共查询到20条相似文献,搜索用时 31 毫秒
1.
Kai F. Yu 《统计学通讯:理论与方法》2013,42(8):2391-2403
Let (θ1,x1),…,(θn,xn) be independent and identically distributed random vectors with E(xθ) = θ and Var(x|θ) = a + bθ + cθ2. Let ti be the linear Bayes estimator of θi and θ~i be the linear empirical Bayes estimator of θi as proposed in Robbins (1983). When Ex and Var x are unknown to the statistician. The regret of using θ~i instead of ti because of ignorance of the mean and the variance is ri = E(θi ? θi)2 ?E(ti ?θi)2. Under appropriate conditions cumulative regret Rn = r1+…rn is shown to have a finite limit even when n tends to infinity. The limit can be explicitly computed in terms of a,b,c and the first four moments of x. 相似文献
2.
Tachen Liang 《统计学通讯:理论与方法》2013,42(3):565-577
We investigate an empirical Bayes testing problem in a positive exponential family having pdf f{x/θ)=c(θ)u(x) exp(?x/θ), x>0, θ>0. It is assumed that θ is in some known compact interval [C1, C2]. The value C1 is used in the construction of the proposed empirical Bayes test δ* n. The asymptotic optimality and rate of convergence of its associated Bayes risk is studied. It is shown that under the assumption that θ is in [C1, C2] δ* n is asymptotically optimal at a rate of convergence of order O(n?1/n n). Also, δ* n is robust in the sense that δ* n still possesses the asymptotic optimality even the assumption that "C1≦θ≦C2 may not hold. 相似文献
3.
Constance Van Eeden 《Revue canadienne de statistique》1995,23(3):245-256
Let X have a gamma distribution with known shape parameter θr;aL and unknown scale parameter θ. Suppose it is known that θ ≥ a for some known a > 0. An admissible minimax estimator for scale-invariant squared-error loss is presented. This estimator is the pointwise limit of a sequence of Bayes estimators. Further, the class of truncated linear estimators C = {θρ|θρ(x) = max(a, ρ), ρ > 0} is studied. It is shown that each θρ is inadmissible and that exactly one of them is minimax. Finally, it is shown that Katz's [Ann. Math. Statist., 32, 136–142 (1961)] estimator of θ is not minimax for our loss function. Some further properties of and comparisons among these estimators are also presented. 相似文献
4.
Let X1,… Xm be a random sample of m failure times under normal conditions with the underlying distribution F(x) and Y1,…,Yn a random sample of n failure times under accelerated condititons with underlying distribution G(x);G(x)=1?[1?F(x)]θ with θ being the unknown parameter under study.Define:Uij=1 otherwise.The joint distribution of ijdoes not involve the distribution F and thus can be used to estimate the acceleration parameter θ.The second approach for estimating θ is to use the ranks of the Y-observations in the combined X- and Y-samples.In this paper we establish that the rank of the Y-observations in the pooled sample form a sufficient statistic for the information contained in the Uii 's about the parameter θ and that there does not exist an unbiassed estimator for the parameter θ.We also construct several estimators and confidence interavals for the parameter θ. 相似文献
5.
D. J. Daley 《Australian & New Zealand Journal of Statistics》1975,17(2):91-93
The largest value of the constant c for which holds over the class of random variables X with non-zero mean and finite second moment, is c=π. Let the random variable (r.v.) X with distribution function F(·) have non-zero mean and finite second moment. In studying a certain random walk problem (Daley, 1976) we sought a bound on the characteristic function of the form for some positive constant c. Of course the inequality is non-trivial only provided that . This note establishes that the best possible constant c =π. The wider relevance of the result is we believe that it underlines the use of trigonometric inequalities in bounding the (modulus of a) c.f. (see e.g. the truncation inequalities in §12.4 of Loève (1963)). In the present case the bound thus obtained is the best possible bound, and is better than the bound (2) |1-?(θ)| ≥ |θEX|-θ2EX2\2 obtained by applying the triangular inequality to the relation which follows from a two-fold integration by parts in the defining equation (*). The treatment of the counter-example furnished below may also be of interest. To prove (1) with c=π, recall that sin u > u(1-u/π) (all real u), so Since |E sinθX|-|E sin(-θX)|, the modulus sign required in (1) can be inserted into (4). Observe that since sin u > u for u < 0, it is possible to strengthen (4) to (denoting max(0,x) by x+) To show that c=π is the best possible constant in (1), assume without loss of generality that EX > 0, and take θ > 0. Then (1) is equivalent to (6) c < θEX2/{EX-|1-?(θ)|/θ} for all θ > 0 and all r.v.s. X with EX > 0 and EX2. Consider the r.v. where 0 < x < 1 and 0 < γ < ∞. Then EX=1, EX2=1+γx2, From (4) it follows that |1-?(θ)| > 0 for 0 < |θ| <π|EX|/EX2 but in fact this positivity holds for 0 < |θ| < 2π|EX|/EX2 because by trigonometry and the Cauchy-Schwartz inequality, |1-?(θ)| > |Re(1-?(θ))| = |E(1-cosθX)| = 2|E sin2θX/2| (10) >2(E sinθX/2)2 (11) >(|θEX|-θ2EX2/2π)2/2 > 0, the inequality at (11) holding provided that |θEX|-θ2EX2/2π > 0, i.e., that 0 < |θ| < 2π|EX|/EX2. The random variable X at (7) with x= 1 shows that the range of positivity of |1-?(θ)| cannot in general be extended. If X is a non-negative r.v. with finite positive mean, then the identity shows that (1-?(θ))/iθEX is the c.f. of a non-negative random variable, and hence (13) |1-?(θ)| < |θEX| (all θ). This argument fans if pr{X < 0}pr{X> 0} > 0, but as a sharper alternative to (14) |1-?(θ)| < |θE|X||, we note (cf. (2) and (3)) first that (15) |1-?(θ)| < |θEX| +θ2EX2/2. For a bound that is more precise for |θ| close to 0, |1-?(θ)|2= (Re(1-?(θ)))2+ (Im?(θ))2 <(θ2EX2/2)2+(|θEX| +θ2EX2-/π)2, so (16) |1-?(θ)| <(|θEX| +θ2EX2-/π) + |θ|3(EX2)2/8|EX|. 相似文献
6.
Minimax Estimation of the Bounded Parameter of Some Discrete Distributions Under LINEX Loss Function
For a class of discrete distributions, including Poisson(θ), Generalized Poisson(θ), Borel(m, θ), etc., we consider minimax estimation of the parameter θ under the assumption it lies in a bounded interval of the form [0, m] and a LINEX loss function. Explicit conditions for the minimax estimator to be Bayes with respect to a boundary supported prior are given. Also for Bernoulli(θ)-distribution, which is not in the mentioned class of discrete distributions, we give conditions for which the Bayes estimator of θ ∈ [0, m], m < 1 with respect to a boundary supported prior is minimax under LINEX loss function. Numerical values are given for the largest values of m for which the corresponding Bayes estimators of θ are minimax. 相似文献
7.
A modified double stage shrinkage estimator has been proposed for the single parameter θ of a distribution function Fθ. It is shown to be locally better in comparison to the usual double stage shrinkage estimator in the sense of smaller mean squared error in a certain neighbourhood of prior estimate θo of θ. 相似文献
8.
D.B. Jensen 《Australian & New Zealand Journal of Statistics》1972,14(1):10-16
Let S (p×p) have a Wishart distribution -with v degrees of freedom and non-centrality matrix θ= [θjK] (p×p). Define θ0= min {| θjk |}, let θ0→∞, and suppose that | θjK | = 0(θo). Then the limiting form of the standardized non-central distribution, as θ while n? remains fixed, is a multivariate Gaussian distribution. This result in turn is used to obtain known asymptotic properties of multivariate chi-square and Rayleigh distributions under somewhat weaker conditions. 相似文献
9.
R. Ponnapalli 《Revue canadienne de statistique》1976,4(1):33-50
Suppose the multinomial parameters pr (θ) are functions of a real valued parameter 0, r = 1,2, …, k. A minimum discrepancy (m.d.) estimator θ of θ is defined as one which minimises the discrepancy function D = Σ nrf(pr/nr), for a suitable function f where nr is the relative frequency in r-th cell, r = 1,2, …, k. All the usual estimators like maximum likelihood (m. l), minimum chi-square (m. c. s.)., etc. are m.d. estimators. All m.d. estimators have the same asymptotic (first order) efficiency. They are compared on the basis of their deficiencies, a concept recently introduced by Hodges and Lehmann [2]. The expression for least deficiency at any θ is derived. It is shown that in general uniformly least deficient estimators do not exist. Necessary and sufficient conditions on pr (0) for m. t. and m. c. s. estimators to be uniformly least deficient are obtained. 相似文献
10.
Yo Sheena † 《Statistics》2013,47(5):371-379
We consider the estimation of Σ of the p-dimensional normal distribution Np (0, Σ) when Σ?=?θ0 Ip ?+?θ1 aa′, where a is an unknown p-dimensional normalized vector and θ0?>?0, θ1?≥?0 are also unknown. First, we derive the restricted maximum likelihood (REML) estimator. Second, we propose a new estimator, which dominates the REML estimator with respect to Stein's loss function. Finally, we carry out Monte Carlo simulation to investigate the magnitude of the new estimator's superiority. 相似文献
11.
M. Samanta 《Australian & New Zealand Journal of Statistics》1986,28(2):164-172
The probability density function (pdf) of a two parameter exponential distribution is given by f(x; p, s?) =s?-1 exp {-(x - ρ)/s?} for x≥ρ and 0 elsewhere, where 0 < ρ < ∞ and 0 < s?∞. Suppose we have k independent random samples where the ith sample is drawn from the ith population having the pdf f(x; ρi, s?i), 0 < ρi < ∞, 0 < s?i < s?i < and f(x; ρ, s?) is as given above. Let Xi1 < Xi2 <… < Xiri denote the first ri order statistics in a random sample of size ni, drawn from the ith population with pdf f(x; ρi, s?i), i = 1, 2,…, k. In this paper we show that the well known tests of hypotheses about the parameters ρi, s?i, i = 1, 2,…, k based on the above observations are asymptotically optimal in the sense of Bahadur efficiency. Our results are similar to those for normal distributions. 相似文献
12.
Andr Plante 《Revue canadienne de statistique》1994,22(1):1-13
Two consistent nonexact-confidence-interval estimation methods, both derived from the consistency-equivalence theorem in Plante (1991), are suggested for estimation of problematic parametric functions with no consistent exact solution and for which standard optimal confidence procedures are inadequate or even absurd, i.e., can provide confidence statements with a 95% empty or all-inclusive confidence set. A belt C(·) from a consistent nonexact-belt family, used with two confidence coefficients (γ = infθ Pθ [ θ ? C(X)] and γ+ = supθ Pθ[θ ? C(X)], is shown to provide a consistent nonexact-belt solution for estimating μ2 -μ1 in the Behrens-Fisher problem. A rule for consistent behaviour enables any confidence belt to be used consistently by providing each sample point with best upper and lower confidence levels [δ+(x) ≥ γ+, δ(x) ≤ γ], which give least-conservative consistent confidence statements ranging from practically exact through informative to noninformative. The rule also provides a consistency correction L(x) = δ+(x)-δ(X) enabling alternative confidence solutions to be compared on grounds of adequacy; this is demonstrated by comparing consistent conservative sample-point-wise solutions with inconsistent standard solutions for estimating μ2/μ1 (Creasy-Fieller-Neyman problem) and $\sqrt {\mu _1^2 + \mu _2^2 }$, a distance-estimation problem closely related to Stein's 1959 example 相似文献
13.
G. J. Mclachlan 《统计学通讯:模拟与计算》2013,42(6):715-726
Providing certain parameters are known, almost any linear map from RP to R1 can be adjusted to yield a consistent and unbiased estimator in the context of estimating the mixing proportion θ on the basis of an unclassified sample of observations taken from a mixture of two p-dimensional distributions in proportions θ and 1-θ. Attention is focused on an estimator proposed recently, θ, which has minimum variance over all such linear maps. Unfortunately, the form of θ depends on the means of the component distributions and the covariance matrix of the mixture distribution. The effect of using appropriate sample estimates for these unknown parameters in forming θ is investigated by deriving the asymptotic mean and variance of the resulting estimator. The relative efficiency of this estimator under normality is derived. Also, a study is undertaken of the performance of a similar type of estimator appropriate in the context where an observed data vector is not an observation from either one or the other onent distributions, but is recorded as an integrated measurement over a surface area which is a mixture of two categories whose characteristics have different statistical distributions.The asymptotic bias in this case is compared with some available practical results. 相似文献
14.
Yoshiji Takagi 《统计学通讯:理论与方法》2014,43(17):3614-3628
We consider the general one-sided hypotheses testing problem expressed as H0: θ1 ? h(θ2) versus H1: θ1 < h(θ2), where h( · ) is not necessary differentiable. The values of the right and the left differential coefficients, h′?( · ) and h′+( · ), at nondifferentiable points play an essential role in constructing the appropriate testing procedures with asymptotic size α on the basis of the likelihood ratio principle. The likelihood ratio testing procedure is related to an intersection–union testing procedure when h′?(θ2) ? h′+(θ2) for all θ2, and to a union–intersection testing procedure when there exists a θ2 such that h′?(θ2) < h′+(θ2). 相似文献
15.
Rhonda magel 《统计学通讯:模拟与计算》2013,42(3):917-925
We consider a test for the equality of k population medians, θi i=1,2,….,k, when it is believed a priori, that θ i: The observations are subject to right censorhip. The distributions of the censoring variables for each population are assumed to be equal. This test is compared with the general k-sample test proposed by Breslow 相似文献
16.
Saul Blumenthal 《统计学通讯:理论与方法》2013,42(4):297-308
Let X1, X2,…,Xn be independent, indentically distributed random variables with density f(x,θ) with respect to a σ-finite measure μ. Let R be a measurable set in the sample space X. The value of X is observable if X ? (X?R) and not otherwise. The number J of observable X’s is binomial, N, Q, Q = 1?P(X ? R). On the basis of J observations, it is desired to estimate N and θ. Estimators considered are conditional and unconditional maximum likelihood and modified maximum likelihood using a prior weight function to modify the likelihood before maximizing. Asymptotic expansions are developed for the [Ncirc]’s of the form [Ncirc] = N + α√N + β + op(1), where α and β are random variables. All estimators have the same α, which has mean 0, variance σ2 (a function of θ) and is asymptotically normal. Hence all are asymptotically equivalent by the usual limit distributional theory. The β’s differ and Eβ can be considered an “asymptotic bias”. Formulas are developed to compare the asymptotic biases of the various estimators. For a scale parameter family of absolutely continuous distributions with X = (0,∞) and R = (T,∞), special formuli are developed and a best estimator is found. 相似文献
17.
In some statistical problems a degree of explicit, prior information is available about the value taken by the parameter of interest, θ say, although the information is much less than would be needed to place a prior density on the parameter's distribution. Often the prior information takes the form of a simple bound, ‘θ > θ1 ’ or ‘θ < θ1 ’, where θ1 is determined by physical considerations or mathematical theory, such as positivity of a variance. A conventional approach to accommodating the requirement that θ > θ1 is to replace an estimator, , of θ by the maximum of and θ1. However, this technique is generally inadequate. For one thing, it does not respect the strictness of the inequality θ > θ1 , which can be critical in interpreting results. For another, it produces an estimator that does not respond in a natural way to perturbations of the data. In this paper we suggest an alternative approach, in which bootstrap aggregation, or bagging, is used to overcome these difficulties. Bagging gives estimators that, when subjected to the constraint θ > θ1 , strictly exceed θ1 except in extreme settings in which the empirical evidence strongly contradicts the constraint. Bagging also reduces estimator variability in the important case for which is close to θ1, and more generally produces estimators that respect the constraint in a smooth, realistic fashion. 相似文献
18.
Thompson (1997) considered a wide definition of p-value and found the Baves p-value for testing a ooint null hypothesis H0: θ= θ0 versus H1: θ ≠ θ0. In this paper, the general case of testing H0: θ ∈ ?0 versus H1: θ ∈ ?c 0 is studied. A generalization of the concept of p-value is given, and it is proved that the posterior predictive p-value based on the posterior odds is (asymptotically) a Bayes p-value. Finally, it is suggested that this posterior predictive p-value could be used as a reference p-value 相似文献
19.
20.
Günter M. T. Weiss 《Revue canadienne de statistique》1985,13(4):317-324
We consider the estimation of a location parameter θ in a one-sample problem. A measure of the asymptotic performance of an estimator sequence {Tn} = T is given by the exponential rate of convergence to zero of the tail probability, which for consistent estimator sequences is bounded by a constant, B (θ, ?), called the Bahadur bound. We consider two consistent estimators: the maximum-likelihood estimator (mle) and a consistent estimator based on a likelihood-ratio statistic, which we call the probability-ratio estimator (pre). In order to compare the local behaviour of these estimators, we obtain Taylor series expansions in ? for B (θ, ?) and the exponential rates of the mle and pre. Finally, some numerical work is presented in which we consider a variety of underlying distributions. 相似文献