Kesten proposed a method for adjusting the coefficients of a scalar stochastic approximation process, and proved w.p. 1 convergence. A family of multidimensional processes for function minimization are treated here. Each method consists of a sequence of truncated one-dimensional procedures of the Kesten type. The methods seem to offer a number of advantages over the usual Kiefer-Wolfowitz procedures, and are more natural analogs of the schemes in common use in deterministic…
Maximum likelihood type robust estimates of regression are defined and their asymptotic properties are investigated both theoretically and empirically. Perhaps the most important new feature is that the number $p$ of parameters is allowed to increase with the number $n$ of observations. The initial terms of a formal power series expansion (essentially in powers of $p/n$) show an excellent agreement with Monte Carlo results, in most cases down to 4 observations per parameter.
A homogeneous random process on the sphere $\{X(P): P \in S_2\}$ is a process whose mean is zero and whose covariance function depends only on the angular distance $\theta$ between the two points, i.e. $E\lbrack X(P)\rbrack \equiv 0$ and $E\lbrack X(P)X(Q)\rbrack = R(\theta)$. Given $T$ independent realizations of a Gaussian homogeneous process $X(P)$, we first derive the exact distribution of the spectral estimates introduced by Jones (1963 b). Further, an estimate $R^{(T)}(\theta)$…
Various selection problems based on sample means have been considered by Bechhofer (Ann. Math. Statist. (1954), 16-39). Parallel selection procedures based on one-sample Hodges-Lehmann estimators (Ann. Math. Statist. (1963), 598-611) are proposed in this paper. These procedures possess more desirable finite sample properties and equivalent asymptotic properties as compared to rival procedures considered by Lehmann (Math. Ann. (1963), 268-275) and Puri and Puri (Ann. Math.…
This paper shows that a quadratic form in a multivariate sample has a certain rank and its nonzero eigenvalues are distinct with probability one under the assumption that the matrix defining the quadratic form satisfies a certain rank condition and that the underlying distribution of the sample is absolutely continuous with respect to Lebesgue measure.
Let $X_1, \cdots, X_n$ be i.i.d. $N(i, 1), i = 0, \pm 1, \pm 2,\cdots$. Hammersley [2] proposed $\lbrack\bar{X}_n\rbrack$, the nearest integer to the sample mean, as an estimator of $i$. It is proved that $d$ is minimax and admissible relative to zero-one loss. However, it is shown that relative to squared error loss, the estimator is neither admissible nor minimax.
Asymptotic expansions of the non-null distributions of the likelihood ratio criteria for testing the equality of a covariance matrix, equality of a mean vector and a covariance matrix, independence between two sets of variates, equality of two covariance matrices, in multivariate normal distributions are derived under the sequence of alternative hypotheses converging to the null hypothesis when the sample size tends to infinity. Numerical accuracies of the asymptotic…
Some new test criteria are proposed for testing various hypotheses concerning covariance matrices. Asymptotic expansions of their null distributions are derived in terms of the $\\chi^2$-distribution.
This paper describes a general method for deriving optimal procedures for problems where the covariance matrices are patterned under both null and alternative hypotheses. The pattern considered in this paper was first suggested by Olkin (1970) and is a generalization of the intraclass correlation model of Wilks (1946) and arises in the study of interchangeable random variables. We prove a theorem showing how we can transform most such problems to products of problems where…
From a finite population units are drawn with varying probabilities with replacement. There is a certain cost for observing a unit. In this paper samples are obtained partly by drawing a fixed number of times, and partly by drawing and observing units until the cost reaches a specified level. Let $X_k$ be the number of times the $k$th unit has been drawn in either case. Consider for a given function $g(\bullet)$ the random variable $Z = \sum_k g(X_k, k)$. Under general…
A general model is proposed for analysis of frequency tables. This model includes conventional log-linear models for complete and incomplete factorial tables and logit models for quantal response analysis. By use of coordinate-free methods of linear algebra and differential calculus, complete minimal sufficient statistics and likelihood equations for the maximum likelihood estimate are derived. The maximum likelihood estimate is shown to be unique if it exists, and necessary…
We consider the general linear model with independent symmetric errors. In this context we propose and examine the large sample behavior of some estimates of the regression parameters. For the location model these statistics are linear combinations of order statistics. In general they depend on a preliminary estimate and the ordered residuals based on it. The asymptotic efficiency of these procedures is independent of the design matrix. Specifically analogues of the median…
In this paper, we have proved a fundamental property of the characteristic function for the random variable $(\partial/\partial\theta) \log f(x \mid \theta)$. Based on this result, we have proved under regularity conditions different from Bahadur's that certain classes of consistent estimators $\{\theta_n^\ast\}$ are asymptotically efficient in Bahadur's sense $\lim_{\varepsilon \rightarrow 0} \lim_{n \rightarrow \infty} \frac{1}{n\varepsilon^2} \log P\theta\{|\theta_n^\ast - \theta| \geqq \varepsilon\} = -\frac{I(\theta)}{2}.$…
Let $X = (X_1, \cdots, X_n)$ where the $X_i: p \times 1$ are independent random vectors, and let $A: n \times n$ be positive semi-definite symmetric. This paper establishes necessary and sufficient conditions that the random matrix $XAX'$ be positive definite w.p.1. The results are applied to cases where $A$ has a particular form or $X_1, \cdots, X_n$ are i.i.d. In particular, it is shown that in the i.i.d. case, the sample covariance matrix $\sigma(X_i - \bar{X})(X_i - \bar{X})'$…
Among all sequential tests with prescribed error probabilities of the null hypothesis $H_0: \theta = -\theta_1$ versus the simple alternative $H_1: \theta = \theta_1$, where $\theta$ is the unknown mean of a normal population, we want to find the test which minimizes the maximum expected sample size. In this paper, we formulate the problem as an optimal stopping problem and find an optimal stopping rule. The analogous problem in continuous time is also studied, where we want…
Some conjectures made by De Waal [Ann. Math. Statist. 43 (1972) 344-347] on the expectation of elementary symmetric functions of the roots of a noncentral Wishart matrix are proved true. The method of proof is through a simple, though perhaps obscure, property of these elementary symmetric functions and simple properties of the Wishart distribution.
Godambe and Joshi (1965) have shown that for any design the Horvitz-Thompson estimator is admissible in the class of all unbiased estimators of a finite population total. In this note we give a simple alternative proof of this result using induction on the size of the population.
For one sample rank order statistics, a law of iterated logarithm and almost sure convergence to Wiener processes are established here. For the one-sample location problem, a sequential test procedure based on rank order statistics is proposed, and with the aid of the earlier results, it is shown that this has power one and arbitrarily small type I error.
The problems of asymptotically optimum spacing of sample quantiles in connection with asymptotically best linear unbiased estimation of the location parameter of an extreme value distribution of Type I or the scale parameter of an extreme value distribution of Type II or III, are essentially the same. Previous results for the scale parameter of an exponential distribution can therefore be used.
The purpose of this note is to show that asymptotically distribution-free (ADF) rank tests proposed by Koul [Ann. Math. Statist. 41 (1970), 1273-1281] for testing subhypotheses in multiple linear regression when based on symmetric scores are asymptotically totally inefficient and can even be inconsistent for symmetric parent distributions. Also, the condition given in his Theorem 2.4 for a test to be ADF is not necessary.
Sufficient conditions for admissibility of the best invariant confidence interval for a translation or scale parameter are given, for a very wide class of loss functions. These conditions result by adapting a theorem of L. D. Brown [2]. Simpler sufficient conditions are found for a subclass of loss functions of special interest. The subclass of losses involves three components. One concerned with coverage of the true value, another concerned with the distance from the…
A rigorous proof is given of the often stated fact that if the variance of an unbiased estimator of a function of a real parameter attains the Cramer-Rao lower bound then the family of distributions must be a one-parameter exponential family.
Maximum likelihood type robust estimates of regression are defined and their asymptotic properties are investigated both theoretically and empirically. Perhaps the most important new feature is that the number p of parameters is allowed to increase with the number n of observations. The initial terms of a formal power series expansion (essentially in powers of p/n) show an excellent agreement with Monte Carlo results, in most cases down to 4 observations per parameter.
A homogeneous random process on the sphere {X(P):P∈S2} is a process whose mean is zero and whose covariance function depends only on the angular distance θ between the two points, i.e. E[X(P)]≡0 and E[X(P)X(Q)]=R(θ). Given T independent realizations of a Gaussian homogeneous process X(P), we first derive the exact distribution of the spectral estimates introduced by Jones (1963 b). Further, an estimate R(T)(θ)…
Let X1,⋯,Xn be i.i.d. N(i,1),i=0,±1,±2,⋯. Hammersley [2] proposed [Xˉn], the nearest integer to the sample mean, as an estimator of i. It is proved that d is minimax and admissible relative to zero-one loss. However, it is shown that relative to squared error loss, the estimator is neither admissible nor minimax.
Some new test criteria are proposed for testing various hypotheses concerning covariance matrices. Asymptotic expansions of their null distributions are derived in terms of the chi2-distribution.
From a finite population units are drawn with varying probabilities with replacement. There is a certain cost for observing a unit. In this paper samples are obtained partly by drawing a fixed number of times, and partly by drawing and observing units until the cost reaches a specified level. Let Xk be the number of times the kth unit has been drawn in either case. Consider for a given function g(∙) the random variable Z=∑kg(Xk,k). Under general…
In this paper, we have proved a fundamental property of the characteristic function for the random variable (∂/∂θ)logf(x∣θ). Based on this result, we have proved under regularity conditions different from Bahadur's that certain classes of consistent estimators {θn∗} are asymptotically efficient in Bahadur's sense limε→0limn→∞nε21logPθ{∣θn∗−θ∣≧ε}=−2I(θ).…
Let X=(X1,⋯,Xn) where the Xi:p×1 are independent random vectors, and let A:n×n be positive semi-definite symmetric. This paper establishes necessary and sufficient conditions that the random matrix XAX′ be positive definite w.p.1. The results are applied to cases where A has a particular form or X1,⋯,Xn are i.i.d. In particular, it is shown that in the i.i.d. case, the sample covariance matrix σ(Xi−Xˉ)(Xi−Xˉ)′…
Among all sequential tests with prescribed error probabilities of the null hypothesis H0:θ=−θ1 versus the simple alternative H1:θ=θ1, where θ is the unknown mean of a normal population, we want to find the test which minimizes the maximum expected sample size. In this paper, we formulate the problem as an optimal stopping problem and find an optimal stopping rule. The analogous problem in continuous time is also studied, where we want…