Empirical partially Bayes methods are considered as a means of improving efficiency in a class of problems in which the number of nuisance parameters increases to infinity. In the method used, the parameter of interest is estimated in an asymptotically unbiased way while James-Stein shrinkage is applied to the nuisance parameter estimates. When the shrinkage estimators are carefully chosen, this yields estimators generally more efficient than maximum likelihood. In the…
The classical chi-square test for independence in a two-way contingency table often rejects the independence hypothesis at an extremely small significance level, particularly when the sample size is large. This paper proposes some alternative distributions to independence, to help interpret the $\chi^2$ statistic in such situations. The uniform alternative, in which every possible contingency table of the given dimension and sample size receives equal probability, leads to…
We introduce two nonparametric multivariate density estimators that are particularly suitable for application in interactive computing environments. These estimators are statistically comparable to kernel methods and computationally comparable to histogram methods. Asymptotic theory of the estimators is presented and examples with univariate and simulated trivariate Gaussian data are illustrated.
We suggest a sieve for the estimation of the spectral density of a Gaussian stationary stochastic process. In contrast to the standard periodogram-based estimates this one aims at exploiting the full Gaussian nature of the process.
In this paper we consider asymptotic approximations to the distribution function $F(x)$ of a linear combination of an estimate in a multivariate linear model. A method is given for obtaining an asymptotic expansion $F_{s - 1}(x)$ of $F(x)$ up to $O(n^{-s + 1})$ and a bound $c_s$ such that $|F(x) - F_{s - 1}(x)| \leq c_s$ uniformly in $x$ and $c_s = O(n^{-s})$.
In a recent paper, Tanner (1983) proves pointwise consistency of a variable bandwidth kernel estimator for the hazard function. In the present note, a simplified proof of uniform consistency of a data-adaptive kernel estimator with certain additional advantages is given.
Glen Meeden · Malay Ghosh · Stephen Vardeman·刊期 1985-06-01 · 卷 13（2）·The Annals of Statistics已编入正刊
若干可容许的非参数估计量及相关有限总体抽样估计量AI 解读
Given a random sample from an unknown distribution $F,$ which is assumed to belong to some nonparametric family of distributions, consider the problem of estimating $\gamma(F),$ some function of $F.$ When the loss function is squared error, admissible estimators are exhibited for a large class of $\gamma$'s. A relationship between these estimators and similar ones in finite population sampling is demonstrated.
A framework is presented for deciding among functional forms when constructing confidence bands in simple linear regression. Using the concept of tautness, definitions of admissibility and completeness are developed. These lead to a characterization of a minimal complete class of band forms. A type of average width optimality within this class is briefly discussed.
The method of maximum likelihood leads to an ill-posed optimization problem in the case of a mixture of normal distributions. Estimation in the univariate case is reformulated using simple constraints into an optimization problem having a strongly consistent, global solution.
Consider a sequential probability ratio test comparing two treatments, where each subject receives only one of the treatments. Each subject's treatment assignment is determined by the flip of a biased coin, where the bias serves to balance the number of patients assigned to each treatment. The asymptotic properties of this test are studied, as the sample size approaches infinity. A renewal theorem is given for the joint distribution of the sample size, the imbalance in…
In a clinical comparison of responses to two treatments, patients are admitted sequentially and given one of the two treatments. The allocation is determined randomly, to decrease the possibility of personal bias in the selection of subjects for the test. To balance the assignments, the probability of receiving one treatment is a function of the proportion of patients previously assigned to that treatment. A local limit theorem for the distribution of the number of patients…
Let $(\mathscr{X, A})$ be a space with a $\sigma$-field, $M = \{P_s; s \in \Theta\}$ be a family of probability measures on $\mathscr{A}$ with $\Theta$ arbitrary, $X_1, \cdots, X_n$ i.i.d. observations on $P_\theta.$ Define $\mu_n(A) = (1/n) \sum^n_{i = 1} I_A(X_i),$ the empirical measure indexed by $A \in \mathscr{A}.$ Assume $\Theta$ is totally bounded when metrized by the $L_1$ distance between measures. Robust minimum distance estimators $\hat{\theta}_n$ are constructed…
It is shown that Hill's estimator (1975) for the exponent of regular variation is asymptotically normal if the number $k_n$ of extreme order statistics used to construct it tends to infinity appropriately with the sample size $n.$ As our main result, we derive a general condition which can be used to determine the optimal $k_n$ explicitly, provided that some prior knowledge is available on the underlying distribution function with regularly varying upper tail. This condition…
Basic decision theory for discrete random variables of the multivariate geometric (power series) type is developed. Some properties of Bayes estimators that carry over in the limit to admissible estimators are obtained. A stepwise generalized Bayes representation of admissible estimators is developed with estimation of the mean of a multivariate Poisson random variable in mind. The development carries over to estimation of the mean of a multivariate negative Binomial random…
Let $X_1, X_2, \cdots$ be i.i.d. random variables with mean $\theta$ and finite, positive variance $\sigma^2,$ depending on unknown parameters $\omega\in\Omega.$ The problem addressed is that of finding a stopping time $t$ for which the risk $R_A(t, \omega) = E_\omega\{A \gamma^2_0(\omega)(\bar{X}_t - \theta)^2 + t\}$ is as small as possible (in a suitable sense), where $A > 0, \gamma_0$ is a positive function on $\Omega$, and $\bar{X}_t = (X_1 + \cdots + X_t)/t.$ For fixed…
An estimator $\hat{\beta}$ of $\beta$ is accurate with accuracy $A$ and confidence $\gamma, 0 < \gamma < 1,$ if $P(\hat{\beta} - \beta \in A) \geq \gamma$ for all $\beta.$ Given a sequence $Y_1, Y_2, \cdots$ of independent vector-valued homoscedastic normally-distributed random variables generated via the general linear model $Y_i = X_i\beta + \varepsilon,$ the $k$-dimensional parameter $\beta$ is accurately estimated using a sequential version of the maximum probability…
Under an age replacement policy, a stochastically failing unit is replaced at failure or after being in service for $t$ units of time, whichever comes first. An important problem is the estimation of $\phi^\ast,$ the optimal replacement time when the form of the failure distribution is unknown. Here, $\phi^\ast$ is optimal in the sense that it is the replacement time that achieves the smallest long-run expected cost. It is shown that substantial cost savings can be effected…
This paper considers the asymptotic minimax property of the sequential probability ratio test (SPRT) when the given distributions $P_{\pm \varepsilon}$ contain a small amount of contamination. Let $\mathscr{P}_{\pm \varepsilon}$ be the neighborhoods of $P_{\pm \varepsilon}.$ Suppose that $P_\varepsilon$ and $P_{-\varepsilon}$ approach each other as $\varepsilon \downarrow 0$ and that $\mathscr{P}_{\pm \varepsilon}$ shrink at an appropriate rate. We prove (under regularity…
We consider a standard Bayes decision situation except that with small probability the data may be irrelevant to the parameter of interest (i.e. the experiment is "biased"). The minimax solution under quite general assumptions is described and discussed.
A new two-sample randomization test is proposed for testing that the joint distribution of two samples is invariant under permutations. The $p$-value of the test has a finite sample minimaxity property over neighborhoods of completely specified alternative distributions. Asymptotically, the test has minimax Bahadur slope against the neighborhoods, which remain fixed as the sample sizes increase. The proposed test also offers the best compromise between robustness against…
A histogram type estimator of the hazard rate of lifetimes which are randomly right-censored is studied. This estimator is based on random spacings of the order statistics of uncensored observations and extends the idea of the histogram density estimation suggested by Van Ryzin (1973) to censored data. In this paper we establish pointwise large sample properties of the estimator including strong consistency, strong uniform consistency on a bounded interval, and two…
The problem of testing for a constant failure rate against alternatives with failure rates involving a single change-point is considered. The asymptotic significance level for tests based on maximal score statistics are shown to involve the solution to a first passage time problem for an Ornstein-Uhlenbeck process. An example illustrates the methodology.
Many population processes in demography, epidemiology and other fields can be represented by a time-continuous Markov chain model with a finite state space. If we have complete information on the life history of a cohort, the intensities of the Markov model may be estimated by the occurrence/exposure rates or by nonparametric techniques. In many situations, however, we have only incomplete information. In this paper we consider the special, but important, case where the…
The problem of testing the underlying distribution in a general regression model is considered when the data are right censored. It is proposed that tests be based on efficient scores from a certain class of parametric alternatives. The connection between tests based on the scores and more traditional approaches using empirical distribution functions is explored. Neyman's smooth test is extended to allow for censored data.
The classical chi-square test for independence in a two-way contingency table often rejects the independence hypothesis at an extremely small significance level, particularly when the sample size is large. This paper proposes some alternative distributions to independence, to help interpret the χ2 statistic in such situations. The uniform alternative, in which every possible contingency table of the given dimension and sample size receives equal probability, leads to…
In this paper we consider asymptotic approximations to the distribution function F(x) of a linear combination of an estimate in a multivariate linear model. A method is given for obtaining an asymptotic expansion Fs−1(x) of F(x) up to O(n−s+1) and a bound cs such that ∣F(x)−Fs−1(x)∣≤cs uniformly in x and cs=O(n−s).
Given a random sample from an unknown distribution F, which is assumed to belong to some nonparametric family of distributions, consider the problem of estimating γ(F), some function of F. When the loss function is squared error, admissible estimators are exhibited for a large class of γ's. A relationship between these estimators and similar ones in finite population sampling is demonstrated.
Let (X,A) be a space with a σ-field, M={Ps;s∈Θ} be a family of probability measures on A with Θ arbitrary, X1,⋯,Xn i.i.d. observations on Pθ. Define μn(A)=(1/n)∑i=1nIA(Xi), the empirical measure indexed by A∈A. Assume Θ is totally bounded when metrized by the L1 distance between measures. Robust minimum distance estimators θ^n are constructed…
It is shown that Hill's estimator (1975) for the exponent of regular variation is asymptotically normal if the number kn of extreme order statistics used to construct it tends to infinity appropriately with the sample size n. As our main result, we derive a general condition which can be used to determine the optimal kn explicitly, provided that some prior knowledge is available on the underlying distribution function with regularly varying upper tail. This condition…
Let X1,X2,⋯ be i.i.d. random variables with mean θ and finite, positive variance σ2, depending on unknown parameters ω∈Ω. The problem addressed is that of finding a stopping time t for which the risk RA(t,ω)=Eω{Aγ02(ω)(Xˉt−θ)2+t} is as small as possible (in a suitable sense), where A>0,γ0 is a positive function on Ω, and Xˉt=(X1+⋯+Xt)/t. For fixed…
An estimator β^ of β is accurate with accuracy A and confidence γ,0<γ<1, if P(β^−β∈A)≥γ for all β. Given a sequence Y1,Y2,⋯ of independent vector-valued homoscedastic normally-distributed random variables generated via the general linear model Yi=Xiβ+ε, the k-dimensional parameter β is accurately estimated using a sequential version of the maximum probability…
Under an age replacement policy, a stochastically failing unit is replaced at failure or after being in service for t units of time, whichever comes first. An important problem is the estimation of ϕ∗, the optimal replacement time when the form of the failure distribution is unknown. Here, ϕ∗ is optimal in the sense that it is the replacement time that achieves the smallest long-run expected cost. It is shown that substantial cost savings can be effected…
This paper considers the asymptotic minimax property of the sequential probability ratio test (SPRT) when the given distributions P±ε contain a small amount of contamination. Let P±ε be the neighborhoods of P±ε. Suppose that Pε and P−ε approach each other as ε↓0 and that P±ε shrink at an appropriate rate. We prove (under regularity…
A new two-sample randomization test is proposed for testing that the joint distribution of two samples is invariant under permutations. The p-value of the test has a finite sample minimaxity property over neighborhoods of completely specified alternative distributions. Asymptotically, the test has minimax Bahadur slope against the neighborhoods, which remain fixed as the sample sizes increase. The proposed test also offers the best compromise between robustness against…