November 06, 2023
When the rate parameter of the exponential distribution is associated with predictors, then main interest will be how to estimate the regression parameter. In this paper, we will investigate how to estimate the parameter on the regression setup of the exponential distribution. To that end, we propose a new estimator, and its asymptotic properties will be discussed.
Keywords: Cramer-von Mises, exponential distribution, minimum distance, survival analysis
In the literature on statistical theory and probability distributions, the exponential distribution is popular for several reasons. For example, it will not be an exaggeration to claim that its distribution function is, de facto, the second simplest, being ranked after that of the uniform distribution. Consequently, this feature of simplicity makes the exponential distribution more practical and useful, and the domain of its application has quickly expanded to many other disciplines by addressing real-world problems. Among those disciplines, the exponential distribution paved the way and laid the foundation for survival analysis, which models the time to the occurrence of a specific event. Having been the most popular with medical science, survival analysis has also drawn attention from many non-medical disciplines, such as social science, engineering, and physics. In survival analysis, hazard rate — which indicates the frequency of deaths — can be expressed as \(f/(1-F)\) with \(f\) and \(F\) being density and distribution functions, respectively.
From the constant rate \(\lambda\), survival analysis derives two critical curves used to understand a population’s lifespan. The probability density functionmeasures the absolute probability of the event occurring at exactly time \(t\). It drops off sharply over time, showing that as time goes on, fewer individuals remain alive to experience the event. On the contrary, the survival function measures the probability that a subject will survive longer than time \(t\). It starts at 1 (100% survival at time zero) and decays smoothly toward 0, which shows the relationship between exponential distribution and the survival distribution.
An interesting questions arises when the unknown rate parameter is associated with some predictors, which extends the one sample exponential distribution to the regression setup. Well-known example will be the Cox proportional hazards (Cox-PH, or simply Cox) model, which assumes the constant rate \(\lambda\) over time. While real-world biological systems or mechanical parts often wear out over time—violating the constant hazard assumption—the exponential distribution remains the vital baseline model. It acts as the mathematical benchmark from which more complex survival models. Minimum distance (MD) method is known to possess many desirable properties, such as asymptotic normality and robustness of the resulting estimator. As [1] extended the MD methodologies from estimation of the location parameter of one sample to that of the parameter on regression setup. Being analogous to what he had done, we will extend the MD method from the single rate parameter estimation to the regression parameter estimation of the Cox proportional hazard model.
During the 1970s and 1980s, many statisticians – e.g., [2], [3], [4], and [5] – have conducted research on the MD estimation since it was proposed by [6]: see also references in [1]. The distance function measures the discrepancy between the observed random sample and the assumptions underlying the theories. More specifically, the distance function — which contains the parameter of interest as an argument – computes the numeric difference between the empirical function and the modeled function, constructed from observed data and assumptions, respectively. Then, the MD estimation method, as the name implies, seeks the optimal value that minimizes the distance function. Researchers investigated the resulting estimator after employing various distance functions. For example, [7] used the Hellinger distance using empirical and modeled density functions. Among many research works on the distance function in the literature of the MD estimation, [8] demonstrated that the MD estimator obtained from the Cramer von-Mises (CvM) distance function exhibits better robustness than those obtained from other distance functions.
In the past two decades, however, no more rigorous research has been conducted; only a few studies have further investigated MD estimation. [9] proposed a novel algorithm to compute the MD estimator, while [10] demonstrated that the MD estimator maintains the desirable asymptotic properties under the assumption of independent observations even when independence doesn’t hold. Applying the MD method to a discrete distribution, [11] demonstrated that the MD estimator still retains asymptotic properties and robustness, thereby comparing favorably with other well-celebrated estimators, including the ML estimator.
One of the fundamental reasons the popularity of the MD method has been waning rapidly is the complexity of its distance function. The empirical distribution function, which is a collection of indicator functions of the observed sample and the parameter of interest, is the main culprit obstructing the search for the optimal solution to the distance function. Since the indicator function is not smooth, it is not differentiable with respect to the parameter; unlike the ML estimation, the closed-form expression for the solution does not exist. Therefore, research on MD estimation should rely on computationally expensive numerical methods to solve the optimization problem. [11] modified the distance function so that the empirical distribution function contains only the observed sample, while the modeled distribution function contains the parameter of interest. As a result, the distance function became smooth and differentiable with respect to the parameter, thereby reducing the computational cost. Findings in [11], however, are limited in that the modified distance function is valid only for a discrete probability distribution, the estimation problem of which is presumed to be less challenging than that of a continuous probability distribution function. In this article, we extend his approach to estimating the regression parameter of this study.
Let \(f\) and \(F\) denote the probability density and distribution functions with a rate parameters, respectively. For example, \(f\) and \(F\) of the exponential distribution with a rate parameter \(\lambda\) will be \(f(y;\lambda):=\lambda e^{-\lambda y}\) and \(F(y;\lambda):=1-e^{-\lambda y}\), respectively, for real \(y\in \mathbb{R}\), while those of the Wiebull distribution with a known shape parameter \(\alpha\) will have \(f(y;\lambda)=\lambda\alpha(\lambda y)^{\alpha-1}e^{-(\lambda y)^{\alpha}}\) and \(F(y;\lambda) = 1-e^{(\lambda y)^{\alpha}}\). Let \(g(\lambda;y):=\partial F(y;\lambda)/\partial \lambda\). Note that \(f(y;\lambda) = g(y;\lambda)\), and hence, \(f\) and \(g\) share many common features such as smoothness and differentiability. In the literature of MD estimation theories, \(f\) has played in the crucial role, such as determining the asymptotic variance of the resulting MD estimator; however, in this study, \(g\) will take the role of \(f\).
Consider a sample of independent observations, \(Y_{1},...,Y_{n}\) whose distribution function is \(F\) with different rate parameters \(\lambda_{1},...\lambda_{n}\). Assume that the rate parameter \(\lambda_{i},\,\,1\le i \le n\) are exponentially associated with a predictor \(\boldsymbol{x}_{i}\in \mathbb{R}^{p}\), that is, \(\lambda_{i}=e^{\boldsymbol{x}_{i}'\boldsymbol{\beta}_{0}}\). Real-world examples of statistical models on this setup include survival analysis, as mentioned in the introduction, where \(Y_{i}\) represents survival time of the \(i\)th individual, while a hazard rate, \(\lambda_i\), is affected by the predictors of the individual, including age, income, etc. To estimate the parameter \(\boldsymbol{\beta}_{0}\in \mathbb{R}^{p}\), we first define the distance function \({\cal L}(\boldsymbol{\beta})\) for \(\boldsymbol{\beta}\in \mathbb{R}^{p}\) with an integrating measure \(H\) as follows: \[\label{eq:distance95function} {\cal L}(\boldsymbol{\beta}) = \sum_{j=1}^p\int\left[ \sum_{i=1}^nd_{ij}\Big\{\textrm{I}(Y_{i}\le y)-F(y;\boldsymbol{x}_{i}'\boldsymbol{\beta}) \Big\} \right]^{2}\,dH(y),\tag{1}\] where \(\textrm{I}(\cdot)\) is an indicator function, and \(d_{ij}\in \mathbb{R},\,1\le i\le n,\,1\le j\le p\) are real numbers. Note that the distance function can be adjusted through the choice of the weights \(d_{ij}\), which is one of the main merits of the MD estimator. For example, [1] demonstrated that proper choice of the weights will give a rise to the most efficient estimator for the given \(F\). In addition, [11] used the uniform weights and showed that the distance function can be stabilized against certain impact from outliers and hence yield more robust estimator than other methods. As will be shown later, choice of \(d_{ij}\) that yields the optimal result will not be fortuitous in this study; rather the weight will be methodically selected.
Next, we define the MD estimator belonging to the neighborhood of the true parameter. More precisely, the MD estimator will solve the following optimization problem \[\label{eq:opt} {\cal L}(\widehat{\boldsymbol{\beta}}) = \inf_{\boldsymbol{\beta}\in\mathbb{R}^{p}}{\cal L}(\boldsymbol{\beta}).\tag{2}\] where infimum is taken over \({\cal N}_{b}(\boldsymbol{\beta}_{0}):=\{\boldsymbol{\beta}\in\mathbb{R}^{p}: \mathbf{A}^{-1}|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}|\le b<\infty\}\) for some \(p\times p\) matrix \(\mathbf{A}\).
Unfortunately, the optimal solution to the above problem doesn’t have any closed-form expression, which is the most stumbling block to finding the MD estimator. However, the concept that the distance function will be uniformly locally asymptotically quadratic (ULAQ) over \({\cal N}(\boldsymbol{\beta}_{0})\) ensures that the MD estimator can be searched by numerical methods using its gradient. Furthermore, it possesses the desirable properties, including asymptotic normality and robustness. In the literature of the MD estimation, it is not exaggeration to state that the ULAQ conditions play the most crucial role to obtain the MD estimator and its asymptotic properties; the successful investigation of the MD estimation hinges on satisfying the ULAQ conditions. The assumptions (U1)-(U3) below state the ULAQ conditions required for this study.
There exist a sequence of \(p\times 1\) random vector \(\boldsymbol{S}_{n}(\boldsymbol{\beta}_{0})\) and a sequence of \(p\times p\) real matrix \(\mathbf{W}_{n}(\boldsymbol{\beta}_{0})\) such that for all \(0<b<\infty\) \[\sup_{|\delta_{n}(\beta-\beta_{0})|\leq b}\left|{\cal L}(\boldsymbol{\beta})- {\cal L}(\boldsymbol{\beta}_{0})-(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})'\boldsymbol{S}_{n}(\boldsymbol{\beta}_{0})-(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})'\mathbf{W}_{n}(\beta_{0})(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})\right|=o_{p}(1).\]
For all \(\varepsilon>0\), there is a \(0<z_{\varepsilon}<\infty\) such that \[\mathbb{P}\big(|{\cal L}(\boldsymbol{\beta}_{0})|\leq z_{\varepsilon} \big)\geq 1-\varepsilon.\]
For all \(\varepsilon>0\) and \(0<c<\infty\), there is a \(0<b<\infty\) and \(N\) – both depending on \(\varepsilon\) and \(c\) – such that \[\mathbb{P}\Big( \inf_{\delta_{n}|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}|>b} |{\cal L}(\boldsymbol{\beta})|> c \Big)\geq 1-\varepsilon, \qquad \textrm{ for all }n\geq N.\]
The following lemma reproduces Theorem 5.4.1 from [1], which gives a clue for the asymptotic distribution of the resulting MD estimator that satisfies the ULAQ conditions.
Lemma 1. Assume that \({\cal L}\) satisfies the ULAQ assumptions (U.1)- (U.3). Let \({\cal B}_{n}:=\mathbf{A}\mathbf{W}_{n}(\boldsymbol{\beta}_{0})\mathbf{A}\) where \(\mathbf{A}\) is used to define \({\cal N}_{b}(\boldsymbol{\beta}_{0})\). Let \(\widehat{\boldsymbol{\beta}}\) denote the MD estimator that solves the optimization problem in (2 ). Then, \[{\cal B}_{n}\mathbf{A}^{-1} (\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_{0}) = -\mathbf{A}\boldsymbol{S}_{n}(\boldsymbol{\beta}_{0})+o_{p}(1).\]
The lemma says the asymptotic distribution of the MD estimator will be determined by that of \(\mathbf{A}\boldsymbol{S}_{n}\); thus, the asymptotic normality of \(\mathbf{A}\boldsymbol{S}_{n}\) – which will be proven in the next section – will vouch for that of the MD estimator.
Consider \(n\) pairs of observations, \((Y_{1}, \boldsymbol{x}_{1}'),...,(Y_{n}, \boldsymbol{x}_{n}')\) where \(\boldsymbol{x}_{i}\in \mathbb{R}^{p}\). Let \(\mathbf{X}\) be an \(n\times p\) design matrix whose \(i\)th row vector is \(\boldsymbol{x}_{i}'\). Using \(d_{ij}\), \(1\le i\le n,\,1\le j\le p\), the same weights in (1 ), define an \(n\times p\) matrix \({\mathbf{D}}:=((d_{ij}))\). As mentioned in the previous section, \({\mathbf{D}}\) will be selected after the meticulous investigation so that we can obtain the optimal MD estimator. It is a matter of course that \({\mathbf{D}}\) will be related with the design matrix \(\mathbf{X}\) as demonstrated in other studies. For example, [1] proved that \({\mathbf{D}}\varpropto \mathbf{X}\) – that is, \({\mathbf{D}}\) is \(\mathbf{X}\) multiplied by a non-singular \(p\times p\) matrix obtained from \(\mathbf{X}\) – will yield the most efficient estimator when estimating the regression parameter of linear models with independent observations. As shown later, the optimal \({\mathbf{D}}\) of this study turns out to be much more complicating than that of [1].
Let \(\boldsymbol{q}_{i}(\boldsymbol{\beta};\cdot):=\partial F(\cdot;\boldsymbol{x}_{i}'\boldsymbol{\beta})/\partial \boldsymbol{\beta}\). For the exponential \(F\), we have \(\boldsymbol{q}_{i}(\boldsymbol{\beta};) = ye^{-y\boldsymbol{x}_{i}'\boldsymbol{\beta}}e^{\boldsymbol{x}_{i}'\boldsymbol{\beta}}\boldsymbol{x}_{i}\). Recall \(\lambda_i=e^{\boldsymbol{x}_{i}'\boldsymbol{\beta}}\) and \(g(\lambda;\cdot)\). Using \(\lambda_{i}\) and \(g\) will simplify \(\boldsymbol{q}_{i}(\boldsymbol{\beta};y) =g(\lambda_{i};y)\lambda_{i}\boldsymbol{x}_{i}\). To denote that \(g(\lambda_{i};y)\lambda_{i}\) is indeed a function of \(\boldsymbol{\beta}\), let \(g(\lambda_{i};y)\lambda_{i}:=g_{i}(\boldsymbol{\beta};y)\). With all these notations, we are ready to state the necessary assumptions for this study. It should be admitted that most of the assumptions have a root in [1].
Let \(\boldsymbol{B}\) denote an \(n\times n\) symmetric, positive definite matrix. Then, \(\mathbf{X}'\boldsymbol{B}\mathbf{X}\) is nonsingular. In addition, there exists a \(p\times p\) nonsingular matrix \(\mathbf{A}\) such that \(\mathbf{A}= (\mathbf{X}'\boldsymbol{B}\mathbf{X})^{-1/2}\).
For all \(1\le j\le p\), the following hold true: \(\sum_{i=1}^nd_{ij}^{2}=1\) and \(\max_{1\le i\le n}d_{ij}=o(1)\).
Let \(\boldsymbol{c}_{i}:=\mathbf{A}\boldsymbol{x}_{i}\). Then \(\max_{1\le i\le n}\|\boldsymbol{c}_{i}\|=o(1)\).
For \(1\le j\le p\), \(\sum_{i=1}^n\|d_{ij}\boldsymbol{c}_{i}\| = O(1)\).
Let \(\lambda_{i}^{0}:=\boldsymbol{x}_{i}'\boldsymbol{\beta}_{0}\). With \(g(\lambda;y)=\partial F(y;\lambda)/\partial \lambda\), the following holds: \[\max_{1\le i\le n} \sup\int_{0}^{\infty}\{\lambda(1-y\lambda)g(\lambda;y)\}^2\,\,dH(y)<\infty,\] where the supremum is taken over \(\lambda\) between \(\lambda_{i}^{0}\) and \(\lambda_{i}\).
Consider \(F(\cdot;\boldsymbol{x}_{i}'\boldsymbol{\beta})\), the df of the random variable \(Y_{i}\). Then, \[\sum_{j=1}^p\int{\sum_{i=1}^nd_{ij}^{2} F(y;\boldsymbol{x}_{i}'\boldsymbol{\beta})\{1-F(y;\boldsymbol{x}_{i}'\boldsymbol{\beta})\}}dH(y) =O(1).\]
Let \(\mathbf{G}_{n}(\boldsymbol{\beta};y)\) be an \(n\times n\) diagonal matrix whose \(i\)th entry is \(g_{i}(\boldsymbol{\beta};y)\). Then a \(p\times p\) matrix \(\boldsymbol{\Gamma}_{n}(\boldsymbol{\beta};y):={\mathbf{D}}'\mathbf{G}_{n}(\boldsymbol{\beta};y)\mathbf{X}\mathbf{A}\) is nonsingular for \(y\in \mathbb{R}\) and \(\boldsymbol{\beta}\in \mathbb{R}^{p}\).
Let \(\boldsymbol{\gamma}_{j}(\boldsymbol{\beta};y)\in \mathbb{R}^{p},\,1\le j\le p\) denote the \(j\)th column vector of \(\boldsymbol{\Gamma}_{n}(\boldsymbol{\beta};y)\). Then the following is true: \[\sum_{j=1}^p\int{\|\boldsymbol{\gamma}_{j}(\boldsymbol{\beta};y)\|^r}dH(y) <\infty, \quad \textrm{ for }r=1,2.\]
Let \(\boldsymbol{e}\in \mathbb{R}^{p}\) be a unit vector, that is, \(\|\boldsymbol{e}\|=1\). Let \(k_{n}(\boldsymbol{e}):=\boldsymbol{e}'\boldsymbol{\Gamma}_{n}\boldsymbol{e}\). Then there exists an \(\alpha>0\) such that \[\liminf_{n}\big\{\inf\{k_{n}(\boldsymbol{e}):\boldsymbol{e}\in \mathbb{R}^{J}\}\big\}\ge \alpha.\]
For all \(1\le k\le n\) and for all unit vectors \(\boldsymbol{e}\in {\mathbb{R}}^{p}\), either \(\boldsymbol{d}_{k}'\boldsymbol{e}\boldsymbol{x}_{k}'\mathbf{A}\boldsymbol{e}\ge 0\) or \(\boldsymbol{d}_{k}'\boldsymbol{e}\boldsymbol{x}_{k}'\mathbf{A}\boldsymbol{e}\le 0\) holds true.
Remark 1. For the exponential \(F\), \(g(\lambda;y) = ye^{-\lambda y}\). With \(H(y)\equiv y\), the integral of the equation in (a.5) will be simplified to \[\int_{0}^{\infty}\{\lambda(1-y\lambda)g(\lambda;y)\}^2\,\,dy\le 2\int_{0}^{\infty}\lambda^2(1+y^2\lambda^2)y^2e^{-2\lambda y}\,\,dy= \frac{4}{\lambda},\] where the inequality readily follows from \((a-b)^{2}\le 2(a^2+b^2)\) for real \(a,b\in \mathbb{R}\), and hence, the assumption (a.5) is equivalent to \(\min_{1\le i\le n}\{\lambda_{i}^{0},\lambda_{i}\}>0\). Using the probability df for the integrating measure, for example, \(H(y)=F(y)\), the left-hand side (LHS) of the equation of the claim will be bounded by \(14/81\), and hence, the assumption will be trivially met, regardless of \(\lambda\).
To conserve the space, let \(F_{i}(\cdot;\boldsymbol{\beta}): = F(\cdot;\boldsymbol{x}_{i}'\boldsymbol{\beta})\). Define \(\boldsymbol{{\cal W}}:=({\cal W}_{1},...,{\cal W}_{j})'\in \mathbb{R}^{p}\) where the \(j\) th entry is \[{\cal W}_{j}(y, \boldsymbol{\beta}):= \sum_{i=1}^nd_{ij}\Big\{\textrm{I}(Y_{i}\le y)-F_{i}(y;\boldsymbol{\beta}) \Big\}.\] Note that the distance function can be written as \[{\cal L}(\boldsymbol{\beta})=\sum_{j=1}^p\int{{\cal W}_{j}(y, \boldsymbol{\beta})^2}dH(y) = \int{\boldsymbol{{\cal W}}(y, \boldsymbol{\beta})'\boldsymbol{{\cal W}}(y, \boldsymbol{\beta})}dH(y) .\] Next, we shall specify \(\boldsymbol{S}_{n}\) and \(\mathbf{W}_{n}\) that satisfy the first ULAQ condition. Intuitively speaking, the candidates for them are the first and second order derivatives of \({\cal L}\) with respect to \(\boldsymbol{\beta}\), that is, \[\begin{align} \boldsymbol{S}_{n}(\boldsymbol{\beta})&=&\frac{\partial {\cal L}}{\partial \boldsymbol{\beta}}=-\sum_{j=1}^p\int{{\cal W}_{j}(y,\boldsymbol{\beta})\sum_{i=1}^nd_{ij}\boldsymbol{q}_{i}(y;\boldsymbol{\beta})}dH(y) , \\ \mathbf{W}_{n}(\boldsymbol{\beta})&=& \frac{\partial^2 {\cal L}}{\partial \boldsymbol{\beta}'\partial \boldsymbol{\beta}}= \sum_{j=1}^p\int{ \sum_{i=1}^n\sum_{k=1}^nd_{ij}d_{kj}\boldsymbol{q}_{i}(y;\boldsymbol{\beta})\boldsymbol{q}_{k}(y;\boldsymbol{\beta})'}dH(y) . \end{align}\] This intuition will turn out to be true in the sequel; see, e.g., Theorem 1. Subsequently, define a quadratic function \[{\cal Q}(\boldsymbol{\beta})={\cal L}(\boldsymbol{\beta}_{0})+2(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})'\boldsymbol{S}_{n}(\boldsymbol{\beta}_{0})+(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})'\mathbf{W}_{n}(\beta_{0})(\boldsymbol{\beta}-\boldsymbol{\beta}_{0}).\] Recall \(\mathbf{G}_{n}\) and \(\boldsymbol{\Gamma}_{n}\) from the assumption (a.7). Replacing \(\boldsymbol{q}_{i}(\boldsymbol{\beta};y)\) with \(g_{i}(\boldsymbol{\beta};y)\boldsymbol{x}_{i}\), rewrite \(\boldsymbol{S}_{n}\) and \(\mathbf{W}_{n}\) in a matrix form as follows: \[\label{eq:SandW95Matrix} \boldsymbol{S}_{n}(\boldsymbol{\beta}) = \mathbf{A}^{-1}\int{ \boldsymbol{\Gamma}_{n}(\boldsymbol{\beta},y)'\boldsymbol{{\cal W}}(y,\boldsymbol{\beta})}dH(y) , \quad \mathbf{W}_{n}(\boldsymbol{\beta}) = \mathbf{A}^{-1}\int{ \boldsymbol{\Gamma}_{n}(\boldsymbol{\beta};y)'\boldsymbol{\Gamma}_{n}(\boldsymbol{\beta};y)}dH(y) \mathbf{A}^{-1}.\tag{3}\] Recall \({\cal N}_{b}(\boldsymbol{\beta}_{0})=\{\boldsymbol{\beta}\in\mathbb{R}^{p}: \mathbf{A}^{-1}|\boldsymbol{\beta}-\boldsymbol{\beta}_{0}|\le b<\infty\}\) and the ULAQ conditions. The first ULAQ condition implies the distance function \({\cal L}\) can be uniformly approximated by the quadratic function \({\cal Q}\) over \({\cal N}_{b}(\boldsymbol{\beta}_{0})\), which is demonstrated by Theorem 1. Before proceeding to the theorem, we shall prove the next lemma, which will be used for the proof of the theorem.
Lemma 2. For \(0<b<\infty\), \[\sup_{\boldsymbol{\beta}\in{\cal N}_{b}(\boldsymbol{\beta}_{0})} \sum_{j=1}^p\int{\left[\sum_{i=1}^nd_{ij}\left\{F_{i}(y;\boldsymbol{\beta})-F_{i}(y; \boldsymbol{\beta}_{0})-(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})'g_{i}(\boldsymbol{\beta}_{0};y)\boldsymbol{x}_{i}\right\}\right]^{2}}dH(y) =o(1).\]
Remark 2. The above lemma is analogue of the assumption (i) of Section 5.5 from [1]. The difference between two originates from the fact that \(f_{i}\) is replaced by \(g_{i}\), as mentioned earlier.
Proof. Let \(\boldsymbol{u} = \mathbf{A}^{-1}(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})\). Note that \(F_{i}(y;\boldsymbol{\beta})-F_{i}(y;\boldsymbol{\beta}_{0})=(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})'\boldsymbol{x}_{i}g_{i}(\widetilde{\boldsymbol{\beta}};y)\), where \(\widetilde{\boldsymbol{\beta}}=c\boldsymbol{\beta}+(1-c)\boldsymbol{\beta}_{0}\) for some \(c\in (0,1)\). Let \(\widetilde{\lambda}_{i}=\boldsymbol{x}_{i}'\widetilde{\boldsymbol{\beta}}\) and \(\lambda_{i}^{0}=\boldsymbol{x}_{i}'\boldsymbol{\beta}_{0}\). Also, observe that \[|g_{i}(\widetilde{\boldsymbol{\beta}};y)-g_{i}(\boldsymbol{\beta}_{0};y)|\le \left|(\widetilde{\boldsymbol{\beta}}-\boldsymbol{\beta}_{0})'\frac{\partial g_{i}(\boldsymbol{\beta};y)}{\partial \boldsymbol{\beta}}\right|\le \|\mathbf{u}\|\cdot \|\mathbf{A}\boldsymbol{x}_{i}\|\cdot |\lambda_{i}^{*}(1-y\lambda_{i}^{*})g(\lambda_{i}^{*};y)|,\] where \(\lambda_{i}^{*}\) lies between \(\lambda_{i}^{0}\) and \(\widetilde{\lambda}_{i}\); the mean value theorem readily implies the first inequality, while the second inequality follows from \(|\boldsymbol{x}_{i}'(\widetilde{\boldsymbol{\beta}}-\boldsymbol{\beta}_{0})|\le |\boldsymbol{x}_{i}'(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})|\). Recall \(\boldsymbol{c}_{ni}\) from the assumption (a.2). Finally, for \(\|\mathbf{u}\|\le b\), \[\begin{align} (\textrm{LHS of the equation}) &\le& \sup_{\|\mathbf{u}\|\le b}\sum_{j=1}^p\int{\left[ \sum_{i=1}^nd_{ij}\mathbf{u}'\boldsymbol{c}_{ni}\{g_{i}(\widetilde{\boldsymbol{\beta}};y)-g_{i}(\boldsymbol{\beta}_{0};y)\} \right]^2}dH(y) ,\\ &\le& p\cdot b^4\left(\sum_{i=1}^n\|d_{ij}\boldsymbol{c}_{ni}\|\right)^2 \max_{1\le i\le n} \sup_{\lambda}\int_{0}^{\infty}\{\lambda(1-y\lambda)g(\lambda;y)\}^2\,\,dH(y) \rightarrow0, \end{align}\] where the convergence to 0 follows from (a.2)-(a.5), thereby completing the proof of the lemma. ◻
Consider a vector-valued function \(\boldsymbol{a}(y):=(a_{1}(y),...,a_{p}(y))\in \mathbb{R}^{p}\) whose entries are functions of \(y\in \mathbb{R}\), that is, \(a_{j}:\mathbb{R}\rightarrow\mathbb{R}\) for all \(1\le j\le p\). Let \(\|\boldsymbol{a}\|_{H}^{2}\) denote its \(L_{2}\) norm \[\|\boldsymbol{a}\|_{H}^{2} := \int{\boldsymbol{a}'(y)\boldsymbol{a}(y)}dH(y) =\sum_{j=1}^p\int{ a_{j}(y)^2}dH(y) .\] Let \(\boldsymbol{\kappa}:=(\kappa_{1},...,\kappa_{p})'\mathbb{R}^{p}\) whose \(j\)th entry is the integrand of the equation in the above lemma, that is, \[\kappa_{j}(y;\boldsymbol{\beta}):= \sum_{i=1}^nd_{ij}\left\{F_{i}(y;\boldsymbol{\beta})-F_{i}(y; \boldsymbol{\beta}_{0})-g_{i}(\boldsymbol{\beta}_{0};y)(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})'\boldsymbol{x}_{i}\right\},\] and hence, the lemma can be written as \(\sup_{\|\mathbf{u}\|\le b} \|\boldsymbol{\kappa}\|_{H}^{2}=o_{p}(1)\). In what follows, we will use the \(L_{2}\) norm notation to conserve space. Next theorem serves to demonstrate that the first ULAQ condition holds.
Theorem 1. Suppose that assumptions (a.1)-(a.8) hold. Then, the distance function \({\cal L}\) in (1 ) satisfies (U.1), that is, for any \(0<b<\infty\), \[\mathbb{E}\Big(\sup|{\cal L}(\boldsymbol{\beta})- {\cal Q}(\boldsymbol{\beta})|\Big)=o(1),\] where the supremum is taken over \({\cal N}_{b}(\boldsymbol{\beta}_{0})\).
Proof. Recall \(\mathbf{G}_{n}\) and \(\boldsymbol{\Gamma}_{n}\) from (a.7), and rewrite \(\boldsymbol{{\cal W}}=\boldsymbol{{\cal W}}(y,\boldsymbol{\beta})\) and \(\boldsymbol{{\cal W}}_{0}=\boldsymbol{{\cal W}}(y,\boldsymbol{\beta}_{0})\). Note that \[\boldsymbol{{\cal W}}= \boldsymbol{{\cal W}}_{0}-(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})'\mathbf{X}'\mathbf{G}_{n}{\mathbf{D}}-\boldsymbol{\kappa},\] and hence, with \(\mathbf{u}=\mathbf{A}^{-1}(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})\), \[\begin{align} |{\cal L}(\boldsymbol{\beta}) - {\cal Q}(\boldsymbol{\beta})|&=& \|\boldsymbol{{\cal W}}\|_{H}^{2} - \|\boldsymbol{{\cal W}}_{0}-(\boldsymbol{\beta}-\boldsymbol{\beta}_{0})'\mathbf{X}'\mathbf{G}_{n}{\mathbf{D}}\|_{H}^{2}\\ &\le& \|\boldsymbol{\kappa}\|_{H}^{2}+2\|\boldsymbol{\kappa}\|_{H}\Big[ \|\boldsymbol{{\cal W}}_{0}\|_{H}+\|\mathbf{u}'\boldsymbol{\Gamma}_{n}'\|_{H} \Big]\longrightarrow 0, \end{align}\] where the inequality follows from applying the Cauchy-Schwarz inequality to the cross product terms after expanding \(\|\boldsymbol{{\cal W}}\|_{H}^{2}\). Note that (a.6) and (a.8) readily imply \(\|\boldsymbol{{\cal W}}_{0}\|_{H}<\infty\) and \[\sup_{\|\mathbf{u}\|\le b}\|\mathbf{u}'\boldsymbol{\Gamma}_{n}\|_{H}^{2}\le b^2\sum_{j=1}^p\int{\|\boldsymbol{\gamma}_{j}(y)\|^2}dH(y) <\infty,\] respectively. Finally, \(\|\boldsymbol{\kappa}\|_{H}^{2}=o(1)\) from Lemma 2 completes the proof of the theorem. ◻
Lemma 3. In addition to the assumptions of Theorem 1, suppose the assumptions (a.9) and (a.10) hold. Then, (U.2)-(U.3) hold true.
Remark 3. The proof of the lemma will be almost the same as that of Lemma 5.5.4 from [1], and hence, we do not include it here.
Ascertaining that the ULAQ conditions are met, we proceed to prove the asymptotic normality of the MD estimator. We first specify the asymptotic distribution of \(\boldsymbol{S}_{n}\) and convergence of \(\mathbf{W}_{n}\) in Lemma 1. Then, the subsequent application of Lemma 1 will yield the desired result, which is another main result of this paper; see, e.g., Theorem 2. Let \(\widetilde{\boldsymbol{\Gamma}}_{n}(\boldsymbol{\beta}):=\int{\boldsymbol{\Gamma}_{n}(\boldsymbol{\beta};y)'\boldsymbol{\Gamma}_{n}(\boldsymbol{\beta};y)}dH(y)\).
Lemma 4. Assume (a.1)-(a.10). In addition, assume that \[\lim_{n\rightarrow\infty}\widetilde{\boldsymbol{\Gamma}}_{n}=\widetilde{\boldsymbol{\Gamma}},\] where \(\widetilde{\boldsymbol{\Gamma}}\) is positive-definite. Then \(\mathbf{A}\boldsymbol{S}_{n}(\boldsymbol{\beta}_{0})\) is asymptotically normally distributed, and \(\mathbf{A}\mathbf{W}_{n}\mathbf{A}\) converges to \(\widetilde{\boldsymbol{\Gamma}}\) as \(n\) approaches \(\infty\).
Proof. The claim for \(\mathbf{W}_{n}\) readily immediately follows from (3 ) and the assumption. Note that for a real-valued function \(\eta:\mathbb{R}\rightarrow\mathbb{R}\), \[\int{\eta(y)\Big\{I(Y_{i}\le y)-F_{i}(y;\boldsymbol{\beta}_{0})\Big\}}dH(y) = \int_{Y_{i}}^{\infty}\eta(y)dH(y)-\mathbb{E}\left(\int_{Y_{i}}^{\infty}\eta(y)dH(y)\right).\] Recall the \(j\)th column vector of \(\boldsymbol{\Gamma}_{n}\): \(\boldsymbol{\gamma}_{j}=(\gamma_{1j},...,\gamma_{pj})'\in\mathbb{R}^{p}\). For any \(\boldsymbol{b}=(b_{1},...,b_{p})'\in \mathbb{R}^{p}\), we have \[\begin{align} \boldsymbol{b}'\mathbf{A}\boldsymbol{S}_{n}(\boldsymbol{\beta}_{0}) &=& \sum_{i=1}^n\sum_{j=1}^pd_{ij} \sum_{l=1}^{p}b_{l}\int{\gamma_{lj}(\boldsymbol{\beta}_{0};y)\Big\{I(Y_{i}\le y)-F_{i}(y;\boldsymbol{\beta}_{0})\Big\}}dH(y) ,\\ &=& \sum_{i=1}^n\sum_{j=1}^pd_{ij} \sum_{l=1}^{p}b_{l}\Big( \psi_{lj}(Y_{i}) - \mathbb{E}\{\psi_{lj}(Y_{i})\} \Big),\\ &=& \sum_{i=1}^n\xi_{i},\qquad (say), \end{align}\] where \(\psi_{lj}(Y_{i}):=\int_{Y_{i}}^{\infty}\gamma_{lj}(\boldsymbol{\beta}_{0};y)dH(y)\). Note that (a.8) implies \(|\psi_{lj}(Y_{i})|<\infty\), which implies \(|\xi_{i}|\) is bounded by \(c\,\max |d_{ij}|\) for some constant \(c<\infty\). We shall show that the Lindeberg-Feller (L-F) condition for \(\boldsymbol{b}'\mathbf{A}\boldsymbol{S}_{n}\) will be satisfied. \(\mathbb{E}{\xi_{i}}=0\) is clear. Let \(\sigma_{i}^{2}=Var(\xi_{i})\) and \(\tau_{n}=\sum_{i=1}^n\sigma_{i}^{2}\). Thus, we have for any \(\epsilon>0\), \[\begin{align} \frac{1}{\tau_{n}^{2}}\sum_{i=1}^n\mathbb{E}[\xi_{i}^{2}:|\xi_{i}|>\epsilon \tau_{n}]&\le & c\tau_{n}^{-2}\left(\max_{1\le i\le n,1\le j\le p}d_{ij}^{2}\right)\sum_{i=1}^n\mathbb{P}(|\xi_{i}|>\epsilon \tau_{n}) \\ &\le & c\epsilon^{-2}\tau_{n}^{-2}\left(\max_{1\le i\le n,1\le j\le p}d_{ij}^{2}\right) \rightarrow 0, \end{align}\] where the second one is immediate after application of the Chevyshev inequality to the summand, while the convergence to zero follows from (a.2), thereby showing that the L-F condition is met. Define an \(n\times n\) diagonal matrix \(\L_{n}\) whose \(i\)th entry is \(F_{i}(y)(1-F_{i}(y))\). Note that \[\begin{align} Var(\boldsymbol{b}'\mathbf{A}\boldsymbol{S}_{n})&=&\boldsymbol{b}'\mathbb{E}\int{\boldsymbol{\Gamma}_{n}'\boldsymbol{{\cal W}}\boldsymbol{{\cal W}}'\boldsymbol{\Gamma}_{n}}dH(y) \boldsymbol{b},\\ &=&\boldsymbol{b}'\int{\boldsymbol{\Gamma}_{n}'{\mathbf{D}}'\boldsymbol{\Lambda}_{n}{\mathbf{D}}\boldsymbol{\Gamma}_{n}}dH(y) \boldsymbol{b},\\ &=&\boldsymbol{b}'\boldsymbol{\Omega}_{n} \boldsymbol{b},\quad (say), \end{align}\] where the second last equality follows from the Fubini’s theorem, and \(\mathbb{E}(\boldsymbol{{\cal W}}\boldsymbol{{\cal W}}')={\mathbf{D}}'\boldsymbol{\Lambda}_{n}{\mathbf{D}}\). Thus, the Cramer-Wold device will yield the asymptotic normality of \(\mathbf{A}\boldsymbol{S}_{n}\) \[\boldsymbol{\Omega}_{n}^{-1/2}\mathbf{A}\boldsymbol{S}_{n}\Rightarrow_{{\cal D}}N(\boldsymbol{0}_{p\times 1}, \mathbf{I}_{p\times p}).\] ◻
Finally, we conclude this section by stating the asymptotic normality of the MD estimator. Recall \(\widetilde{\boldsymbol{\Gamma}}_{n}(\boldsymbol{\beta}_{0})\) and \(\boldsymbol{\Omega}_{n}(\boldsymbol{\beta}_{0})\) in Lemma 4. Define \(\boldsymbol{\Sigma}_{n}(\boldsymbol{\beta}_{0}) := \widetilde{\boldsymbol{\Gamma}}_{n}^{-1}\boldsymbol{\Omega}_{n}\widetilde{\boldsymbol{\Gamma}}_{n}^{-1}\).
Theorem 2. Suppose the assumptions of Lemma 4 hold. Then the MD estimator \(\widehat{\boldsymbol{\beta}}\) in (2 ) will be asymptotically normally distributed \[\boldsymbol{\Sigma}_{n}^{-1/2}\mathbf{A}^{-1}(\widehat{\boldsymbol{\beta}}-\boldsymbol{\beta}_{0})\Rightarrow_{{\cal D}}N(\boldsymbol{0}_{p\times 1}, \mathbf{I}_{p\times p}).\]
Proof. Note that the ULAQ conditions are met by Theorem 1 and Lemma 3, and hence, Lemma 1 accompanied by Lemma 4 will immediately imply the claim, thereby completing the proof of the theorem. ◻
This study applied the MD estimation with the CvM-type distance function, along with a different approach proposed by [11], to estimate the parameter of the regression setup of the exponential distribution and demonstrated that the MD estimator still retains desirable properties, such as asymptotic normality and robustness.The constant rate over time, which is a bit strong and rare assumption in real-world application, is the definite limit of this study. Thus, further extensions of the current study by releasing the constant rate assumption will form the basis for future research.