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})\}} S_{lj}^{(j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots, j_{l-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}))}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})\nonumber\\ &=& \mathop{\mathrm{argmax}}_{j \in \{2,\dots,n\}\setminus\{j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots, j_{l-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})\}} S_{lj}^{(j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots, j_{l-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}))}(\boldsymbol{x}^{*})\nonumber\\ &=& \mathop{\mathrm{argmax}}_{j \in \{2,\dots,n\}\setminus\{j_1(\boldsymbol{x}^{*}), \dots, j_{l-1}(\boldsymbol{x}^{*})\}} S_{lj}^{(j_1(\boldsymbol{x}^{*}), \dots, j_{l-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*})\nonumber\\ &=& j_l(\boldsymbol{x}^{*}),\nonumber \end{align}\] for all \(l=2,\dots,\min\{t_0,t\}-1\). This completes the proof of Lemma 3. ◻
Proof of Lemma 4
Proof. First observe that, when \(t_0=1\), the conclusion \(t_0\le t\) holds trivially, since \(t\ge 1\). On the other hand, if \(t=1\), then the procedure stops at the first stage when \(\boldsymbol{x}^{*}\) is observed, so \(S_{1j}(\boldsymbol{x}^{*})\le C_1^G\) for all \(j=1,\dots,n\). In that case, if \(t_0>1\), then at stage 1, some coordinate \(k\ne 1\) must be rejected when \(\boldsymbol{x}^{*}+r_0\boldsymbol{g}\) is observed, so \(S_{1k}(\boldsymbol{x}^{*}+r_0\boldsymbol{g})>C_1^G\). But by Corollary 1, \(S_{1k}(\boldsymbol{x}^{*}+r_0\boldsymbol{g})=S_{1k}(\boldsymbol{x}^{*})\), a contradiction. Hence \(t_0=1\), and therefore \(t_0\le t\).
Thus it remains to consider the case \(t>1\) and \(t_0>1\).
Since \(t>1\), the process does not stop in the first \((t-1)\) steps without rejecting \(H_{01}\) when \(\boldsymbol{x}^{*}\) is observed. Consequently, we must have \[\begin{align} S_{lj_l(\boldsymbol{x}^{*})}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{l-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*}) > C^{G}_{l}\nonumber \end{align}\] for all \(l=1,\dots,t-1\), where \(j_l(\boldsymbol{x}^{*}) \neq 1\) for each \(l\), and \[\begin{align} S_{tj_{t}(\boldsymbol{x}^{*})}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*}) \leqslant C^{G}_{t}, \nonumber \end{align}\] which implies that \[\begin{align} S_{tj}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*}) \leqslant C^{G}_{t} \nonumber \end{align}\] for all \(j \in \{1,\dots,n\}\setminus\{j_1(\boldsymbol{x}^{*}),\dots,j_{t-1}(\boldsymbol{x}^{*})\}\). Since \(j_l(\boldsymbol{x}^{*}) \neq 1\) for all \(l=1,\dots,t-1\), the index \(1\) remains active at stage \(t\).
On the contrary, let us now assume that \(t_0>t\). Then \[\begin{align} \label{LEM95STOPPING95STAGE95EQ1} S_{tj_t(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})}^{(j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots,j_{t-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}))}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})> C^{G}_{t}, \end{align}\tag{14}\] otherwise the process would have stopped at stage \(t\) without rejecting \(H_{01}\) when \(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}\) is observed, which would be a contradiction.
Let \[k=j_t(\boldsymbol{x}^{*}+r_0\boldsymbol{g}).\notag\]
Since we are assuming that \(t_0>t\), the hypothesis \(H_{01}\) is not rejected at stage \(t\) when \(\boldsymbol{x}^{*}+r_0\boldsymbol{g}\) is observed. Hence \(k\neq 1\). Moreover, by Corollary 2, \[\label{LEM95STOPPING95STAGE95EQ2} j_l(\boldsymbol{x}^{*}+r_0\boldsymbol{g})=j_l(\boldsymbol{x}^{*}), \;\textrm{ for } l=1,\dots,t-1.\tag{15}\]
Therefore \(k\) remains active at stage \(t\) when \(\boldsymbol{x}^{*}\) is observed. That is, \(H_{0k}\) is not rejected, at least, until stage \(t\) when \(\boldsymbol{x}^{*}\) is observed. Since \(k\neq 1\), using Lemma 2 we obtain \[\begin{align} \label{LEM95STOPPING95STAGE95EQ3} U_{tk}^{(j_1(\boldsymbol{x}^{*}+r_0\boldsymbol{g}),\dots, j_{t-1}(\boldsymbol{x}^{*}+r_0\boldsymbol{g}))} (\boldsymbol{x}^{*}+r_0\boldsymbol{g}) &=& U_{tk}^{(j_1(\boldsymbol{x}^{*}+r_0\boldsymbol{g}),\dots, j_{t-1}(\boldsymbol{x}^{*}+r_0\boldsymbol{g}))} (\boldsymbol{x}^{*}) \nonumber\\ &=& U_{tk}^{(j_1(\boldsymbol{x}^{*}),\dots, j_{t-1}(\boldsymbol{x}^{*}))} (\boldsymbol{x}^{*}). \end{align}\tag{16}\]
It is important to observe that, due to 15 , the locally adaptive transformation functions corresponding to the generalized residual statistics in 16 also coincide at \(\boldsymbol{x}^{*}+r_0\boldsymbol{g}\) and \(\boldsymbol{x}^{*}\). Therefore, since each generalized residual score is obtained through the same strictly increasing transformation applied to the absolute value of the corresponding residual statistic, 16 coupled with 14 , implies \[\begin{align} \label{LEM95STOPPING95STAGE95EQ4} S_{tk}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t-1}(\boldsymbol{x}^{*}))} (\boldsymbol{x}^{*}) &=& S_{tk}^{(j_1(\boldsymbol{x}^{*}+r_0\boldsymbol{g}),\dots, j_{t-1}(\boldsymbol{x}^{*}+r_0\boldsymbol{g}))} (\boldsymbol{x}^{*}+r_0\boldsymbol{g}) \nonumber\\ &>& C_t^G. \end{align}\tag{17}\]
But \(k\) is active at stage \(t\) when \(\boldsymbol{x}^{*}\) is observed. Since the procedure stops at stage \(t\) under \(\boldsymbol{x}^{*}\), we must have \[S_{tk}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*}) \le C_t^G,\nonumber\] which contradicts 17 . This completes the proof of Lemma 4. ◻
Proof of Lemma 5
Proof. Since \(\phi_1(\boldsymbol{x}^{*})=0\), the point \(\boldsymbol{x}^{*}\) belongs to the acceptance region of \(H_{01}\). Hence, the testing procedure must stop before rejecting \(H_{01}\). Let the testing procedure stop at some stage \(t\) without rejecting \(H_{01}\) when \(\boldsymbol{x}^{*}\) is observed.
Again, since \(\phi_1(\boldsymbol{x}^{*}+r_0\boldsymbol{g})=1\), the point \(\boldsymbol{x}^{*}+r_0\boldsymbol{g}\) belongs to the rejection region of \(H_{01}\). Let the testing procedure reject \(H_{01}\) at some stage \(t_0\) when \(\boldsymbol{x}^{*}+r_0\boldsymbol{g}\) is observed.
Let us first consider the situation when \(t_0>1\).
By Lemma 4, we have \(t_0\le t\), and hence the indices \(j_1(\boldsymbol{x}^{*}),\ldots,j_{t_0}(\boldsymbol{x}^{*})\) are well-defined.
Observe first that if \[j_{t_0}(\boldsymbol{x}^{*})=1,\nonumber\] the test fails to reject \(H_{01}\) at stage \(t_{0}\) when \(\boldsymbol{x}^{*}\) is observed. Consequently, \[\begin{align} S_{t_{0}1}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t_{0}-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*}) &=& S_{t_{0}j_{t_0}(\boldsymbol{x}^{*})}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t_{0}-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*})\nonumber\\ &\leqslant& C_{t_0}^G \nonumber\\ &<& S_{t_{0}j_{t_0}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})}^{(j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots, j_{t_{0}-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}))} (\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})\nonumber\\ &=& S_{t_{0}1}^{(j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots, j_{t_{0}-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}))} (\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})\nonumber\\ &=& S_{t_{0}1}^{(j_1(\boldsymbol{x}^{*}),\dots, j_{t_{0}-1}(\boldsymbol{x}^{*}))} (\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}), \nonumber \end{align}\] where the strict inequality in the above chain of inequalities follows from the fact that \(H_{01}\) is rejected at stage \(t_0\) when \(\boldsymbol{x}^{*}+r_0\boldsymbol{g}\) is observed.
Otherwise, if \[j_{t_0}(\boldsymbol{x}^{*})\neq 1,\nonumber\] then using Corollary 1 together with Corollary 2 we have \[\begin{align} S_{t_{0}1}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t_{0}-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*}) &<& S_{t_{0}j_{t_{0}}(\boldsymbol{x}^{*})}^{(j_1(\boldsymbol{x}^{*}),\dots, j_{t_{0}-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*})\nonumber\\ &=& S_{t_{0}j_{t_{0}}(\boldsymbol{x}^{*})}^{(j_1(\boldsymbol{x}^{*}),\dots, j_{t_{0}-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}) \nonumber\\ &=& S_{t_{0}j_{t_{0}}(\boldsymbol{x}^{*})}^{(j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots, j_{t_{0}-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}))} (\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}) \nonumber\\ &\leqslant& S_{t_{0}1}^{(j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}),\dots, j_{t_{0}-1}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}))} (\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}) \nonumber\\ &=& S_{t_{0}1}^{(j_1(\boldsymbol{x}^{*}),\dots, j_{t_{0}-1}(\boldsymbol{x}^{*}))} (\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}). \nonumber \end{align}\]
Thus, in either case, it is shown that \[\begin{align} S_{t_{0}1}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t_{0}-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*}) &<& S_{t_{0}1}^{(j_1(\boldsymbol{x}^{*}),\dots, j_{t_{0}-1}(\boldsymbol{x}^{*}))} (\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}). \nonumber \end{align}\]
By Corollary 2, the adaptive histories up to stage \((t_0-1)\) coincide at \(\boldsymbol{x}^{*}\) and \(\boldsymbol{x}^{*}+r_0\boldsymbol{g}\). Hence the locally adaptive transformation function associated with the statistic \[S^{(j_1(\boldsymbol{x}^{*}),\ldots,j_{t_0-1}(\boldsymbol{x}^{*}))}_{t_0 1}\notag\] is the same at the two points \(\boldsymbol{x}^{*}\) and \(\boldsymbol{x}^{*}+r_0\boldsymbol{g}\). Therefore, the preceding comparison of the corresponding \(S\)-statistics may be equivalently translated into a comparison of the absolute residual statistics.
Since for fixed \(j_1,\dots,j_{t_{0}-1},\) \(S_{t_{0}1}^{(j_1,\dots,j_{t_{0}-1})}\) is a strictly increasing function of \(|U_{t_{0}1}^{(j_1,\dots,j_{t_{0}-1})}|,\) it follows that \[\begin{align} |U_{t_{0}1}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t_{0}-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*})| &< |U_{t_{0}1}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t_{0}-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})|.\label{LEM95SHIFT95PROPERTY95EQ1} \end{align}\tag{18}\]
We now show that \[U^{(j_1(\boldsymbol{x}^{*}),\ldots,j_{t_0-1}(\boldsymbol{x}^{*}))}_{t_01} (\boldsymbol{x}^{*}+r_0\boldsymbol{g})>0.\notag\] On the contrary, suppose the above claim is not true. Since \[U^{(j_1(\boldsymbol{x}^{*}),\ldots,j_{t_0-1}(\boldsymbol{x}^{*}))}_{t_01} (\boldsymbol{x}^{*}+r\boldsymbol{g})\notag\] is strictly increasing in \(r\) and \(r_0>0\), we must have \[U^{(j_1(\boldsymbol{x}^{*}),\ldots,j_{t_0-1}(\boldsymbol{x}^{*}))}_{t_01} (\boldsymbol{x}^{*}) < U^{(j_1(\boldsymbol{x}^{*}),\ldots,j_{t_0-1}(\boldsymbol{x}^{*}))}_{t_01} (\boldsymbol{x}^{*}+r_0\boldsymbol{g}) \le 0.\notag\] The preceding step coupled with Remark 3 would imply \[|U^{(j_1(\boldsymbol{x}^{*}),\ldots,j_{t_0-1}(\boldsymbol{x}^{*}))}_{t_01} (\boldsymbol{x}^{*})| > |U^{(j_1(\boldsymbol{x}^{*}),\ldots,j_{t_0-1}(\boldsymbol{x}^{*}))}_{t_01} (\boldsymbol{x}^{*}+r_0\boldsymbol{g})|,\notag\] contradicting 18 .
Thus we obtain \[U^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t_0-1}(\boldsymbol{x}^{*}))}_{t_01} (\boldsymbol{x}^{*}+r_0\boldsymbol{g})>0.\notag\]
But for given \((j_1,\dots,j_{t_{0}-1})\), the function \(U_{t_{0}1}^{(j_1,\dots,j_{t_{0}-1})}(\boldsymbol{x}^{*}+r\boldsymbol{g})\) is strictly increasing in \(r\). Hence for all \(r > r_0,\) we have \[\begin{align} \label{LEM95SHIFT95PROPERTY95EQ2} U_{t_{0}1}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t_{0}-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*}+r\boldsymbol{g}) &>& U_{t_{0}1}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t_{0}-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})>0. \end{align}\tag{19}\]
We now complete the proof now based on a contrapositive argument. Recall that, we need to show \(\phi_1(\boldsymbol{x}^{*}+r\boldsymbol{g})=1\) for all \(r > r_0.\) On the contrary, suppose this is not true. Then there exists some \(r_1 > r_0\) such that \(\phi_1(\boldsymbol{x}^{*}+r_1\boldsymbol{g})=0.\) Let \(t_1\) denote the step at which the testing procedure must stop without rejecting \(H_{01}\) when \(\boldsymbol{x}^{*}+r_1\boldsymbol{g}\) is observed.
Let us now view \(\boldsymbol{x}^{*}+r_{1}\boldsymbol{g}\) as the new acceptance point and observe that \[\notag \boldsymbol{x}^{*}+r_{0}\boldsymbol{g} = (\boldsymbol{x}^{*}+r_{1}\boldsymbol{g}) - (r_{1}-r_{0})\boldsymbol{g},\] where \(r_{1}-r_{0}>0\). Thus, relative to the new base point \(\boldsymbol{x}^{*}+r_{1}\boldsymbol{g}\), the point \(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}\) becomes a rejection point obtained through a nonzero perturbation along the admissibility direction \(\boldsymbol{g}\) with perturbation parameter \(-(r_{1}-r_{0})\). Since Lemma 3 and Lemma 4 together with Corollary 2 remain valid for arbitrary nonzero perturbations, the preceding arguments continue to hold under this reversed perturbation as well. Consequently \[t_0 \le t_1. \notag\] Similarly, applying Corollary 2 with the same base point \(\boldsymbol{x}^{*}+r_{1}\boldsymbol{g}\) and perturbation \(-(r_{1}-r_{0})\) together with the assumption \(t_{0}> 1\), we obtain \[j_l(\boldsymbol{x}^{*} + r_1 \boldsymbol{g}) = j_l(\boldsymbol{x}^{*} + r_0 \boldsymbol{g}) = j_l(\boldsymbol{x}^{*}) \notag\] for all \(l = 1,\ldots,t_0-1\).
Applying the preceding arguments with the new base point \(\boldsymbol{x}^{*}+r_1\boldsymbol{g}\) and the reversed perturbation \(-(r_{1}-r_{0})\), we obtain \[\begin{align} |U_{t_{0}1}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t_{0}-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*}+r_1\boldsymbol{g})| &<& |U_{t_{0}1}^{(j_1(\boldsymbol{x}^{*}),\dots,j_{t_{0}-1}(\boldsymbol{x}^{*}))}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})|\nonumber \end{align}\] which contradicts (19 ). Therefore one must have \(\phi_1(\boldsymbol{x}^{*}+r\boldsymbol{g})=1\) for all \(r>r_0\), when \(t_0>1\).
Next consider the case when \(t_0=1\). Then, by definition, \[j_1(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})=1,\notag\] and hence \[S_{11}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})>C_1^G.\notag\]
Since \(\phi_1(\boldsymbol{x}^{*})=0\), the procedure does not reject \(H_{01}\) when \(\boldsymbol{x}^{*}\) is observed. Therefore, \[S_{11}(\boldsymbol{x}^{*})\le C_1^G.\notag\]
Consequently, \[S_{11}(\boldsymbol{x}^{*})<S_{11}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g}).\notag\]
Now, by Corollary 1, \[S_{1j}(\boldsymbol{x}^{*}+r \boldsymbol{g})=S_{1j}(\boldsymbol{x}^{*}) \quad \text{for all } j\in\{2,\dots,n\} \text{ and all } r\in\mathbb{R}.\nonumber\] Thus, along the admissibility direction \(\boldsymbol{g}\), the competing first-stage statistics remain invariant, while only the statistic corresponding to coordinate \(1\) evolves with \(r\).
Since \(S_{11}\) is a strictly increasing function of \(|U_{11}|\), and \(U_{11}(\boldsymbol{x}^{*}+r\boldsymbol{g})\) is strictly increasing in \(r\), Remark 3 implies that \[U_{11}(\boldsymbol{x}^{*}+r_0\boldsymbol{g})>0.\notag\]
Hence, for all \(r>r_0\), \[\label{LEM95SHIFT95PROPERTY95EQ4} U_{11}(\boldsymbol{x}^{*}+r\boldsymbol{g})>U_{11}(\boldsymbol{x}^{*}+r_0\boldsymbol{g})>0.\tag{20}\]
Since \(t_{0}=1\), 20 above implies for all \(r>r_0\), \[\label{LEM95SHIFT95PROPERTY95EQ5} S_{11}(\boldsymbol{x}^{*}+r\boldsymbol{g})>S_{11}(\boldsymbol{x}^{*}+r_0\boldsymbol{g})> C^{G}_{1}.\tag{21}\]
Since \(t_0=1\), using the preceding arguments together with Corollary 1, for all \(r > r_0\), and for all \(j\in\{2,\dots,n\}\), we obtain \[\begin{align} \label{LEM95SHIFT95PROPERTY95EQ6} S_{11}(\boldsymbol{x}^{*}+r\boldsymbol{g}) &>& S_{11}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})\nonumber\\ &\geqslant& S_{1j}(\boldsymbol{x}^{*}+r_{0}\boldsymbol{g})\nonumber\\ &=& S_{1j}(\boldsymbol{x}^{*})\nonumber\\ &=& S_{1j}(\boldsymbol{x}^{*}+r\boldsymbol{g}). \end{align}\tag{22}\]
21 and 22 together imply that every \(\boldsymbol{x}^{*}+r\boldsymbol{g}\) will be a point of rejection for \(H_{01}\) for all \(r > r_0\), that is, \(\phi_1(\boldsymbol{x}^{*}+r\boldsymbol{g})=1\) for all \(r > r_0\) when \(t_0=1\). This completes the proof of Lemma 5. ◻