On upper bounds for the characteristic values of the covariance matrix for multinomial,dirichlet and multivariate hypergeometric distributions

For the characteristic values T1 of the matrix V:=Diag(p)-pp^T with p=(p1,...,pk), p1≥p2≥...≥pk≥pk+1>0 and p1+p2+...+pk+pk+1=1 the inequalities p1≥τ1≥p2≥τ2≥...≥pk≥τk>0 are given by RONNING (1982). These inequalities give, if p and pk+1 are unknown, the upper bound 1≥T1. However, in this note the bound 1/2≥T1 is derived. V is proportional to the covariance matrix for multinomial, Dirichlet and multivariate hypergeometric distributions. A statistical application for the multinomial distribution is given.     

Confidence intervals for the largest mean from k correlated normal populations

Let X= (X₁,…, X_k)’ be a k-variate (k ≥ 2) normal random vector with unknown population mean vector μ = (μ₁ ,…, μ_k)’ and covariance matrix Σ of order k and let μ_[1] ≤ … ≤ μ_[k] be the ordered values of the μ ’ s. No prior knowledge of the pairing of the μ_[i] with the X_j. (or μ_[i] with the σ_j ²) is assumed for any i and j (1 ≤ i, j ≤ k). Based on a random sample of N independent vector observations on X, this paper considers both upper and lower (one-sided) and two-sided 100γ% (0 < γ < 1) confidence intervals for μ_[k] and μ_[1], the largest and the smallest mean, respectively, when Σ is known and when Σ is equal to σ²R with common unknown variance σ² > 0 and correlation matrix R known, respectively. An optimum two-sided confidence interval via finding the shortest length from this class is also considered. Necessary tables and computer program to actually apply these procedures are provided.     

Dispersive ordering—Some applications and examples

A basic concept for comparing spread among probability distributions is that of dispersive ordering. Let X and Y be two random variables with distribution functions F and G, respectively. Let F ⁻¹ and G ⁻¹ be their right continuous inverses (quantile functions). We say that Y is less dispersed than X (Y≤ _disp X) if G ⁻¹(β)−G ⁻¹(α)≤F ⁻¹(β)−F ⁻¹(α), for all 0<α≤β<1. This means that the difference between any two quantiles of G is smaller than the difference between the corresponding quantiles of F. A consequence of Y≤ _disp X is that |Y ₁−Y ₂| is stochastically smaller than |X ₁−X ₂| and this in turn implies var(Y)≤var(X) as well as E[|Y ₁−Y ₂|]≤E[|X ₁−X ₂|], where X ₁, X ₂ (Y ₁, Y ₂) are two independent copies of X(Y). In this review paper, we give several examples and applications of dispersive ordering in statistics. Examples include those related to order statistics, spacings, convolution of non-identically distributed random variables and epoch times of non-homogeneous Poisson processes. This work was supported in part by KOSEF through Statistical Research Center for Complex Systems at Seoul National University. Subhash Kochar is thankful to Dr. B. Khaledi for many helpful discussions.     

On pre-test estimation of parametric functions in the general Gauss-Markov model with quadratic risk

The approach to preliminary test estimation based on comparing the weighted quadratic risk function of two competing estimators of β under the linear regression model {y,Xβ, σ² I} is extended to the case when a given vector of parametric functions κ=Kβ is to be estimated under the general Gauss-Markov model.     

Approximation of density functions by orthogonal series with grouped data

Summary Letg(x) andf(x) be continuous density function on (a, b) and let {ϕ_j} be a complete orthonormal sequence of functions onL ₂(g), which is the set of squared integrable functions weighted byg on (a, b). Suppose that over (a, b). Given a grouped sample of sizen fromf(x), the paper investigates the asymptotic properties of the restricted maximum likelihood estimator of density, obtained by setting all but the firstm of the ϑ_j’s equal to0. Practical suggestions are given for performing estimation via the use of Fourier and Legendre polynomial series. Research partially supported by: CNR grant, n. 93. 00837. CT10.     

Comparing several exponential populations with more than one control

Suppose there are k ₁ (k ₁ ≥ 1) test treatments that we wish to compare with k ₂ (k ₂ ≥ 1) control treatments. Assume that the observations from the ith test treatment and the jth control treatment follow a two-parameter exponential distribution and , where θ is a common scale parameter and and are the location parameters of the ith test and the jth control treatment, respectively, i = 1, . . . ,k ₁; j = 1, . . . ,k ₂. In this paper, simultaneous one-sided and two-sided confidence intervals are proposed for all k ₁ k ₂ differences between the test treatment location and control treatment location parameters, namely , and the required critical points are provided. Discussions of multiple comparisons of all test treatments with the best control treatment and an optimal sample size allocation are given. Finally, it is shown that the critical points obtained can be used to construct simultaneous confidence intervals for Pareto distribution location parameters.     

A discrete distribution associated with a pure birth process

A discrete distribution associated with a pure birth process starting with no individuals, with birth rates λ ⁿ =λ forn=0, 2, …,m−1 and λ ⁿ forn≥m is considered in this paper. The probability mass function is expressed in terms of an integral that is very convenient for computing probabilities, moments, generating functions and others. Using this representation, the mean and the k-th factorial moments of the distribution are obtained. Some nice characterizations of this distribution are also given.     

Measuring robustness for weighted distributions: Bayesian perspective

There are many situations where the usual random sample from a population of interest is not available, due to the data having unequal probabilities of entering the sample. The method of weighted distributions models this ascertainment bias by adjusting the probabilities of actual occurrence of events to arrive at a specification of the probabilities of the events as observed and recorded. We consider two different classes of contaminated or mixture of weight functions, Γ _a ={w(x):w(x)=(1−ε)w ₀(x)+εq(x),q∈Q} and Γ _g ={w(x):w(x)=w ₀ ^1−ε (x)q ^ε(x),q∈Q} wherew ₀(x) is the elicited weighted function,Q is a class of positive functions and 0≤ε≤1 is a small number. Also, we study the local variation of ϕ-divergence over classes Γ _a and Γ _g . We devote on measuring robustness using divergence measures which is based on the Bayesian approach. Two examples will be studied.     

Admissible tests for the mean with additional information

Let X be a po-normal random vector with unknown µ and unknown covariance matrix ∑ and let X be partitioned as X = (X^′ ₍₁₎, …, X^′ _(r))′ where X_(j)is a subvector of X with dimension p_jsuch that ∑^r _j=1Pj = P0. Some admissible tests are derived for testing H₀: μ = 0 versus H₁: μ ¦0 based on a sample drawn from the whole vector X of dimension p and r additional samples drawn from X₍₁₎, X₍₂₎, …, X(r) respectively, All (r+1) samples are assumed to be independent. The distribution of some of the tests' statistics involved are also derived.     

Estimation of system reliability

In this paper, we estimate the reliability of a system with k components. The system functions when at least s (1≤s≤k) components survive a common random stress. We assume that the strengths of these k components are subjected to a common stress which is independent of the strengths of these k components. If (X ₁,X ₂,…,X _k ) are strengths of k components subjected to a common stress (Y), then the reliability of the system or system reliability is given byR=P[Y<X _(k−s+1)] whereX _(k−s+1) is (k−s+1)-th order statistic of (X ₁,…,X _k ). We estimate R when (X ₁,…,X _k ) follow an absolutely continuous multivariate exponential (ACMVE) distribution of Hanagal (1993) which is the submodel of Block (1975) and Y follows an independent exponential distribution. We also obtain the asymptotic normal (AN) distribution of the proposed estimator.     

On the robustness of the linear hypothesis test procedure in the singular linear model with implied restrictions

The linear hypothesis test procedure is considered in the restricted linear modelsM _r = {y, Xβ |Rβ = 0, σ ²V} andM _r ^* = {y, Xβ |ARβ = 0, σ ²V}. Necessary and sufficient conditions are derived under which the statistic providing anF-test for the linear hypothesisH ₀:Kβ=0 in the modelM _r ^* (M_r) continues to be valid in the modelM _r (M _r ^* ); the results obtained cover the case whereM _r ^* is replaced by the general Gauss-Markov modelM = {y, Xβ, σ ²V}.     

Distributions of patterns of two successes separated by a string of <Emphasis Type="Italic">k</Emphasis>-2 failures

Let Z ₁, Z ₂, . . . be a sequence of independent Bernoulli trials with constant success and failure probabilities p = Pr(Z _t  = 1) and q = Pr(Z _t  = 0) = 1 − p, respectively, t = 1, 2, . . . . For any given integer k ≥ 2 we consider the patterns E₁{\mathcal{E}_{1}}: two successes are separated by at most k−2 failures, E₂{\mathcal{E}_{2}}: two successes are separated by exactly k −2 failures, and E₃{\mathcal{E}_{3}} : two successes are separated by at least k − 2 failures. Denote by N_n,k⁽ⁱ⁾{ N_{n,k}^{(i)}} (respectively M_n,k⁽ⁱ⁾{M_{n,k}^{(i)}}) the number of occurrences of the pattern E_i{\mathcal{E}_{i}} , i = 1, 2, 3, in Z ₁, Z ₂, . . . , Z _n when the non-overlapping (respectively overlapping) counting scheme for runs and patterns is employed. Also, let T_r,k⁽ⁱ⁾{T_{r,k}^{(i)}} (resp. W_r,k⁽ⁱ⁾){W_{r,k}^{(i)})} be the waiting time for the r − th occurrence of the pattern E_i{\mathcal{E}_{i}}, i = 1, 2, 3, in Z ₁, Z ₂, . . . according to the non-overlapping (resp. overlapping) counting scheme. In this article we conduct a systematic study of N_n,k⁽ⁱ⁾{N_{n,k}^{(i)}}, M_n,k⁽ⁱ⁾{M_{n,k}^{(i)}}, T_r,k⁽ⁱ⁾{T_{r,k}^{(i)}} and W_r,k⁽ⁱ⁾{W_{r,k}^{(i)}} (i = 1, 2, 3) obtaining exact formulae, explicit or recursive, for their probability generating functions, probability mass functions and moments. An application is given.     

Locally best invariant and locally minimax test of independence

Let X₁ X₂ … X_N be independent normal p-vectors with common mean vector $$ = ($$) and common nonsingular covariance matrix $$ = Diag ($sG_i) [(1–p) I + pE] Diag ($sG_i), $sG_i> 0, i = 1… p, 1>p>=1/p–1. Write r_ij = sample correlation between the i th and the j th variable i j = 1,… p. It has been proved that for testing the hypothesis H₀ : p = 0 against the alternative H₁ : p>0 where $$ and $sG₁,…, $sG_p are unknown, the test which rejects H₀ for large value of $$ r_ij is locally best invariant for every $aL: 0 > $aL > 1 and locally minimax as p $$ 0 in the sense of Giri and Kiefer, 1964, for every $aL: 0 > $aL $$ $aL₀ > 1 where$aL₀ = P_p=0 $$.     

NON-EXISTENCE OF AN EXACT β-CONFIDENCE INTERVAL FOR A SCALE PARAMETER*

Abstract There are given k (≥22) independent distributions with c.d.f.'s F(x;θ_j) indexed by a scale parameter θ_j, j = 1,…, k. Let θ_[i] (i = 1,…, k) denote the ith smallest one of θ₁,…, θ_k. In this paper we wish to show that, under some regularity conditions, there does not exist an exact β-level (0≤β1) confidence interval for the ith smallest scale parameter θ_i based on k independent samples. Since the log transformation method may not yield the desired results for the scale parameter problem, we will treat the scale parameter case directly without transformation. Application is considered for normal variances. Two conservative one-sided confidence intervals for the ith smallest normal variance and the percentage points needed to actually apply the intervals are provided.     

Improvement of the Liu estimator in linear regression model

In the presence of stochastic prior information, in addition to the sample, Theil and Goldberger (1961) introduced a Mixed Estimator for the parameter vector β in the standard multiple linear regression model (T,Xβ,σ² I). Recently, the Liu estimator which is an alternative biased estimator for β has been proposed by Liu (1993). In this paper we introduce another new Liu type biased estimator called Stochastic restricted Liu estimator for β, and discuss its efficiency. The necessary and sufficient conditions for mean squared error matrix of the Stochastic restricted Liu estimator to exceed the mean squared error matrix of the mixed estimator will be derived for the two cases in which the parametric restrictions are correct and are not correct. In particular we show that this new biased estimator is superior in the mean squared error matrix sense to both the Mixed estimator and to the biased estimator introduced by Liu (1993).     

An algorithm for computing principal points with respect to a loss function in the unidimensional case

In Flury (1990) the k principal points of a random vector X are defned as the points p(1),..., p(k) minimizing EX–p(i)²; i=1,..., k. We extend this concept to that of k principal points with respect to a loss function L, and present an algorithm for their computation in the univariate case.     

ON ASYMPTOTIC OPTIMALITY OF SOME TESTS FOR EXPONENTIAL DISTRIBUTIONS

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; ρ_i, s?_i), 0 < ρ_i < ∞, 0 < s?_i < s?_i < and f(x; ρ, s?) is as given above. Let X_i1 < X_i2 <… < X_iri denote the first r_i order statistics in a random sample of size n_i, drawn from the ith population with pdf f(x; ρ_i, s?_i), i = 1, 2,…, k. In this paper we show that the well known tests of hypotheses about the parameters ρ_i, 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.     

Improved estimators for the selected location parameters

_i , i = 1, 2, ..., k be k independent exponential populations with different unknown location parameters θ _i , i = 1, 2, ..., k and common known scale parameter σ. Let Y _i denote the smallest observation based on a random sample of size n from the i-th population. Suppose a subset of the given k population is selected using the subset selection procedure according to which the population π _i is selected iff Y _i ≥Y ₍₁₎−d, where Y ₍₁₎ is the largest of the Y _i 's and d is some suitable constant. The estimation of the location parameters associated with the selected populations is considered for the squared error loss. It is observed that the natural estimator dominates the unbiased estimator. It is also shown that the natural estimator itself is inadmissible and a class of improved estimators that dominate the natural estimator is obtained. The improved estimators are consistent and their risks are shown to be O(kn ⁻²). As a special case, we obtain the coresponding results for the estimation of θ₍₁₎, the parameter associated with Y ₍₁₎. Received: January 6, 1998; revised version: July 11, 2000