A two stage procedure for selecting δ∗-optimal means in the normal model

Suppose π₁,…,π_k are k normal populations with π_i having unknown mean μ_i and unknown variance σ². The population π_i will be called δ^?-optimal (or good) if μ_i is within a specified amountδ? of the largest mean. A two stage procedure is proposed which selects a subset of the k populations and guarantees with probability at least P^? that the selected subset contains only δ?-optimal π_i ’s. In addition to screening out non-good populations the rule guarantees a high proportion of sufficiently good π_i’S will be selected.     

A class of fixed-width confidence intervals for a ranked normal mean

Suppose that we are given k(≥ 2) independent and normally distributed populations π₁, …, π_k where π_i has unknown mean μ_i and unknown variance σ² _i (i = 1, …, k). Let μ_[i] (i = 1, …, k) denote the i^th smallest one of μ₁, …, μ_k. A two-stage procedure is used to construct lower and upper confidence intervals for μ_[i] and then use these to obtain a class of two-sided confidence intervals on μ_[i] with fixed width. For i = k, the interval given by Chen and Dudewicz (1976) is a special case. Comparison is made between the class of two-sided intervals and a symmetric interval proposed by Chen and Dudewicz (1976) for the largest mean, and it is found that for large values of k at least one of the former intervals requires a smaller total sample size. The tables needed to actually apply the procedure are provided.     

A Subset Selection Procedure for Selecting the Exponential Population Having the Longest Mean Lifetime When the Guarantee Times are the Same

ABSTRACT

Consider k(≥ 2) independent exponential populations Π₁, Π₂, …, Π _k , having the common unknown location parameter μ ∈ (?∞, ∞) (also called the guarantee time) and unknown scale parameters σ₁, σ₂, …σ _k , respectively (also called the remaining mean lifetimes after the completion of guarantee times), σ _i  > 0, i = 1, 2, …, k. Assume that the correct ordering between σ₁, σ₂, …, σ _k is not known apriori and let σ_[i], i = 1, 2, …, k, denote the ith smallest of σ _j s, so that σ_[1] ≤ σ_[2] ··· ≤ σ_[k]. Then Θ _i  = μ + σ _i is the mean lifetime of Π _i , i = 1, 2, …, k. Let Θ_[1] ≤ Θ_[2] ··· ≤ Θ_[k] denote the ranked values of the Θ _j s, so that Θ_[i] = μ + σ_[i], i = 1, 2, …, k, and let Π_(i) denote the unknown population associated with the ith smallest mean lifetime Θ_[i] = μ + σ_[i], i = 1, 2, …, k. Based on independent random samples from the k populations, we propose a selection procedure for the goal of selecting the population having the longest mean lifetime Θ_[k] (called the “best” population), under the subset selection formulation. Tables for the implementation of the proposed selection procedure are provided. It is established that the proposed subset selection procedure is monotone for a general k (≥ 2). For k = 2, we consider the loss measured by the size of the selected subset and establish that the proposed subset selection procedure is minimax among selection procedures that satisfy a certain probability requirement (called the P*-condition) for the inclusion of the best population in the selected subset.     

On selecting a subset which contains all populations better than a control

Let π₁…, π_k denote k(≥ 2) populations with unknown means μ₁ , …, μ_k and variances σ₁ ² , …, σ_k ² , respectively and let π_o denote the control population having mean μ_o and variance σ_o ² . It is assumed that these populations are normally distributed with correlation matrix {ρ_ij}. The goal is to select a subset, of populations of π₁ , …, π_k which contains all the populations with means larger than or equal to the mean of the control one. Procedures are given for selecting such a subset so that the probability that all the populations with means larger than or equal to the mean of the control one are included in the selected subset is at least equal to a predetermined value P?(l/k < P? < 1). The goal treated here is a first step screening procedure that allows the experimenter to choose a subset and withhold judgement about which one has the largest mean. Then, if the one with the largest mean is desired it can be chosen from the selected subset on the basis of cost and other considerations. Percentage points are also included.     

Evaluation of a method for setting confidence intervals on the common mean of two normal populations

Let X₁, , X₂, …, X be distributed N(µ, σ² _x), let Y₁, Y₂, …, Y"_n be distributed N(µ, σ² _y), and let X , X , … X_m, Y₁, Y₂, …, Y_n be mutually independent. In this paper a method for setting confidence intervals on the common mean µ is proposed and evaluated.     

An exact decomposition for a class of exponential dispersion models: Application to sequential testing

We define the Wishart distribution on the cone of positive definite matrices and an exponential distribution on the Lorentz cone as exponential dispersion models. We show that these two distributions possess a property of exact decomposition, and we use this property to solve the following problem: given q samples (y_il,… y_iNj), i = l,…,q, from a N(μ_i,Σ_i,) distribution, test H:Σ₁ = Σ₂ = … = σ_q. Using the exact decomposition property, the classical test statistic for H, involving q parameters p_i = (N_i, - l)/2, i = 1,…,q, is replaced by a sequence of q - l test statistics for the sequence of tests H_i,:σ₁ =σ₂ = … =σ_i given that H_i-1 is true, i = 2,…,q. Each one of these test statistics involves two parameters only, p._i-1 = p₁ + … + p_i-1 and p_i. We also use the exact decomposition property to test equality of the “direction parameters” for q sample points from the exponential distribution on the Lorentz cone. We give a table of critical values for the distribution on the three-dimensional Lorentz cone. Tables of critical values in higher dimensions can easily be computed following the same method as in dimension three.     

Inequalities for the partial sums of elliptical order statistics related to genetic selection

Let X₁,…,X_n be exchangeable normal variables with a common correlation p, and let X₍₁₎ > … > X_(n) denote their order statistics. The random variable σⁿ_i=_n—_k+1x_i, called the selection differential by geneticists, is of particular interest in genetic selection and related areas. In this paper we give results concerning a conjecture of Tong (1982) on the distribution of this random variable as a function of ρ. The same technique used can be applied to yield more general results for linear combinations of order statistics from elliptical distributions.     

A RENEWAL THEOREM IN MULTIDIMENSIONAL TIME

Let Y_l, Y₂,… be i.i.d., positive, integer-valued random variables with means, μ. Let the sequences {Y_ij, j= 1,2,…}, i= 1,…, r be independent copies of {Y₁, Y₂,…}. For n={n₁,…, n_r.}, n₁≥1, let S_n=S?ⁿ¹_k1=1= 1 …S?^nr_kr=1 Y_ik1… Y_rkr. We show that S?^N_k=1S?^∞_k1=1…S?^∞_nr=1 P[[S_n= k] ? [μ^-r N log^r-1 (N)/(r-1)!] as N →∞.     

On the asymptotic expectation and variance of the number of boundary crossings

Let X₁,X₂,… be independent and identically distributed nonnegative random variables with mean μ, and let S_n = X₁ + … + X_n. For each λ > 0 and each n ≥ 1, let A_n be the interval [λn^Y, ∞), where γ > 1 is a constant. The number of times that S_n is in A_n is denoted by N. As λ tends to zero, the asymtotic behavior of N is studied. Specifically under suitable conditions, the expectation of N is shown to be (μλ^?1)^β + o(λ^?β/2 where β = 1/(γ-1) and the variance of N is shown to be (μλ^?1)^β(βμ¹)²σ² + o(λ^?β) where σ² is the variance of X_n.     

Estimating population size with truncated sampling

Let X₁, X₂,…,X_n 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 + o_p(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.     

Locally minimax tests for multiple correlations

Let X be a normally distributed p-dimensional column vector with mean μ and positive definite covariance matrix σ. and let X α, α = 1,…, N, be a random sample of size N from this distribution. Partition X as ( X ₁, X ₍₂₎', X '₍₃₎)', where X₁ is one-dimension, X₍₂₎ is p₂- dimensional, and so 1 + p₁ + p₂ = p. Let ρ₁ and ρ be the multiple correlation coefficients of X₁ with X₍₂₎ and with ( X '₍₂₎, X '₍₃₎)', respectively. Write ρ2/2 = ρ² - ρ2/1. We shall cosider the following two problems     

Robustness of the posterior mean in normal hierarchical models

We consider the problem of robustness in hierarchical Bayes models. Let X = (X₁,X₂, … ,X_p)^τ be a random vector, the X₁ being independently distributed as N(θ₁,σ²) random variables (σ² known), while the θ₁ are thought to be exchangeable, modelled as i.i.d, N(μ,τ²). The hyperparameter µ is given a noninformative prior distribution π(μ) = 1 and τ² is assumed to be independent of µ having a distribution g(τ²) lying in a certain class of distributions g. For several g's, including e-contaminations classes and density ratio classes we determine the range of the posterior mean of θ₁ as g ranges over g.     

Some robust tests of independence in symmetrical multivariate distributions

Let X =(x)^ij=(11¹, …, X,)_T, i = l, …n, be an n X random matrix having multivariate symmetrical distributions with parameters μ, Σ. The p-variate normal with mean μ and covariance matrix is a member of this family. Let be the squared multiple correlation coefficient between the first and the succeeding p₁ components, and let p² = + be the squared multiple correlation coefficient between the first and the remaining p₁ + p₂ =p – 1 components of the p-variate normal vector. We shall consider here three testing problems for multivariate symmetrical distributions. They are (A) to test p² =0 against; (B) to test against =0, 0; (C) to test against p² =0, We have shown here that for problem (A) the uniformly most powerful invariant (UMPI) and locally minimax test for the multivariate normal is UMPI and is locally minimax as p² 0 for multivariate symmetrical distributions. For problem (B) the UMPI and locally minimax test is UMPI and locally minimax as for multivariate symmetrical distributions. For problem (C) the locally best invariant (LBI) and locally minimax test for the multivariate normal is also LBI and is locally minimax as for multivariate symmetrical distributions.     

Orderings for series and parallel systems comprising heterogeneous exponentiated Weibull-geometric components

Let X₁, …, X_n be independent random variables with X_i ～ EWG(α, β, λ_i, p_i), i = 1, …, n, and Y₁, …, Y_n be another set of independent random variables with Y_i ～ EWG(α, β, γ_i, q_i), i = 1, …, n. The results established here are developed in two directions. First, under conditions p₁ = ??? = p_n = q₁ = ??? = q_n = p, and based on the majorization and p-larger orders between the vectors of scale parameters, we establish the usual stochastic and reversed hazard rate orders between the series and parallel systems. Next, for the case λ₁ = ??? = λ_n = γ₁ = ??? = γ_n = λ, we obtain some results concerning the reversed hazard rate and hazard rate orders between series and parallel systems based on the weak submajorization between the vectors of (p₁, …, p_n) and (q₁, …, q_n). The results established here can be used to find various bounds for some important aging characteristics of these systems, and moreover extend some well-known results in the literature.     

Small samples confidencce intervals on common mean of two normal distributions variances

Let X₁,X₂,…,X_m be distributed normally with mean μ and variance σ² _X; Let Y₁,Y₂,…,Y_n be distributed normally with mean μ and variance σ² _Y; let X₁,X₂,…,X_m,Y₁,Y₂,…,Y_n be jointly independent. There have been several papers written concerning point estimation of μ for this problem, but very little is available in the literature concerning confidence intervals on the common mean μ. In this paper a method is proposed that results in a confidence interval with confidence coefficient essentially equal to a prescribed value 1 - α. The method is evaluated and compnred with other methods through the expected length of the confidence interval.     

Testing independence with additional information

Let X = (X^j : j = 1,…, n) be n row vectors of dimension p independently and identically distributed multinomial. For each j, X^j is partitioned as X^j = (X^j₁, X^j₂, X^j₃), where p_i is the dimension of X^j_i with p₁ = 1,p₁+p₂+p₃ = p. In addition, consider vectors Y^j_i, i = 1,2j = 1,…,n_i that are independent and distributed as X¹_i. We treat here the problem of testing independence between X¹₁ and X¹₃ knowing that X¹₁ and X¹₂ are uncorrected. A locally best invariant test is proposed for this problem.     

A note on the admissibility of hammersley's estimator of an integer mean

Let X₁, X₂, …, X_n be identically, independently distributed N(i,1) random variables, where i = 0, ±1, ±2, … Hammersley (1950) showed that d = [X?_n], the nearest integer to the sample mean, is the maximum likelihood estimator of i. Khan (1973) showed that d is minimax and admissible with respect to zero-one loss. This note now proves a conjecture of Stein to the effect that in the class of integer-valued estimators d is minimax and admissible under squared-error loss.     

A CHARACTERIZATION OF GEOMETRIC DISTRIBUTIONS THROUGH CONDITIONAL INDEPENDENCE

Let X₁,…, X_n be mutually independent non-negative integer-valued random variables with probability mass functions f_i(x) > 0 for z= 0,1,…. Let E denote the event that {X₁≥X₂≥…≥X_n}. This note shows that, conditional on the event E, X_i-X_i+ 1 and X_i+ 1 are independent for all t = 1,…, k if and only if X_i (i= 1,…, k) are geometric random variables, where 1 ≤k≤n-1. The k geometric distributions can have different parameters θ_i, i= 1,…, k.