The paper deals with approximate ancillarity as discussed by Efron and Hinkley (1978). In the multivariate i.i.d. case we derive the second-order Edgeworth expansion of the MLE given a normalized version of the second derivative of the log-likelihood at its maximum. The expansion agrees with the one derived by Amari (1982a) for curved exponential families, but holds for any family satisfying the regularity conditions given in the paper. It is shown that the Fisher…
T. W. Anderson · Charles Stein · Asad Zaman·刊期 1985-06-01 · 卷 13（2）·The Annals of Statistics已编入正刊
方向参数的最佳不变估计AI 解读
Let $X$ be an $n \times k$ random matrix whose coordinates are independently normally distributed with common variance $\sigma^2$ and means given by $EX = e\mu' + \theta\lambda',$ where $e$ is the vector in $R^n$ having all coordinates equal to $1, \theta \in R^n,$ and $\mu, \lambda \in R^k$ with $\sum^k_{j = 1} \lambda^2_j = 1.$ The problem is to estimate $\lambda$, say by $\hat{\lambda},$ with loss function $1 - (\lambda'\hat{\lambda})^2$ when $\mu, \theta,$ and $\sigma^2$…
The number of local maxima of the Cauchy location likelihood function which are not global maxima is asymptotically Poisson distributed with mean parameter $1/\\pi.$
The classical problem of choice of number of classes in testing goodness of fit is considered for a class of alternatives, for the chi-square and likelihood ratio statistics. Pitman and Bahadur efficiencies are used to compare the two statistics and also to analyse the effect for each statistic of changing the number of classes for the case where the number of classes increases asymptotically with the number of observations. Overall, the results suggest that if the class of…
Let $(X, Y)$ be a pair of random variables such that $X = (X_1, \cdots, X_J)$ and let $f$ by a function that depends on the joint distribution of $(X, Y).$ A variety of parametric and nonparametric models for $f$ are discussed in relation to flexibility, dimensionality, and interpretability. It is then supposed that each $X_j \in \lbrack 0, 1\rbrack,$ that $Y$ is real valued with mean $\mu$ and finite variance, and that $f$ is the regression function of $Y$ on $X.$ Let $f^\ast,$…
Multidimensional asymptotic normality of the kernel quantile estimator is established under fairly general conditions on the underlying distribution function and on the kernel. Sharpening these assumptions, one can utilize the proof to achieve also a bound for the rate of convergence which entails the comparison of the kernel estimator with the empirical quantile on the basis of their covering probabilities.
R. A. Boyles · A. W. Marshall · F. Proschan·刊期 1985-03-01 · 卷 13（1）·The Annals of Statistics已编入正刊
平均失效率递增分布的最大似然估计不一致性AI 解读
Marshall and Proschan (1965) showed that the MLE for a life distribution with increasing failure rate is strongly consistent. In this note we show that the MLE for a life distribution with increasing failure rate average is not consistent; in fact, maximum likelihood estimation in the IFRA case yields estimators of the average failure rate and of the distribution function which, in general, converge a.s. to values other than the true values. In the decreasing failure rate…
Several authors have proposed empirical Bayes tests (EBT) for the continuous one-parameter exponential family for the case that the prior distribution is completely unspecified. They investigated the convergence rate of the (unconditional) Bayes risk, and gave upper bounds for this convergence rate. In this paper it is proposed to study the convergence of the conditional Bayes risk. A method is presented which makes it possible to derive the exact convergence rate of the…
Consider a trigonometric regression of order $m$ or a polynomial regression of degree $m$. Explicit $D_s$-optimal designs are given for some subsets of the coefficients. Lauter type optimal designs are given for various models involving the order or the degree. The designs are calculated using canonical moments.
Generalized linear models are used for regression analysis in a number of cases, including categorical responses, where the classical assumptions are violated. The statistical analysis of such models is based on the asymptotic properties of the maximum likelihood estimator. We present mild general conditions which, respectively, assure weak or strong consistency or asymptotic normality. Most of the previous work has been concerned with natural link functions. In this case…
A class of goodness-of-fit tests based on sums of squares of $L$-statistics is proposed for testing a composite parametric location and/or scale null hypothesis versus a general parametric alternative. It is shown that such tests can be constructed optimally to have the same asymptotic power against sequences of local alternatives as the generalized likelihood ratio statistics [G.L.R.S.] and, in fact, under suitable regularity conditions to be asymptotically equivalent to…
In Wald's statistical decision theory, the criterion of domination (or uniform betterness) is defined with respect to a specific loss. In practice, however, the exact form of a loss function is difficult to specify. Hence, it is important to study the domination criterion simultaneously under a class of loss functions. In this paper we focus on estimation problems. We mainly investigate the possibility of domination simultaneously under the class of loss functions $L(|\theta - \delta|)$…
Admissible linear estimators $Mx + \gamma$ must be pointwise limits of Bayes estimators. Using properties of Bayes estimators preserved by taking limits, the structure of $M$ and $\gamma$ can be determined. Among $M, \gamma$ with this structure, a necessary and sufficient condition for admissibility is obtained. This condition is applied to the case of linear (mixture) models. It is shown that only the most trivial such models admit linear estimators of full rank which are…
For a continuous and diagonally symmetric multivariate distribution, incorporating the idea of preliminary test estimators, a variant form of the James-Stein type estimation rule is used to formulate some shrinkage estimators of location based on rank statistics and $U$-statistics. In an asymptotic setup, the relative risks for these shrinkage estimators are shown to be smaller than their classical counterparts.
Consider the problem of estimating an unknown distribution function $F$ from the class of all distribution functions given a random sample of size $n$ from $F$. It is proved that the empirical distribution function is admissible for the loss functions $L(F, \hat{F}) = \int (\hat{F}(t) - F(t))^2(F(t))^\alpha(1 - F(t))^b dW(t)$ for any $a < 1$ and $b < 1$ and finite measure $W$. Related results for simultaneous estimation of distribution functions and for finite population…
A new estimator for smoothing towards log-linear models for Poisson means is introduced. The estimator is a generalization of one of Peng (1975). The theoretical basis for the choice of estimator is developed from an approximation to the mean square error of any estimator of Poisson means. The new estimator and its competitors are evaluated in simulations. The method has widespread application in contingency table analysis.
A sequential fixed-width confidence interval for the mean of a $U$-statistic, having coverage probability approximately equal to preassigned $\alpha$, is presented. The main result, Theorem 2, shows that the sequential procedure is asymptotically efficient in the sense of Chow and Robbins (1965) and assumes only finiteness of the second moment of the kernel, the weakest possible condition. The paper follows naturally from Sproule (1974) and Sproule (1969), the primary…
Let $\mathscr{P}$ be a finite population with $N \geq 1$ elements; for each $e \in \mathscr{P}$, let $X_e$ and $Y_e$ be independent, positive random variables with unknown distribution functions $F$ and $G$; and suppose that the pairs $(X_e, Y_e)$ are i.i.d. We consider the problem of estimating $F, G$, and $N$ when the data consist of those pairs $(X_e, Y_e)$ for which $e \in \mathscr{P}$ and $Y_e \leq X_e$. The nonparametric maximum likelihood estimators (MLEs) of $F$ and $G$…
A key identity for the product-limit estimator due to Aalen and Johansen (1978) and Gill (1980) is shown to be a consequence of the exponential formula of Doleans-Dade (1970). The basic counting processes in the censored data problem are shown to converge jointly to Poisson processes under "heavy-censoring": $G_n \rightarrow_d \delta_0$, but $n(1 - G_n) \rightarrow \alpha$ where $G_n$ is the censoring distribution. The Poisson limit theorem for counting processes implies…
The paper considers the Kaplan-Meier estimator $F^{\mathrm{KM}}_n$ from a combinatoric viewpoint. Under the assumption that the estimated distribution $F$ and the censoring distribution $G$ are continuous, the combinatoric results are used to show that $\int |\theta(z)| dF^{\mathrm{KM}}_n(z)$ has expectation not larger than $\int |\theta(z)| dF(z)$ for any sample size $n$. This result is then coupled with Chebychev's inequality to demonstrate the weak convergence of the…
A conjecture of Prentice is established which states that for censored linear rank test, exact scores based on conditional expectations can be replaced by approximate scores obtained by evaluating the score function at an estimate of the survival function. We show that under minimal conditions, asymptotically equivalent tests are obtained when either the Kaplan-Meier, Altshuler, or moment estimator of the survival function is used. Asymptotic normality is also established…
A general procedure for multistage modification of pivotal statistics is developed to improve the normal approximation. Bootstrapping a first stage modified statistic is shown to be equivalent, in terms of asymptotic order, to the normal approximation of a second stage modification. Explicit formulae are given for some basic cases involving independent random samples and samples drawn without replacement. The Hodges-Lehmann deficiency is calculated to compare the regular $t$-…
Bootstrap tests and confidence regions for functions of the population covariance matrix have the desired asymptotic levels, provided model restrictions, such as multiple eigenvalues in the covariance matrix, are taken into account in designing the bootstrap algorithm.
Let X be an n×k random matrix whose coordinates are independently normally distributed with common variance σ2 and means given by EX=eμ′+θλ′, where e is the vector in Rn having all coordinates equal to 1,θ∈Rn, and μ,λ∈Rk with ∑j=1kλj2=1. The problem is to estimate λ, say by λ^, with loss function 1−(λ′λ^)2 when μ,θ, and σ2…
The number of local maxima of the Cauchy location likelihood function which are not global maxima is asymptotically Poisson distributed with mean parameter 1/pi.
Let (X,Y) be a pair of random variables such that X=(X1,⋯,XJ) and let f by a function that depends on the joint distribution of (X,Y). A variety of parametric and nonparametric models for f are discussed in relation to flexibility, dimensionality, and interpretability. It is then supposed that each Xj∈[0,1], that Y is real valued with mean μ and finite variance, and that f is the regression function of Y on X. Let f∗,…
Consider a trigonometric regression of order m or a polynomial regression of degree m. Explicit Ds-optimal designs are given for some subsets of the coefficients. Lauter type optimal designs are given for various models involving the order or the degree. The designs are calculated using canonical moments.
Optimal Goodness-of-Fit Tests for Location/Scale Families of Distributions Based on the Sum of Squares of L-Statistics
A class of goodness-of-fit tests based on sums of squares of L-statistics is proposed for testing a composite parametric location and/or scale null hypothesis versus a general parametric alternative. It is shown that such tests can be constructed optimally to have the same asymptotic power against sequences of local alternatives as the generalized likelihood ratio statistics [G.L.R.S.] and, in fact, under suitable regularity conditions to be asymptotically equivalent to…
In Wald's statistical decision theory, the criterion of domination (or uniform betterness) is defined with respect to a specific loss. In practice, however, the exact form of a loss function is difficult to specify. Hence, it is important to study the domination criterion simultaneously under a class of loss functions. In this paper we focus on estimation problems. We mainly investigate the possibility of domination simultaneously under the class of loss functions L(∣θ−δ∣)…
Admissible linear estimators Mx+γ must be pointwise limits of Bayes estimators. Using properties of Bayes estimators preserved by taking limits, the structure of M and γ can be determined. Among M,γ with this structure, a necessary and sufficient condition for admissibility is obtained. This condition is applied to the case of linear (mixture) models. It is shown that only the most trivial such models admit linear estimators of full rank which are…
For a continuous and diagonally symmetric multivariate distribution, incorporating the idea of preliminary test estimators, a variant form of the James-Stein type estimation rule is used to formulate some shrinkage estimators of location based on rank statistics and U-statistics. In an asymptotic setup, the relative risks for these shrinkage estimators are shown to be smaller than their classical counterparts.
Consider the problem of estimating an unknown distribution function F from the class of all distribution functions given a random sample of size n from F. It is proved that the empirical distribution function is admissible for the loss functions L(F,F^)=∫(F^(t)−F(t))2(F(t))α(1−F(t))bdW(t) for any a<1 and b<1 and finite measure W. Related results for simultaneous estimation of distribution functions and for finite population…
Sequential Nonparametric Fixed-Width Confidence Intervals for U-Statistics
A sequential fixed-width confidence interval for the mean of a U-statistic, having coverage probability approximately equal to preassigned α, is presented. The main result, Theorem 2, shows that the sequential procedure is asymptotically efficient in the sense of Chow and Robbins (1965) and assumes only finiteness of the second moment of the kernel, the weakest possible condition. The paper follows naturally from Sproule (1974) and Sproule (1969), the primary…
Let P be a finite population with N≥1 elements; for each e∈P, let Xe and Ye be independent, positive random variables with unknown distribution functions F and G; and suppose that the pairs (Xe,Ye) are i.i.d. We consider the problem of estimating F,G, and N when the data consist of those pairs (Xe,Ye) for which e∈P and Ye≤Xe. The nonparametric maximum likelihood estimators (MLEs) of F and G…
A key identity for the product-limit estimator due to Aalen and Johansen (1978) and Gill (1980) is shown to be a consequence of the exponential formula of Doleans-Dade (1970). The basic counting processes in the censored data problem are shown to converge jointly to Poisson processes under "heavy-censoring": Gn→dδ0, but n(1−Gn)→α where Gn is the censoring distribution. The Poisson limit theorem for counting processes implies…
The paper considers the Kaplan-Meier estimator FnKM from a combinatoric viewpoint. Under the assumption that the estimated distribution F and the censoring distribution G are continuous, the combinatoric results are used to show that ∫∣θ(z)∣dFnKM(z) has expectation not larger than ∫∣θ(z)∣dF(z) for any sample size n. This result is then coupled with Chebychev's inequality to demonstrate the weak convergence of the…
A general procedure for multistage modification of pivotal statistics is developed to improve the normal approximation. Bootstrapping a first stage modified statistic is shown to be equivalent, in terms of asymptotic order, to the normal approximation of a second stage modification. Explicit formulae are given for some basic cases involving independent random samples and samples drawn without replacement. The Hodges-Lehmann deficiency is calculated to compare the regular t-…