Let $\mathbf{B} = (N_1, \cdots, N_k)$ be a multivariate counting process and let $\mathscr{F}_t$ be the collection of all events observed on the time interval $\lbrack 0, t\rbrack.$ The intensity process is given by $\Lambda_i(t) = \lim_{h \downarrow 0} \frac{1}{h}E(N_i(t + h) - N_i(t) \mid \mathscr{F}_t)\quad i = 1, \cdots, k.$ We give an application of the recently developed martingale-based approach to the study of $\mathbf{N}$ via $\mathbf{\Lambda}.$ A statistical model…
Convergence with probability one of a recursive stochastic approximation algorithm is considered. Some extensions of previous results for the Robbins-Monro and the Kiefer-Wolfowitz procedures are given. An inportant feature of the approach taken here is that the convergence analysis can be directly extended to more complex algorithms.
An adaptive approach is considered as an alternative to the classical indifference-zone formulation of the problems of ranking and selection. With a fixed $\gamma^\ast$, the proposed procedure calls for the termination of sampling when the estimated probability of correct selection exceeds $\gamma^\ast$ for the first time. Asymptotic properties of this procedure are proved as $\gamma^\ast \rightarrow 1$, and Monte Carlo results show that the procedure is well behaved even…
A set of recurrences is developed for computing the probability content of any prior or posterior confidence bands for a distribution function assuming the parameter of the prior Dirichlet process known. When the prior parameter is unknown and is estimated from the sample, nonparametric estimates of the probability content of the posterior bands are obtained.
It is known that the autocorrelation function of a stationary discrete-time scalar process can be uniquely characterized by the so-called partial autocorrelation function, which is a sequence of numbers less or equal to one in magnitude. We show here that the matrix covariance function of a multivariate stationary process can be characterized by a sequence of matrix partial correlations, having singular values less than or equal to one in magnitude. This characterization can…
An estimation procedure for stochastic processes based on the minimization of a sum of squared deviations about conditional expectations is developed. Strong consistency, asymptotic joint normality and an iterated logarithm rate of convergence are shown to hold for the estimators under a variety of conditions. Special attention is given to the widely studied cases of stationary ergodic processes and Markov processes with are asymptotically stationary and ergodic. The…
This paper considers the estimation of a location parameter $\theta$ in a one-sample problem. The asymptotic performance of a sequence of estimates $\{T_n\}$ is measured by the exponential rate of convergence to 0 of $\max \{P_\theta(T_n < \theta - a), P_\theta(T_n > \theta + a)\}, \text{say} e(a).$ This measure of asymptotic performance is equivalent to one considered by Bahadur (1967). The optimal value of $e(a)$ is given for translation invariant estimates. Some…
In the simple linear regression problem $\{Y_i = \alpha + \beta x_i + e_i i = 1,\cdots, n, e_i$ i.i.d. $\sim F$ continuous, $x_1 \leqq \cdots \leqq x_n$ known, $\alpha, \beta$ unknown$\}$ we investigate the following type of estimator: To each $s_{ij} = (Y_j - Y_i)/(x_j - x_i)$ with $x_i < x_j$ attach weight $w_{ij}$ and as estimator for $\beta$ consider the median of this weight distribution over the $s_{ij}$. A confidence interval for $\beta$ is found by taking certain…
Asymptotic normality is established for multivariate linear rank statistics of general type in the non-i.i.d. case covering null hypotheses as well as almost arbitrary alternatives. The functions generating the regression constants and the scores are allowed to have a finite number of discontinuities of the first kind, and to tend to infinity near 0 and 1. The proof is based on properties of empirical df's in the non-i.i.d. case and is patterned on the 1958 Chernoff-Savage…
Interval estimation of variance components is studied for the unbalanced one-way random effects model. An easily calculated function, $W$, of the harmonic mean of the class sizes and of the sample variance of the class means is found to be important. The exact distribution of $W$ is found and is shown to be excellently approximated by a chi-square distribution. The random variable $W$ is used to construct interval estimates for (i) the between classes variance component and…
The notion of Bahadur efficiency for test statistics is extended to the sequential case and illustrated in the specific context of testing one-sided hypotheses about a normal mean. An analog of Bahadur's theorem on the asymptotic optimality of the likelihood ratio statistic is seen to hold in the normal case. Some possible definitions of attained level for a sequential experiment are considered.
Nonparametric estimators are proposed for transition probabilities in partial Markov chains relative to multiple decrement models. The estimators are generalizations of the product limit estimator. We study the bias of the estimators, prove a strong consistency result and derive asymptotic normality of the estimators considered as stochastic processes. We also compute their efficiency relative to the maximum likelihood estimators in the case of constant forces of transition.
We study a class of decision rules based on an adaptive partitioning of an Euclidean observation space. The class of partitions has a computationally attractive form, and the related decision rule is invariant under strictly monotone transformations of coordinate axes. We provide sufficient conditions that a sequence of decision rules be asymptotically Bayes risk efficient as sample size increases. The sufficient conditions involve no regularity assumptions on the underlying…
In the discrimination problem the random variable $\theta$, known to take values in $\{1,\cdots, M\}$, is estimated from the random vector $X$. All that is known about the joint distribution of $(X, \theta)$ is that which can be inferred from a sample $(X_1, \theta_1),\cdots, (X_n, \theta_n)$ of size $n$ drawn from that distribution. A discrimination rule is any procedure which determines a decision $\hat{\theta}$ for $\theta$ from $X$ and $(X_1, \theta_1),\cdots, (X_n, \theta_n)$…
This paper considers likelihood ratio tests for testing hypotheses that a collection of parameters satisfy some order restriction. The first problem considered is to test a hypothesis specifying an order restriction on a collection of means of normal distributions. Equality of the means is the subhypothesis of the null hypothesis which yields the largest type I error probability (i.e., is least favorable). Furthermore, the distribution of $T = -\ln$ (likelihood ratio) is…
The main result of this paper is that $g$-inverses are not needed for computing optimum designs when the singularity of the information matrix is unavoidable. They are, of course, needed for the analysis. It will be shown that it is possible to augment the experimental region so that procedures for computing optimum designs for $s$ out $k$ parameters $(s < k)$ which are developed for the nonsingular case may also be used for the singular case.
Limit processes are obtained for the sequences of partial sums of polynomial regression residuals. Properties of linear and quadratic functionals on the sequences are discussed. Distribution theory for Cramer-von Mises type functionals is obtained. An indication is given of the relevance of these results to the problem of testing for change of regression at unknown times.
Ann R. Cohen Brandwein · William E. Strawderman·刊期 1978-03-01 · 卷 6（2）·The Annals of Statistics已编入正刊
球对称单峰分布位置参数在二次损失下的极小极大估计AI 解读
Families of minimax estimators are found for the location parameter of a $p$-variate $(p \geqq 3)$ spherically symmetric unimodal distribution with respect to general quadratic loss. The estimators of James and Stein, Baranchik, Bock and Strawderman are all considered for this general problem. Specifically, when the loss is general quadratic loss given by $L(\delta, \theta) = (\delta - \theta)'D(\delta - \theta)$ where $D$ is a known $p \times p$ positive definite matrix,…
The asymptotic form of the Bayes acceptance region is derived for testing simple null-hypotheses in multiparameter exponential families. This result suggests a reasonable definition for tests which might be called "almost-Bayes". The rate at which the risk of the Bayes test converges to zero is obtained, showing the nature of its dependence on the prior distribution and providing a basis for comparison of almost-Bayes procedures. Concluding remarks contain a brief discussion…
A decision maker is seen to be coherent in the sense of de Finetti if, and only if, his probabilities are computed in accordance with some finitely additive prior. If a bounded loss function is specified, then a decision rule is extended admissible (i.e., not uniformly dominated) if and only if it is Bayes for some finitely additive prior. However, if an improper countably additive prior is used, then decisions need not cohere and decision rules need not be extended…
Let $T_n$ be an $M$-estimator with defining function $\psi$ and preliminary estimate of scale $s_n$. Without loss of generality, let $s_n \rightarrow 1$ and take $E\psi(X/\xi) = 0$. Under various conditions, it is shown that any consistent version of $T_n$ is almost surely to order $O(n^{-1} \log_2 n)$ a linear combination of $n^{-1} \sum^n_1 \psi(X_i)$ and $s_n$. Only in the case $EX_1\psi'(X_1) = 0$ does the contribution of $S_n$ vanish; it is shown how this affects the…
A nonparametric minimum Hellinger distance estimator of location is introduced and shown to be asymptotically efficient at every symmetric density with finite Fisher information. Under small, possibly asymmetric, perturbations in such a density, the estimator is asymptotically robust in a technical sense which extends Hajek's concept of "regularity." A numerical example illustrates the computational feasibility of the estimator and its resistance to an arbitrary single…
We study the asymptotic power functions of tests for heteroscedasticity and nonlinearity in the linear model which were proposed by Anscombe and introduce and study some competitors robust against gross errors.
Let B=(N1,⋯,Nk) be a multivariate counting process and let Ft be the collection of all events observed on the time interval [0,t]. The intensity process is given by Λi(t)=limh↓0h1E(Ni(t+h)−Ni(t)∣Ft)i=1,⋯,k. We give an application of the recently developed martingale-based approach to the study of N via Λ. A statistical model…
An adaptive approach is considered as an alternative to the classical indifference-zone formulation of the problems of ranking and selection. With a fixed γ∗, the proposed procedure calls for the termination of sampling when the estimated probability of correct selection exceeds γ∗ for the first time. Asymptotic properties of this procedure are proved as γ∗→1, and Monte Carlo results show that the procedure is well behaved even…
This paper considers the estimation of a location parameter θ in a one-sample problem. The asymptotic performance of a sequence of estimates {Tn} is measured by the exponential rate of convergence to 0 of max{Pθ(Tn<θ−a),Pθ(Tn>θ+a)},saye(a). This measure of asymptotic performance is equivalent to one considered by Bahadur (1967). The optimal value of e(a) is given for translation invariant estimates. Some…
In the simple linear regression problem {Yi=α+βxi+eii=1,⋯,n,ei i.i.d. ∼F continuous, x1≦⋯≦xn known, α,β unknown} we investigate the following type of estimator: To each sij=(Yj−Yi)/(xj−xi) with xi<xj attach weight wij and as estimator for β consider the median of this weight distribution over the sij. A confidence interval for β is found by taking certain…
Interval estimation of variance components is studied for the unbalanced one-way random effects model. An easily calculated function, W, of the harmonic mean of the class sizes and of the sample variance of the class means is found to be important. The exact distribution of W is found and is shown to be excellently approximated by a chi-square distribution. The random variable W is used to construct interval estimates for (i) the between classes variance component and…
In the discrimination problem the random variable θ, known to take values in {1,⋯,M}, is estimated from the random vector X. All that is known about the joint distribution of (X,θ) is that which can be inferred from a sample (X1,θ1),⋯,(Xn,θn) of size n drawn from that distribution. A discrimination rule is any procedure which determines a decision θ^ for θ from X and (X1,θ1),⋯,(Xn,θn)…
This paper considers likelihood ratio tests for testing hypotheses that a collection of parameters satisfy some order restriction. The first problem considered is to test a hypothesis specifying an order restriction on a collection of means of normal distributions. Equality of the means is the subhypothesis of the null hypothesis which yields the largest type I error probability (i.e., is least favorable). Furthermore, the distribution of T=−ln (likelihood ratio) is…
The main result of this paper is that g-inverses are not needed for computing optimum designs when the singularity of the information matrix is unavoidable. They are, of course, needed for the analysis. It will be shown that it is possible to augment the experimental region so that procedures for computing optimum designs for s out k parameters (s<k) which are developed for the nonsingular case may also be used for the singular case.
Families of minimax estimators are found for the location parameter of a p-variate (p≧3) spherically symmetric unimodal distribution with respect to general quadratic loss. The estimators of James and Stein, Baranchik, Bock and Strawderman are all considered for this general problem. Specifically, when the loss is general quadratic loss given by L(δ,θ)=(δ−θ)′D(δ−θ) where D is a known p×p positive definite matrix,…
On Almost Sure Expansions for M-Estimates
Let Tn be an M-estimator with defining function ψ and preliminary estimate of scale sn. Without loss of generality, let sn→1 and take Eψ(X/ξ)=0. Under various conditions, it is shown that any consistent version of Tn is almost surely to order O(n−1log2n) a linear combination of n−1∑1nψ(Xi) and sn. Only in the case EX1ψ′(X1)=0 does the contribution of Sn vanish; it is shown how this affects the…