The paper discusses statistics that can be used to test whether two failure times, say $X_1$ and $X_2$, are independent. The two variables are subject to right censoring so that what is observed is $Y_i = \min(X_i, Z_i)$ and $\delta_i = I(X_i = Y_i)$, where $(Z_1, Z_2)$ are censoring times independent of $(X_1, X_2)$. Statistics that generalize the Spearman rank correlation and the log-rank correlation are considered, as well as general linear rank statistics. The…
This article presents a family of estimators of the survival function based on right-censored observations which admit the possibility that the censoring variables may not be independent of the true failure variables. This family is obtained by generalizing the self-consistent property (Efron, 1967) of the product limit estimator (Kaplan and Meier, 1958). By assuming a Dirichlet process prior distribution of the observable random vectors, nonparametric Bayesian estimators of…
The estimation problem of the quantiles $\xi + b\sigma$ of an exponential distribution with unknown location-scale parameter $(\xi, \sigma)$ is considered. We establish the admissibility of the traditional (best equivariant) estimator for quadratic loss when $n^{-1} \leq b \leq 1 + n^{-1}$ where $n$ is the sample size. For $b > 1 + n^{-1}$ a class of minimax procedures is found. This class contains generalized Bayes rules and one of them is shown to be admissible within the…
A new class of solutions to a general differential inequality often encountered in multiparameter estimation problems is obtained. Using these solutions as guidelines, improved estimators for the scale parameters as well as the natural parameters of independent gamma distributions are obtained for a large class of weighted quadratic losses. The improved estimators have an empirical Bayes interpretation. They also permit an exact analytical representation of the risk…
This paper examines the asymptotic properties of compromise estimators. By this we mean an estimation method which compromises between a finite number of sampling situations in a small sample optimal way. We develop the asymptotic theory of such estimators in the location problem and show that under a specific choice of a pair of sampling situations the compromise estimator is asymptotically robust in Huber's sense.
Tests for a null hypothesis whose specification involves an unknown nuisance parameter may be obtained by inverting a bootstrap confidence region for the parameter being tested or by constructing a simulated null distribution for the test statistic. The power of either test against certain alternatives involving the same unknown nuisance parameter can itself be estimated by simulation.
Gaussian Markov distributions are characterised by zeros in the inverse of their covariance matrix and we describe the conditional independencies which follow from a given pattern of zeros. Describing Gaussian distributions with given marginals and solving the likelihood equations with covariance selection models both lead to a problem for which we present two cyclic algorithms. The first generalises a published algorithm for covariance selection whilst the second is…
For an arbitrary one parameter exponential family density it is shown how to construct a mixing distribution (prior) on the parameter in such a way that the resulting mixture distribution is a two (or more) parameter exponential family. Reweighted infinitely divisible distributions are shown to be the parametric mixing distributions for which this occurs. As an illustration conditions are given under which a parametric mixture of negative exponentials is in the exponential…
Since their introduction by Dwass (1957) and Barnard (1963), Monte Carlo tests have attracted considerable attention. The aim of this paper is to give a unified approach that covers the case of an arbitrary null distribution in order to study the statistical properties of Monte Carlo tests under the null hypothesis and under the alternative. For finite samples we obtain bounds for the power of the Monte Carlo test with the original test that allow determination of the…
For the canonical problem of estimating a multivariate normal mean under squared-error-loss, this article addresses the problem of selecting a minimax shrinkage estimator when vague or conflicting prior information suggests that more than one estimator from a broad class might be effective. For this situation a new class of alternative estimators, called multiple shrinkage estimators, is proposed. These estimators use the data to emulate the behavior and risk properties of…
A general asymptotic theory is developed for the maximum likelihood estimator based on a partial likelihood. Conditions are given for consistency and asymptotic normality, and a method is provided for the calculation of the asymptotic efficiency of the estimator. The implications of the general theory are examined in special cases such as inference in stochastic processes, Cox regression models, and AR processes with missing segments.
here K is a kernel function (usually a symmetric probability density function) and h is a smoothing parameter or bandwidth, the value of which determines how much the data are smoothed to produce the density estimate. The first published paper specifically on density estimation was Rosenblatt (1956); however, density estimates were suggested several years earlier by Fix and Hodges (1951) in a technical report. It is regrettable that this report was never published, since it…
Boscovich's (1757) proposal to estimate the parameters of a linear model by minimizing the sum of absolute deviations subject to the constraint that the mean residual be zero is considered. The asymptotic theory of the estimator confirms a remark of Edgeworth who called it a "remarkable hybrid" between $\mathscr{l}_1$ and $\mathscr{l}_2$ methods.
For $x \in \mathbb{R}$ let $N_\alpha(x) := m\alpha, \operatorname{iff} x \in (\alpha m - \alpha/2, \alpha m + \alpha/2\rbrack$. For a sample $X_1,\ldots, X_n$ we mainly study the asymptotic properties of the estimators $\bar{N}_\alpha := 1/n\sum^n_{i = 1} N_\alpha(X_i)$ and $S^2_\alpha := 1/(n - 1)\sum^n_{i = 1}(N_\alpha(X_i) - \overline{N}_\alpha)^2$ for $\alpha = \alpha_n \rightarrow 0,$ as $n \rightarrow \infty.$ For example, if $E(X^2) < \infty, E(e^{itX}) = o(|t|^{-k}),(|r| \rightarrow\infty)$…
Stein's general technique for improving upon the best invariant unbiased and minimax estimators of the normal covariance matrix is described. The technique is to obtain solutions to a certain differential inequality involving the eigenvalues of the sample covariance matrix. Several improved estimators are obtained by solving the differential inequality. These estimators shrink or expand the sample eigenvalues depending on their magnitude. A scale invariant, adaptive minimax…
Sequential selections are to be made from two stochastic processes, or "arms." At each stage the arm selected for observation depends on past observations. The objective is to maximize the expected sum of the first $n$ observations. For arm 1 the observations are identically distributed with probability measure $P$, and for arm 2 the observations have probability measure $Q; P$ is a Dirichlet process and $Q$ is known. An equivalent problem is deciding sequentially when to…
Jayalakshmi Natarajan · William E. Strawderman·刊期 1985-12-01 · 卷 13（4）·The Annals of Statistics已编入正刊
二次损失下多元正态均值的两阶段序贯估计AI 解读
In estimating a multivariate normal mean under quadratic loss, this paper looks into the existence of two-stage sequential estimators that are better both in risk (mean square error) and sample size than the usual estimator of a given fixed sample size. In other words, given any sample size $n$, we are looking for two-stage sequential estimators truncated at $n$, with a positive probability of stopping earlier and risk lower than that of the sample mean based on $n$ observati…
Strong consistency of least-squares estimates in stochastic regression models is established under the assumption that the underlying model can be reparametrized so that the new design vectors are weakly correlated. An application to fixed-width interval estimation in stochastic approximation schemes is also discussed.
For the linear regression model, $y_i = \mathbf{x}_i\mathbf{\beta} + \varepsilon_i$ with fixed $\mathbf{x}_i s$, the asymptotic normality of $(\hat{\mathbf{\beta}}, \hat{\sigma})$ which minimizes the Huber-Dutter loss function, $\sum\sigma\rho\{(y_i - \mathbf{x}_i\mathbf{\beta})/\sigma\} + A_n\sigma$, is established under rather general conditions.
The well known pure error-lack of fit test, which can be used to assess the adequacy of a linear regression model, is generalized to accommodate the case of nonreplication. The asymptotic null distribution of the proposed test statistic is derived. Also, the proposed test statistic is shown to be asymptotically comparable under general alternatives to the test statistic obtained in the case of replication. Consistency properties associated with pseudo lack of fit and pure…
Kernel estimators of an unknown multivariate regression function are investigated. A bandwidth-selection rule is considered, which can be formulated in terms of cross validation. Under mild assumptions on the kernel and the unknown regression function, it is seen that this rule is asymptotically optimal.
For data $\theta + \varepsilon_i, i = 1, \ldots, n$ where $\varepsilon_i$ are i.i.d. $\sim F$ with the median of $F$ equal to $0$ but $F$ otherwise unknown, it is desired to estimate $\theta$. In Doss (1985) priors are put on the pair $(F, \theta)$, the marginal posterior distribution of $\theta$ is computed, and the mean of the posterior is taken as the estimate of $\theta$. In the present paper a frequentist point of view is adopted. The consistency properties of the Bayes…
Let $X_i, i = 1, \ldots, n$ be i.i.d. $\sim F_\theta$, where $F_\theta(x) = F(x - \theta)$ for some $F$ that has median equal to 0. $F$ is assumed unknown or only partially known, and the problem is to estimate $\theta$. Priors are put on the pair $(F, \theta)$. The priors on $F$ concentrate all their mass on c.d.f.s with median equal to 0. These priors include "Dirichlet-type" priors. The marginal posterior distribution of $\theta$ given $X_1, \ldots, X_n$ is computed. The…
This paper proposes an estimator for the mean of a positive random variable given a sample from that random variable. The positive mean estimation problem is common, but there are few results for this problem in the literature. This paper shows the proposed estimator to be less sensitive to outlying values than the sample mean and determines its asymptotic mean squared error.
The paper discusses statistics that can be used to test whether two failure times, say X1 and X2, are independent. The two variables are subject to right censoring so that what is observed is Yi=min(Xi,Zi) and δi=I(Xi=Yi), where (Z1,Z2) are censoring times independent of (X1,X2). Statistics that generalize the Spearman rank correlation and the log-rank correlation are considered, as well as general linear rank statistics. The…
The estimation problem of the quantiles ξ+bσ of an exponential distribution with unknown location-scale parameter (ξ,σ) is considered. We establish the admissibility of the traditional (best equivariant) estimator for quadratic loss when n−1≤b≤1+n−1 where n is the sample size. For b>1+n−1 a class of minimax procedures is found. This class contains generalized Bayes rules and one of them is shown to be admissible within the…
Boscovich's (1757) proposal to estimate the parameters of a linear model by minimizing the sum of absolute deviations subject to the constraint that the mean residual be zero is considered. The asymptotic theory of the estimator confirms a remark of Edgeworth who called it a "remarkable hybrid" between l1 and l2 methods.
For x∈R let Nα(x):=mα,iffx∈(αm−α/2,αm+α/2]. For a sample X1,…,Xn we mainly study the asymptotic properties of the estimators Nˉα:=1/n∑i=1nNα(Xi) and Sα2:=1/(n−1)∑i=1n(Nα(Xi)−Nα)2 for α=αn→0, as n→∞. For example, if E(X2)<∞,E(eitX)=o(∣t∣−k),(∣r∣→∞)…
Sequential selections are to be made from two stochastic processes, or "arms." At each stage the arm selected for observation depends on past observations. The objective is to maximize the expected sum of the first n observations. For arm 1 the observations are identically distributed with probability measure P, and for arm 2 the observations have probability measure Q;P is a Dirichlet process and Q is known. An equivalent problem is deciding sequentially when to…
In estimating a multivariate normal mean under quadratic loss, this paper looks into the existence of two-stage sequential estimators that are better both in risk (mean square error) and sample size than the usual estimator of a given fixed sample size. In other words, given any sample size n, we are looking for two-stage sequential estimators truncated at n, with a positive probability of stopping earlier and risk lower than that of the sample mean based on n observati…
For the linear regression model, yi=xiβ+εi with fixed xis, the asymptotic normality of (β^,σ^) which minimizes the Huber-Dutter loss function, ∑σρ{(yi−xiβ)/σ}+Anσ, is established under rather general conditions.
For data θ+εi,i=1,…,n where εi are i.i.d. ∼F with the median of F equal to 0 but F otherwise unknown, it is desired to estimate θ. In Doss (1985) priors are put on the pair (F,θ), the marginal posterior distribution of θ is computed, and the mean of the posterior is taken as the estimate of θ. In the present paper a frequentist point of view is adopted. The consistency properties of the Bayes…
Let Xi,i=1,…,n be i.i.d. ∼Fθ, where Fθ(x)=F(x−θ) for some F that has median equal to 0. F is assumed unknown or only partially known, and the problem is to estimate θ. Priors are put on the pair (F,θ). The priors on F concentrate all their mass on c.d.f.s with median equal to 0. These priors include "Dirichlet-type" priors. The marginal posterior distribution of θ given X1,…,Xn is computed. The…