A generalization of a Bernoulli process which incorporates a dependence structure was given by Klotz (1972, 1973), in which he considered $X_1, X_2, \cdots, X_n$ as a stationary two-state Markov chain with state space $\{0, 1\}$. The parameters of the process are $p = P(X_i = 1)$ and $\lambda$, which measures the degree of persistence in the chain. Klotz was unable to solve the equations arising from the full likelihood for the M.L.E.'s of $p$ and $\lambda$, so proposed and…
Let $\theta$ be the natural parameter of a continuous one-parameter exponential family. An empirical Bayes test is constructed for testing $\theta \leqq 0$ against $\theta > 0$ with a piecewise linear loss function. Since the problem is monotone, Bayes tests for a given prior distribution can be characterized by a single parameter, e.g., the size of the test under $\theta = 0$. Therefore the construction of an empirical Bayes test can be reduced to the construction of an…
It is shown that under certain regularity conditions the Bernstein-v. Mises approximation of the posterior distributions is of order $O(n^{-\\frac{1}{2}})$ with high probability.
Lower and upper bounds on the power of a Smirnov test for symmetry $H_0: \bar{F} = F$ versus $H_1: \bar{F} \geqq F, \sup_x\lbrack\bar{F}(x) - F(x)\rbrack = \Delta > 0$ are obtained exactly or estimated for selected values of sample size $N$, level $\alpha$, and asymmetry $\Delta$. Furthermore the asymptotic power of the test as $N^{\frac{1}{2}}\Delta_N \rightarrow c$ is shown to be bounded by $\Phi(c - k_\alpha)$ and 1 if $c \geqq k_\alpha$ and by $\alpha$ and $2\Phi(c - k_\alpha)$…
By means of a general weak convergence theorem some invariance principles are proven for the multivariate sequential empirical process and for the multivariate rank order process w.r.t. stronger metrics than the generalized Skorohod metric. The underlying random variables are neither assumed to be independent nor to be stationary. These results are then applied to derive convergence of the weighted empirical cumulatives and for the weighted rank order process. Finally by a…
Some concepts of relative and absolute efficiency for sequential tests are considered. These are sequential analogs of Hodges-Lehmann and Chernoff efficiencies. Using these criteria, several sequential tests for the mean of a normal distribution are evaluated. Among them are Wald's SPRT, Anderson's triangular boundary, Bayes and APO tests and a repeated significance test. Truncated versions of these tests are also considered. Asymptotic expressions for the (expected)…
Anthony Olsen · Justus Seely · David Birkes·刊期 1976-09-01 · 卷 4（5）·The Annals of Statistics已编入正刊
两个方差分量的不变二次无偏估计AI 解读
For a normally distributed mixed model with two unknown variance components $\theta_1$ and $\theta_2$, a tractable characterization is given for the admissible estimators within the class $\tilde{\mathscr{N}}_\delta$ of invariant quadratic unbiased estimators for $\delta_1\theta_1 + \delta_2\theta_2$. Here the term admissible is used with reference only to the class $\tilde{\mathscr{N}}_\delta$. This characterization is based on a result for general linear models which…
A measurable cross-section for orbits of a sample space under a free (exact) transformation group is shown to exist under topological regularity conditions. This is used to represent the sample space as essentially the product of a maximal invariant and an equivariant part, which implies Stein's representation for the density of the maximal invariant.
A. K. Chattopadhyay · K. C. S. Pillai · Hung C. Li·刊期 1976-07-01 · 卷 4（4）·The Annals of Statistics已编入正刊
矩阵函数积分的极大化及两个矩阵特征根分布的渐近展开AI 解读
The noncentral distribution of latent roots arising in several situations in multivariate analysis involves the integration of a hypergeometric function of matrix variates over a group of orthogonal matrices in the real case and that of unitary matrices in the complex case. In this paper the subgroup of the orthogonal group (unitary group) for which the integrand is maximized has been found under mild restrictions. The results of earlier authors (Anderson, Chang, James, Li…
In this note we discuss multivariate linear models from the coordinate-free point of view, as earlier done by Eaton (1970). We generalize the result of Eaton by allowing for missing observations. This leads to models of the kind $EY \in L, Cov Y \in\{P(I \otimes \sum)P'\}$ where $P$ is a diagonal mapping. The paper starts by deriving the conditions for existence of Gauss-Markov estimators (GME) of $EY$ in models where the covariance-mappings are not necessarily nonsingular.…
This paper gives rates of convergence for the $L_1$ distances between the distributions of standardized linear rank statistics and the standard normal random variable. These rates are $O(N^{-\frac{1}{2}})$ under various conditions on the score function and the distributions of the underlying observations.
Sanov's theorem on the asymptotic behavior of probabilities of large deviations for an empirical probability distribution is established under different conditions than previously given by Sanov, Hoadley and Stone. The new conditions are based on likelihood ratio approximations rather than on multinomial approximations. It is shown that these conditions are strictly more general than those of Stone.
Conditions are given for the attainment of the Hammersley-Chapman-Robbins bound for the variance of an unbiased estimator, in both regular and nonregular cases. Comparisons are made between this bound and the Bhattacharyya system of bounds for a wide class of distributions and parametric functions. Sufficient conditions are provided to determine when one bound is sharper than the other one.
The limiting distributions are obtained for two estimators of variance components: C. R. Rao's MINQUE, and an estimator produced by an iterative procedure, referred to as I-MINQUE. Limits are taken as the number of independent and identically distributed vector observations on the model assumed gets large. This approach provides the asymptotics of interest when an experiment with a large number of observations can be thought of as independent replications of a smaller…
This article discusses the sampling of stationary discrete-time stochastic processes at fixed but unequally spaced time points. The pattern of the sampling times is periodic with a cycle of $p$ time units. One of the major problems is to determine given $p$ the minimum number of sampling points required per cycle in order to estimate the covariances at all lags. The second problem is to find a pattern of distribution for the sampling points within the cycle which will allow…
L. D. Brown · Arthur Cohen · W. E. Strawderman·刊期 1976-07-01 · 卷 4（4）·The Annals of Statistics已编入正刊
严格单调似然比下的完全类定理及其应用AI 解读
Suppose a random variable has a density belonging to a one parameter family which has strict monotone likelihood ratio. For inference regarding the parameter (or a monotone function of the parameter) consider the loss function to be bowl shaped for each fixed parameter and also to have each action be a "point of increase" or a "point of decrease" for some value of the parameter. Under these conditions, given any nonmonotone decision procedure, a unique monotone procedure is…
A theory of sequential Bayes interval estimation procedures for a single parameter is developed for the case where the loss for using an interval $I$ is a linear combination of the length of $I$, the indicator of noncoverage of $I$, and the number of observations taken. A class of stopping rules $\{t(c): c > 0\}$ is shown to be asymptotically pointwise optimal (A.P.O.) and asymptotically optimal (A.O.) for the confidence interval problem as the cost $c$ per observation tends…
In the usual linear regression model the sample regression coefficients converge with probability one to the population regression coefficients when the dependent variables are normally distributed and the inverse of the second-order moment matrix of the independent variables converges to the zero matrix.
Let $x_1, x_2,\cdots$ be independent random variables which under $P_\theta$ have probability density function of the form $P_\theta\{x_k \in dx\} = \exp(\theta x - \Psi(\theta)) dH(x)$, where $\Psi$ is normalized so that $\Psi(0) = \Psi'(0) = 0.$ Let $a \leqq 0 < b, s_n = \sum^n_1 x_k$, and $T = \inf \{n: s_n \not\in (a, b)\}.$ For $u < 0$, an unbiased Monte Carlo estimate of $P_u(s_T \geqq b)$ is the average of independent $P_\theta$-realizations of $I_{\{s_T \geqq b\}} \exp\{(u - \theta)s_T - T(\Psi(u) - \Psi(\theta))\}$…
If $m$ objects $x_1, x_2, \cdots, x_m$ are compared pairwise, then let $s_{ij}$ denote the number of times $x_i$ beats $x_j$ in $n_{ij}$ independent comparisons. In a ranking, if $x_i$ precedes $x_j$ then one may require that the probability of $x_i$ beating $x_j$ be at least $\frac{1}{2}$. Such a ranking is called weak stochastic ranking. Let $I(R)$ be the set of all pairs $(i, j)$ such that $x_i$ precedes $x_j$ in the ranking $R$ in spite of the paired comparison outcomes…
Let $X = (X_1, \cdots, X_p)^t$ be a $p$-variate normal random vector with unknown mean $\theta = (\theta_1, \cdots, \theta_p)^t$ and unknown positive definite diagonal covariance matrix $A$. Assume that estimates $V_i$ of the variances $A_i$ are available, and that $V_i/A_i$ is $\chi^2_{n_i}$. Assume also that all $X_i$ and $V_i$ are independent. It is desired to estimate $\theta$ under the quadratic loss $\lbrack\sum^p_{i=1} q_i(\delta_i - \theta_i)^2\rbrack/\lbrack\sum^p_{i=1} q_i A_i\rbrack,\quad\text{where} q_i > 0, i = 1, \cdots, p.$…
A generalization of a Bernoulli process which incorporates a dependence structure was given by Klotz (1972, 1973), in which he considered X1,X2,⋯,Xn as a stationary two-state Markov chain with state space {0,1}. The parameters of the process are p=P(Xi=1) and λ, which measures the degree of persistence in the chain. Klotz was unable to solve the equations arising from the full likelihood for the M.L.E.'s of p and λ, so proposed and…
Let θ be the natural parameter of a continuous one-parameter exponential family. An empirical Bayes test is constructed for testing θ≦0 against θ>0 with a piecewise linear loss function. Since the problem is monotone, Bayes tests for a given prior distribution can be characterized by a single parameter, e.g., the size of the test under θ=0. Therefore the construction of an empirical Bayes test can be reduced to the construction of an…
It is shown that under certain regularity conditions the Bernstein-v. Mises approximation of the posterior distributions is of order O(n−frac12) with high probability.
Lower and upper bounds on the power of a Smirnov test for symmetry H0:Fˉ=F versus H1:Fˉ≧F,supx[Fˉ(x)−F(x)]=Δ>0 are obtained exactly or estimated for selected values of sample size N, level α, and asymmetry Δ. Furthermore the asymptotic power of the test as N21ΔN→c is shown to be bounded by Φ(c−kα) and 1 if c≧kα and by α and 2Φ(c−kα)…
For a normally distributed mixed model with two unknown variance components θ1 and θ2, a tractable characterization is given for the admissible estimators within the class N~δ of invariant quadratic unbiased estimators for δ1θ1+δ2θ2. Here the term admissible is used with reference only to the class N~δ. This characterization is based on a result for general linear models which…
In this note we discuss multivariate linear models from the coordinate-free point of view, as earlier done by Eaton (1970). We generalize the result of Eaton by allowing for missing observations. This leads to models of the kind EY∈L,CovY∈{P(I⊗∑)P′} where P is a diagonal mapping. The paper starts by deriving the conditions for existence of Gauss-Markov estimators (GME) of EY in models where the covariance-mappings are not necessarily nonsingular.…
L1 Rates of Convergence for Linear Rank Statistics
This paper gives rates of convergence for the L1 distances between the distributions of standardized linear rank statistics and the standard normal random variable. These rates are O(N−21) under various conditions on the score function and the distributions of the underlying observations.
This article discusses the sampling of stationary discrete-time stochastic processes at fixed but unequally spaced time points. The pattern of the sampling times is periodic with a cycle of p time units. One of the major problems is to determine given p the minimum number of sampling points required per cycle in order to estimate the covariances at all lags. The second problem is to find a pattern of distribution for the sampling points within the cycle which will allow…
A theory of sequential Bayes interval estimation procedures for a single parameter is developed for the case where the loss for using an interval I is a linear combination of the length of I, the indicator of noncoverage of I, and the number of observations taken. A class of stopping rules {t(c):c>0} is shown to be asymptotically pointwise optimal (A.P.O.) and asymptotically optimal (A.O.) for the confidence interval problem as the cost c per observation tends…
Let x1,x2,⋯ be independent random variables which under Pθ have probability density function of the form Pθ{xk∈dx}=exp(θx−Ψ(θ))dH(x), where Ψ is normalized so that Ψ(0)=Ψ′(0)=0. Let a≦0<b,sn=∑1nxk, and T=inf{n:sn∈(a,b)}. For u<0, an unbiased Monte Carlo estimate of Pu(sT≧b) is the average of independent Pθ-realizations of I{sT≧b}exp{(u−θ)sT−T(Ψ(u)−Ψ(θ))}…
If m objects x1,x2,⋯,xm are compared pairwise, then let sij denote the number of times xi beats xj in nij independent comparisons. In a ranking, if xi precedes xj then one may require that the probability of xi beating xj be at least 21. Such a ranking is called weak stochastic ranking. Let I(R) be the set of all pairs (i,j) such that xi precedes xj in the ranking R in spite of the paired comparison outcomes…
Let X=(X1,⋯,Xp)t be a p-variate normal random vector with unknown mean θ=(θ1,⋯,θp)t and unknown positive definite diagonal covariance matrix A. Assume that estimates Vi of the variances Ai are available, and that Vi/Ai is χni2. Assume also that all Xi and Vi are independent. It is desired to estimate θ under the quadratic loss [∑i=1pqi(δi−θi)2]/[∑i=1pqiAi],whereqi>0,i=1,⋯,p.…