Estimating a restricted normal mean

Let X ₁, X ₂, ..., X _n be a random sample from a normal distribution with unknown mean μ and known variance σ ². In many practical situations, μ is known a priori to be restricted to a bounded interval, say [−m, m] for some m > 0. The sample mean , then, becomes an inadmissible estimator for μ. It is also not minimax with respect to the squared error loss function. Minimax and other estimators for this problem have been studied by Casella and Strawderman (Ann Stat 9:870–878, 1981), Bickel (Ann Stat 9:1301–1309, 1981) and Gatsonis et al. (Stat Prob Lett 6:21–30, 1987) etc. In this paper, we obtain some new estimators for μ. The case when the variance σ ² is unknown is also studied and various estimators for μ are proposed. Risk performance of all estimators is numerically compared for both the cases when σ ² may be known and unknown.     

Multivariate multistage methodologies for simultaneous all pairwise comparisons

We consider the problem of constructing simultaneous fixed-width confidence intervals for all pairwise treatment differences μ₁−μ _J , in the presence ofk(≥2) independent populationsN _p (μ₁,Σ), 1≤i≠j≤k. Appropriate purely sequential, accelerated sequential and three-stage sampling strategies have been developed and variousfirst-order asymptotic properties are then derived when Σ _pxp is completely unknown, but positive definite (p.d.). In the two special cases when the largest component variance in Σ is a known multiple of one of the variances or Σ=σ² H where σ(>0) is unknown, butH _pxp is known and p.d., the original multistage sampling strategies are specialized. Under such special circumstances, associatedsecond-order characteristics are then developed. It is to be noted that our present formulation and the methodologies fill important voids in the context of multivariate multiple comparisons which is a challenging area that has not yet been fully explored. Moderate sample performances of the proposed techniques were very encouraging and detailed remarks on these were included in Mukhopadhyay and Aoshima (1997).     

Determination of uniformly most powerful tests in discrete sample spaces

Uniformly most powerful (UMP) tests are known to exist in one-parameter exponential families when the hypothesis H ₀ and the alternative hypothesis H ₁ are given by (i) H ₀ : θ≤θ₀, H ₁ : θ>θ₀, and (ii) H ₀ : θ≤θ₁ or θ≥θ₂, H ₁ : θ₁<θ<θ₂, where θ₁<θ₂.  Likewise, uniformly most powerful unbiased (UMPU) tests do exist when the hypotheses H ₀ and H ₁ take the form (iii) H ₀ : θ₁≤θ≤θ₂, H ₁ : θ<θ₁ or θ>θ₂, where θ₁<θ₂, and (iv) H ₀ : θ=θ₀, H ₁:θ≠θ₀.  To determine tests in case (i), only one critical value c and one randomization constant γ have to be computed. In cases (ii) through (iv) tests are determined by two critical values c ₁, c ₂ and two randomization constants γ₁, γ₂. Unlike determination of tests in case (i), computation of critical values and randomization constants in the remaining cases is rather difficult, unless distributions are symmetric. No straightforward method to determine two-sided UMP tests in discrete sample spaces seems to be known. The purpose of this note is to disclose a distribution independent principle for the determination of UMP tests in cases (ii) through (iv). Received: March 1999     

A bound on the expected overshoot for some concave boundaries

LetX ₁,X ₂,… be i.i.d. with finite meanμ>0,S _n =X ₁+…+X _n . Forf(n)=n ^β ,c>0 we consider the stopping timesT _c =inf{n:S _n >c+f(n)} with overshootR _c =S _{T c} −(c+f(T _c )). For 0<β<1 we give a bound for sup _c≥0 ER _c in the spirit of Lorden’s well-known inequality forf=0.     

Characterization of some types of completeness resp. Total completeness and their conservation under direct products

Some notions ofL _p (μ)-completeness resp. totally L _p (μ)-completeness (1≦p≦∞) are characterized for families of probability distributions dominated by aσ-finite measureμ and their conservation with respect to direct products is proved. Furthermore, it is shown that totallyL _∞(μ)-completeness does not implyL ₁(μ)-completeness and that there are families of probability distributions in the i.i.d. case induced by the order statistic, which are L₁(μ)-complete but not totallyL _∞(μ)-complete.     

Multi-stage point and interval estimation of the largest mean ofK normal populations and the associated second-order properties

Summary We havek independent normal populations with unknown meansμ ₁, …,μ _k and a common unknown varianceσ ². Both point and interval estimation procedures for the largest mean are proposed by means of sequential and three-stage procedures. For the point estimation problem, we require that the maximal risk be at mostW, a preassigned positive number. For the other problem, we wish to construct a fixed-width confidence interval having the confidence coefficient at least 1-α, a preassigned number between zero and one. Asymptotic second order expansions are provided for various characteristics, such as average sample size, associated risks etc., for the suggested multi-stage estimation procedures.     

Multivariate L1 mean

The center of a univariate data set {x ₁,…,x _n} can be defined as the point μ that minimizes the norm of the vector of distances y′=(|x ₁−μ|,…,|x _n−μ|). As the median and the mean are the minimizers of respectively the L ₁- and the L ₂-norm of y, they are two alternatives to describe the center of a univariate data set. The center μ of a multivariate data set {x ₁,…,x _n} can also be defined as minimizer of the norm of a vector of distances. In multivariate situations however, there are several kinds of distances. In this note, we consider the vector of L ₁-distances y′₁=(∥x ₁- μ∥₁,…,∥x _n- μ∥₁) and the vector of L ₂-distances y′₂=(∥x ₁- μ∥₂,…,∥x _n-μ∥₂). We define the L ₁-median and the L ₁-mean as the minimizers of respectively the L ₁- and the L ₂-norm of y ₁; and then the L ₂-median and the L ₂-mean as the minimizers of respectively the L ₁- and the L ₂-norm of y ₂. In doing so, we obtain four alternatives to describe the center of a multivariate data set. While three of them have been already investigated in the statistical literature, the L ₁-mean appears to be a new concept. Received January 1999     

Tests of significance for the average square deviation from target in a finite lot

Consider lots of discrete items 1, 2, …,N with quality characteristicsx ₁,x ₂, …,x _N . Leta be a target value for item quality. Lot quality is identified with the average square deviation from target per item in the lot (lot average square deviation from target). Under economic considerations this is an appropriate lot quality indicator if the loss respectively the profit incurred from an item is a quadratic function ofx _i −a. The present paper investigates tests of significance on the lot average square deviationz under the following assumptions: The lot is a subsequence of a process of production, storage, transport; the random quality characteristics of items resulting from this process are i.i.d. with normal distributionN(μ, σ ²); the target valuea coincides with the process meanμ.     

Comparison of test vs. control treatments using distance optimality criterion

We consider the problem of comparison of one test treatment (τ₀) with a set of v control treatments (τ₁, τ₂, …, τ_v) using distance optimality [DS-optimality] criterion introduced by Sinha (1970) in some treatment-connected design settings. It turns out that the nature of DS-optimal designs is quite similar to that for the usual A−, D− and E− optimality criteria. However, the optimality problem is quite complicated in most situations. First we deal with the CRD model and derive DS-optimal allocations for a given set of treatments. The results are almost identical to the A-optimal allocations for such problems. Then we consider a block design set-up and examine the nature of DS-optimal designs. In the process, we introduce the method of weighted coverage probability and maximize the resulting expression to obtain an optimal design. Received: December 1999     

One-Sample Bayesian Predictive Interval of Future Ordered Observations for the Pareto Distribution

Nigm et al. (2003, statistics 37: 527–536) proposed Bayesian method to obtain predictive interval of future ordered observation Y _(j) (r < j≤ n ) based on the right type II censored samples Y ₍₁₎ < Y ₍₂₎ < ... < Y _(r) from the Pareto distribution. If some of Y ₍₁₎ < ... < Y _(r-1) are missing or false due to artificial negligence of typist or recorder, then Nigm et al.’s method may not be an appropriate choice. Moreover, the conditional probability density function (p.d.f.) of the ordered observation Y _(j) (r < j ≤ n ) given Y ₍₁₎ <Y ₍₂₎ < ... < Y _(r) is equivalent to the conditional p.d.f. of Y _(j) (r < j ≤ n ) given Y _(r). Therefore, we propose another Bayesian method to obtain predictive interval of future ordered observations based on the only ordered observation Y _(r), then compares the length of the predictive intervals when using the method of Nigm et al. (2003, statistics 37: 527–536) and our proposed method. Numerical examples are provided to illustrate these results.     

Distributions determined by conditioning on a pair of order statistics

LetX ₁,X ₂, ...,X _n (n≥3) be a random sample on a random variableX with distribution functionF having a unique continuous inverseF ⁻¹ over (a,b), −∞≤a<b≤∞ the support ofF. LetX _1:n <X _2:n <...<X _n:n be the corresponding order statistics. Letg be a nonconstant continuous function over (a,b). Then for some functionG over (a, b) and for some positive integersr ands, 1<r+1<s≤n

Minimax estimators andΓ-minimax estimators for a bounded normal mean under the lossl p (θ, d)=|θ-d|p

Summary Let the random variableX be normal distributed with known varianceσ ²>0. It is supposed that the unknown meanθ is an element of a bounded intervalΘ. The problem of estimatingθ under the loss functionl _p (θ, d)=|θ-d| ^p p≥2 is considered. In case the length of the intervalθ is sufficiently small the minimax estimator and theΓ(β, τ)-minimax estimator, whereΓ(β, τ) represents special vague prior information, are given.     

Locally minimax test of independence in elliptically symmetrical distributions with additional observations

Summary LetX=(X _ij )=(X ₁, ...,X _n )’,X’ _i =(X _i1, ...,X _ip )’,i=1,2, ...,n be a matrix having a multivariate elliptical distribution depending on a convex functionq with parameters, 0,σ. Let ϱ²=ϱ ₂ ^-2 be the squared multiple correlation coefficient between the first and the remainingp ₂+p ₃=p−1 components of eachX _i . We have considered here the problem of testingH ₀:ϱ²=0 against the alternativesH ₁:ϱ ₁ ^-2 =0, ϱ ₂ ^-2 >0 on the basis ofX andn ₁ additional observationsY ₁ (n ₁×1) on the first component,n ₂ observationsY ₂(n ₂×p ₂) on the followingp ₂ components andn ₃ additional observationsY ₃(n ₃×p ₃) on the lastp ₃ components and we have derived here the locally minimax test ofH ₀ againstH ₁ when ϱ ₂ ^-2 →0 for a givenq. This test, in general, depends on the choice ofq of the familyQ of elliptically symmetrical distributions and it is not optimality robust forQ.     

On the strong uniform consistency of a new kernel density estimator

Summary A new multivariate kernel probability density estimator is introduced and its strong uniform consistency is proved under certain regularity conditions. This result is then applied particularly to a kernel estimator whose mean vector and covariance matrix areμ _n andV _n, respectively, whereμ _n is an unspecified estimator of the mean vector andV _n, up to a multiplicative constant, the sample covariance matrix of the probability density to be estimated, respectively. Work supported by the Natural Sciences and Engineering Research Council of Canada and by the Fonds F.C.A.R. of the Province of Quebec.     

On kalman filtering,posterior mode estimation and fisher scoring in dynamic exponential family regression

Summary Dynamic exponential family regression provides a framework for nonlinear regression analysis with time dependent parametersβ ₀,β ₁, …,β _t, …, dimβ _t=p. In addition to the familiar conditionally Gaussian model, it covers e.g. models for categorical or counted responses. Parameters can be estimated by extended Kalman filtering and smoothing. In this paper, further algorithms are presented. They are derived from posterior mode estimation of the whole parameter vector (β′₀, …,β′_t) by Gauss-Newton resp. Fisher scoring iterations. Factorizing the information matrix into block-bidiagonal matrices, algorithms can be given in a forward-backward recursive form where only inverses of “small”p×p-matrices occur. Approximate error covariance matrices are obtained by an inversion formula for the information matrix, which is explicit up top×p-matrices. Heinz Leo Kaufmann, my friend and coauthor for many years, died in a tragical rock climbing accident in August 1989. This paper is dedicated to his memory.     

A note on the maximum deviation of the scale-contaminated normal to the best normal distribution

In this paper we consider the case of the scale-contaminated normal (mixture of two normals with equal mean components but different component variances: (1−p)N(μ,σ²)+pN(μ,τ²) with σ and τ being non-negative and 0≤p≤1). Here is the scale error and p denotes the amount with which this error occurs. It's maximum deviation to the best normal distribution is studied and shown to be montone increasing with increasing scale error. A closed-form expression is derived for the proportion which maximizes the maximum deviation of the mixture of normals to the best normal distribution. Implications to power studies of tests for normality are pointed out. Received May 2001     

A note on generalized wald’s method

Let {v _n(θ)} be a sequence of statistics such that whenθ =θ ₀,v _n(θ ₀) N _p(0,Σ), whereΣ is of rankp andθ εR ^d. Suppose that underθ =θ ₀, {Σ _n} is a sequence of consistent estimators ofΣ. Wald (1943) shows thatv _n ^T (θ ₀)Σ _n ⁻¹ v _n(θ ₀) x ²(p). It often happens thatv _n(θ ₀) N _p(0,Σ) holds butΣ is singular. Moore (1977) states that under certain assumptionsv _n ^T (θ ₀)Σ _n ⁻ v _n(θ ₀) x ²(k), wherek = rank (Σ) andΣ _n ⁻ is a generalized inverse ofΣ _n. However, Moore’s result as stated is incorrect. It needs the additional assumption that rank (Σ _n) =k forn sufficiently large. In this article, we show that Moore’s result (as corrected) holds under somewhat different, but easier to verify, assumptions. Research partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell University.     

Criterion-robust optimal product designs for discrimination between nested response surface models

Consider the problem of discriminating between two rival response surface models and estimating parameters in the identified model. To construct designs serving for both model discrimination and parameter estimation, the M _γ-optimality criterion, which puts weight γ (0≤γ≤1) for model discrimination and 1 − γ for parameter estimation, is adopted. The corresponding M _γ-optimal product design is explicitly derived in terms of canonical moments. With the application of the maximin principle on the M _γ-efficiency of any M _γ'-optimal product design, a criterion-robust optimal product design is proposed.     

Testing whetherF is more IFR thanG

A distributionF is said to be “more IFR” than another distributionG ifG ⁻¹ F is convex. WhenF(0) =G(0) = 0, the problem of testingH ₀ :F(x) =G (θx) for someθ > 0 andx ⩾ 0, against the alternativeH _A:F is more IFR thanG, is considered in this paper. Both cases, whenG is completely specified (one-sample case) and when it is not specified but a random sample form it is available (two-sample case) are considered. The proposed tests are based onU-statistics. The asymptotic relative efficiency of the tests are compared with several other tests and the test statistics remain asymptotically normal under certain dependency assumptions. Research supported in part by a grant from the US Air Force Office of Scientific Research.     

Estimating survivor function using optimally selected order statistics

This paper deals with the estimation of survivor function using optimally selected order statistics when the sample sizen is large. We use the estimates (μ*,σ*) based on the optimum set of order statistics for largen and fixedk (≤n) such that the estimate has optimum variance property. The asymptotic relative efficiency of such an estimator is compared with the one based on the complete sample. The general theory of the problem and specific details with respect to a two-parameter Normal, Logistic, Exponential and Pareto distributions is considered as an example.