A multivariate extension is made of Stein's result (1964) on the estimation of the normal variance. Here the generalized variance $|\Sigma|$ is being estimated from a Wishart random matrix $S: p \times p \sim W(n, \Sigma)$ and an independent normal random matrix $X: p \times k \sim N(\xi, \Sigma \otimes 1_k)$ with $\xi$ unknown. The main result is that the minimax, best affine equivariant estimator $((n + 2 - p)!/(n + 2)!)|S|$ is dominated by $\min\{((n + 2 - p)!/(n + 2)!)|S|, ((n + k + 2 - p)!/(n + k + 2)!)|S + XX'|\}$…
In recent work by O'Reilly, a necessary and sufficient condition for the existence of an unbiased estimate of the distribution function of a "future" observation was given. The result was obtained under the condition that the model had full rank. Here, the result is generalized to any number of future observations and the full rank condition is relaxed. The corresponding uniformly minimum variance unbiased estimator is identified from the density estimates given by Ghurye…
The sample paths of cumulative sums of induced order statistics obtained from $n$ independent two-dimensional random vectors, when appropriately normalized, converge weakly (as $n$ increases indefinitely) to the sum of a Brownian motion with time change and an integrated Brownian bridge which is independent of the Brownian motion. Applications in regression analysis are given.
In the present paper, an integer-valued version $(T_N)$ of the Kendall rank correlation coefficient is considered. Under the hypothesis of independence, a local limit theorem with the Edgeworth expansion for $T_N$ is proved and an asymptotic expansion of the distribution function of $T_N$ is derived.
This paper provides a proof of the asymptotic sufficiency of the ranks in the two sample situation when the observations are censored at the $r$th order statistic. Several important conclusions may then be drawn concerning optimal tests.
A sequence of empirical Bayes estimators is defined for estimating a distribution function. The sequence is shown to be asymptotically optimal relative to a Ferguson Dirichlet process prior. Exact risk expressions are derived and the rate, at which the overall expected loss approaches the minimum Bayes risk, is exhibited. The empirical Bayes approach, based on the Dirichlet process, is also applied to the problem of estimating the mean of a distribution.
The purpose of this paper is to obtain weaker sufficient conditions than those given by Robbins [6] for the asymptotic optimality of empirical Bayes procedures. In so doing a general method for providing asymptotic optimality is given which because of its simplicity should prove quite useful not only for the empirical Bayes problem but for other related asymptotic problems as well. Three areas of application are indicated herein.
A sequential version of Chernoff-Savage linear rank statistics is introduced as a basis for inference. The principal result is an invariance principle for two-sample rank statistics, i.e., under a fixed alternative the sequence of sequential linear rank statistics converges weakly to a Wiener process. The domain of application of the theorem is quite broad and includes score functions which tend to infinity at the end points much more rapidly than that of the normal scores…
This paper deals with distribution-free tests for independence under the constraint that the population has a bivariate discrete distribution. The locally most powerful conditional test, given the marginal empirical distributions, is derived. The unconditional asymptotic distribution of the conditional test statistic standardized by the conditional mean and variance is also given under the hypothesis of independence and under contiguous alternatives. Furthermore, some…
Suppose that for each number $t$ in [0, 1] there is a distribution with distribution function $F_t(\bullet)$ which has $p$th percentile $\xi(t)$. Consider the problem of estimating $\xi(\bullet)$ under the assumption that $\xi(\bullet)$ is monotone. An estimator which is analogous to the median regression estimator considered in Cryer, Robertson, Wright and Casady (1972), is studied. Asymptotic properties including consistency and law of the iterated logarithm results are…
F. Y. Edgeworth's 1908-9 investigation is examined for its contribution to knowledge of the sampling properties of maximum likelihood and related estimates, especially asymptotic efficiency. The nature and extent of his progress and anticipation of R. A. Fisher are described. Fisher's relevant work is briefly examined in relation to Edgeworth's and to the Cramer-Rao inequality.
Empirical Bayes estimators, asymptotically optimal with rates, are proposed. In the component problem there is a pair $(X, \omega)$ of real valued random variables. The Lebesgue density of $X$, conditional on $\omega$, is of the form $u(x)C(\omega)e^{\omega x}$. Based on a realization of $X$, the problem is squared error loss estimation of $\omega$. Let $G$ be a prior distribution on $\omega$, and $R(G)$ be the Bayes optimal risk wrt $G$. Using $(X_1, \cdots, X_n)$, the…
Uniformly most accurate level $1 - \alpha$ confidence procedures for a linear function $\mu + \lambda\sigma^2$ with known $\lambda$ for the parameters of a normal distribution defined by Land were previously shown for both the one-sided and two-sided procedures to be always intervals for $\nu \geqq 2, \nu$ being the number of degrees of freedom for estimating $\sigma^2$. These results are shown in this paper to hold also in the case $\nu = 1$. During the course of the…
The Pearson-Fisher $\chi^2$ statistic is asymptotically chi-square under the null hypothesis with $M - m - 1$ degrees of freedom where $M =$ number of cells and $m =$ dimension of parameter. The Chernoff-Lehmann statistic is a weighted sum of chi-squares and the Kambhampati statistic is $\chi^2$ with $M - 1$ degrees of freedom. The approximate Bahadur slopes of the tests based on these statistics are computed. It is shown that the Kambhampati test always dominates the…
For $N = 1, 2, \cdots$ let $X_{1N}, X_{2N}, \cdots, X_{NN}$ be independent rv's having continuous df's $F_{1N}, F_{2N}, \cdots, F_{NN}$. For the set $X_{1N}, X_{2N}, \cdots, X_{NN}$, let us denote by $X_{1:N} \leqq X_{2:N} \leqq \cdots \leqq X_{N:N}$ the order statistics, by $\mathbb{F}_N$ the empirical df and by $\bar{F}_N$ the averaged df, i.e. $\bar{F}_N(x) = N^{-1}\sum^N_{n=1} F_{nN}(x)$ for $x\epsilon(-\infty, \infty)$. It is shown that for each $\varepsilon > 0$ there…
Conditions to ensure the asymptotic normality of rank statistics having a scores generating function with finitely many jumps are obtained. These conditions are derived by studying rank statistics with a scores generating function $J$ such that $J(u) = 1$ or $0$ as $u \geqq s$ or $u < s$ for a fixed $s, 0 < s < 1$. No differentiability conditions are imposed on the underlying distribution functions at the jump points of the scores generating function.
A central limit theorem is proved for the sample serial covariances of an ergodic, stationary, purely nondeterministic process whose linear innovations have their first four moments as for a sequence of independent random variables. The necessary and sufficient condition for the theorem is then that the spectra be square integrable.
The general mixed linear model can be written $y = X\alpha + Zb$, where $\alpha$ is a vector of fixed effects and $b$ is a vector of random variables. Assume that $E(b) = 0$ and that $\operatorname{Var} (b) = \sigma^2D$ with $D$ known. Consider the estimation of $\lambda_1'\alpha + \lambda_2'\beta$, where $\lambda_1'\alpha$ is estimable and $\beta$ is the realized, though unobservable, value of $b$. Among linear estimators $c + r'y$ having $E(c + r'y) \equiv E(\lambda_1'\alpha + \lambda_2'b)$…
The approximate Bahadur slope of the Chernoff-Lehmann $\chi^2$-test-of-fit to a scale-location family on $R^k$ is computed. The goal is to select cells (whose number is independent of sample size) to maximize this slope. The supremum is found and is shown to be a maximum only in trivial cases. If the $\sup$ is finite there is always a best selection for a fixed number of cells. Equally likely cells are shown to be admissible when the alternative is large.
Percentage points are given for the asymptotic distributions of the goodness-of-fit statistics $W^2, U^2$ and $A^2$, for the cases where the distribution tested is (a) normal, with mean or variance, or both, unknown; (b) exponential, with scale parameter unknown. Some exact means and variances are also given. The distributions can be expressed as a sum of weighted chi-square variables; the weights are calculated, and the higher cumulants can then be found. The first four…
Let $X$ be an $n$-dimensional random vector with density $f(x - \theta)$. It is desired to estimate $\theta_1$, under a strictly convex loss $L(\delta - \theta_1)$. If $F$ is a generalized Bayes prior density, the admissibility of the corresponding generalized Bayes estimator, $\delta_F$, is considered. An asymptotic approximation to $\delta_F$ is found. Using this approximation, it is shown that if (i) $f$ has enough moments, (ii) $L$ and $F$ are smooth enough, and (iii) $F(\theta) \leqq K(|\theta_1| + \sum^n_{i=2} \theta_i^2)^{(3-n)/2}$…
Let $X = (X_1, \cdots, X_n)$ be an $n$-dimensional random vector with density $f(x - \theta)$. Assume the loss incurred in estimating $\theta$ by $d$ is $L(d - \theta)$ and is convex. Let $F$ be a generalized prior density. The question of inadmissibility of generalized Bayes estimators of $\theta$ is considered. As in Brown (1974c), the crucial role played by the "moment structure" in determining inadmissibility results is indicated (moments defined as the quantities $m_{i,j(1), \cdots, j(l)} = \int \prod^l_{k=1} x_{j(k)} \cdot \lbrack(\partial/\partial x_i)L(x)\rbrack f(x) dx)$…
Associated with any Borel gambling model $G$ or dynamic programming model $D$ is a corresponding class of stochastic processes $M(G)$ or $M(D)$. Say that $G(D)$ is regular if there is a $D(G)$ with $M(D) = M(G)$. Necessary and sufficient conditions for regularity are given, and it is shown how to modify any model slightly to achieve regularity.
Approximation of the covariance matrix $\mathbf{\Sigma}$ of $T$ consecutive observations from a second-order stationary process with continuous positive spectral density $f(\lambda) = \lbrack\sigma^2/(2\pi)^2\rbrack|\sum^\infty_{j=0}\delta_je^{i\lambda j}|^2$ is considered. If $\mathbf{\Sigma}^\ast$ is the covariance matrix corresponding to a process with spectral density $1/\lbrack(2\pi)^2f(\lambda)\rbrack$, then $\mathbf{\Sigma}^\ast - \mathbf{\Sigma}^{-1} \geqq 0$. A…
A multivariate extension is made of Stein's result (1964) on the estimation of the normal variance. Here the generalized variance ∣Σ∣ is being estimated from a Wishart random matrix S:p×p∼W(n,Σ) and an independent normal random matrix X:p×k∼N(ξ,Σ⊗1k) with ξ unknown. The main result is that the minimax, best affine equivariant estimator ((n+2−p)!/(n+2)!)∣S∣ is dominated by min{((n+2−p)!/(n+2)!)∣S∣,((n+k+2−p)!/(n+k+2)!)∣S+XX′∣}…
The sample paths of cumulative sums of induced order statistics obtained from n independent two-dimensional random vectors, when appropriately normalized, converge weakly (as n increases indefinitely) to the sum of a Brownian motion with time change and an integrated Brownian bridge which is independent of the Brownian motion. Applications in regression analysis are given.
In the present paper, an integer-valued version (TN) of the Kendall rank correlation coefficient is considered. Under the hypothesis of independence, a local limit theorem with the Edgeworth expansion for TN is proved and an asymptotic expansion of the distribution function of TN is derived.
This paper provides a proof of the asymptotic sufficiency of the ranks in the two sample situation when the observations are censored at the rth order statistic. Several important conclusions may then be drawn concerning optimal tests.
Suppose that for each number t in [0, 1] there is a distribution with distribution function Ft(∙) which has pth percentile ξ(t). Consider the problem of estimating ξ(∙) under the assumption that ξ(∙) is monotone. An estimator which is analogous to the median regression estimator considered in Cryer, Robertson, Wright and Casady (1972), is studied. Asymptotic properties including consistency and law of the iterated logarithm results are…
Empirical Bayes estimators, asymptotically optimal with rates, are proposed. In the component problem there is a pair (X,ω) of real valued random variables. The Lebesgue density of X, conditional on ω, is of the form u(x)C(ω)eωx. Based on a realization of X, the problem is squared error loss estimation of ω. Let G be a prior distribution on ω, and R(G) be the Bayes optimal risk wrt G. Using (X1,⋯,Xn), the…
Uniformly most accurate level 1−α confidence procedures for a linear function μ+λσ2 with known λ for the parameters of a normal distribution defined by Land were previously shown for both the one-sided and two-sided procedures to be always intervals for ν≧2,ν being the number of degrees of freedom for estimating σ2. These results are shown in this paper to hold also in the case ν=1. During the course of the…
The Pearson-Fisher χ2 statistic is asymptotically chi-square under the null hypothesis with M−m−1 degrees of freedom where M= number of cells and m= dimension of parameter. The Chernoff-Lehmann statistic is a weighted sum of chi-squares and the Kambhampati statistic is χ2 with M−1 degrees of freedom. The approximate Bahadur slopes of the tests based on these statistics are computed. It is shown that the Kambhampati test always dominates the…
For N=1,2,⋯ let X1N,X2N,⋯,XNN be independent rv's having continuous df's F1N,F2N,⋯,FNN. For the set X1N,X2N,⋯,XNN, let us denote by X1:N≦X2:N≦⋯≦XN:N the order statistics, by FN the empirical df and by FˉN the averaged df, i.e. FˉN(x)=N−1∑n=1NFnN(x) for xϵ(−∞,∞). It is shown that for each ε>0 there…
Conditions to ensure the asymptotic normality of rank statistics having a scores generating function with finitely many jumps are obtained. These conditions are derived by studying rank statistics with a scores generating function J such that J(u)=1 or 0 as u≧s or u<s for a fixed s,0<s<1. No differentiability conditions are imposed on the underlying distribution functions at the jump points of the scores generating function.
The general mixed linear model can be written y=Xα+Zb, where α is a vector of fixed effects and b is a vector of random variables. Assume that E(b)=0 and that Var(b)=σ2D with D known. Consider the estimation of λ1′α+λ2′β, where λ1′α is estimable and β is the realized, though unobservable, value of b. Among linear estimators c+r′y having E(c+r′y)≡E(λ1′α+λ2′b)…
The approximate Bahadur slope of the Chernoff-Lehmann χ2-test-of-fit to a scale-location family on Rk is computed. The goal is to select cells (whose number is independent of sample size) to maximize this slope. The supremum is found and is shown to be a maximum only in trivial cases. If the sup is finite there is always a best selection for a fixed number of cells. Equally likely cells are shown to be admissible when the alternative is large.
Percentage points are given for the asymptotic distributions of the goodness-of-fit statistics W2,U2 and A2, for the cases where the distribution tested is (a) normal, with mean or variance, or both, unknown; (b) exponential, with scale parameter unknown. Some exact means and variances are also given. The distributions can be expressed as a sum of weighted chi-square variables; the weights are calculated, and the higher cumulants can then be found. The first four…
Let X be an n-dimensional random vector with density f(x−θ). It is desired to estimate θ1, under a strictly convex loss L(δ−θ1). If F is a generalized Bayes prior density, the admissibility of the corresponding generalized Bayes estimator, δF, is considered. An asymptotic approximation to δF is found. Using this approximation, it is shown that if (i) f has enough moments, (ii) L and F are smooth enough, and (iii) F(θ)≦K(∣θ1∣+∑i=2nθi2)(3−n)/2…
Let X=(X1,⋯,Xn) be an n-dimensional random vector with density f(x−θ). Assume the loss incurred in estimating θ by d is L(d−θ) and is convex. Let F be a generalized prior density. The question of inadmissibility of generalized Bayes estimators of θ is considered. As in Brown (1974c), the crucial role played by the "moment structure" in determining inadmissibility results is indicated (moments defined as the quantities mi,j(1),⋯,j(l)=∫∏k=1lxj(k)⋅[(∂/∂xi)L(x)]f(x)dx)…
Associated with any Borel gambling model G or dynamic programming model D is a corresponding class of stochastic processes M(G) or M(D). Say that G(D) is regular if there is a D(G) with M(D)=M(G). Necessary and sufficient conditions for regularity are given, and it is shown how to modify any model slightly to achieve regularity.
Approximation of the covariance matrix Σ of T consecutive observations from a second-order stationary process with continuous positive spectral density f(λ)=[σ2/(2π)2]∣∑j=0∞δjeiλj∣2 is considered. If Σ∗ is the covariance matrix corresponding to a process with spectral density 1/[(2π)2f(λ)], then Σ∗−Σ−1≧0. A…