ON TESTING AGAINST RESTRICTED ALTERNATIVES FOR THE VARIANCES OF GAUSSIAN MODELS

Let π₁,…,π_p be p independent normal populations with means μ₁…, μ_p and variances σ²₁,…, σ²_p respectively. Let X(n_i) be a simple random sample of size n_i from π_i, i = 1,…,p. Given the simple random samples X(n₁),…, X(n_p) from π₁,…,π_p respectively, a test has been proposed for testing the homogeneity of variances H₀: σ²₁=…σ²_p, against the restricted alternative, H₁: σ²₁≥…≥σ²_p, with at least one strict inequality. Some properties of the test are discussed and critical values are tabulated.     

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.     

ON TRANSITION PROBABILITIES OF SKIP-FREE MARKOV CHAINS

Consider an ergodic Markov chain X(t) in continuous time with an infinitesimal matrix Q = (q_ij) defined on a finite state space {0, 1,…, N}. In this note, we prove that if X(t) is skip-free positive (negative, respectively), i.e., q_ij, = 0 for j > i+ 1 (i > j+ 1), then the transition probability p_ij(t) = Pr[X(t)=j | X(0) =i] can be represented as a linear combination of p_0N(t) (p^(m)_(N0)(t)), 0 ≤ m ≤N, where f^(m)(t) denotes the mth derivative of a function f(t) with f⁽⁰⁾(t) =f(t). If X(t) is a birth-death process, then p_ij(t) is represented as a linear combination of p_0N^(m)(t), 0 ≤m≤N - |i-j|.     

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.     

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.     

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.     

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 $$ $aL₀ > 1 where$aL₀ = P_p=0 $$.     

Extreme probabilities for contingency tables under row and column independence with application to fisher's exact test

Theorerms are proved for the maxima and minima of IIR_i!/IIC_j!/T!II_yij ! over r× c contingcncy tables Y=(y_ij) with row sums R₁,…,R_r, column sums C₁,…,C_c, and grand total T. These results are imlplemented into the network algorithm of Mehta and Patel (1983) for computing the P-value of Fisher's exact test for unordered r×c contingency tables. The decrease in the amount of computing time can be substantial when the column sums are very different.     

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 →∞.     

Pooling operators with the marginalization property

In this paper, we consider the problem of combining a number of opinions which have been expressed as probability measures P₁, …, P_n, over some space. It is shown that a pooling formula which has the marginalization property of McConway (1981) must be of the form T = Σⁿ_i=₁W_i P_i + (1 - Σⁿ_i =₁W_i)Q, where Q is an arbitrary measure and W₁, …, W_n ϵ [—1,1] are weights such that| Σ_J Σ _j w_j | ≤ 1 for every subset J of {1, …, n}. If, in addition, T is required to preserve the independence of arbitrary events A and B whenever these events are independent under each P_i, then either T = P_i for some 1 ≤ i ≤ n or T = Q, in which case Q takes values in {0, l}.     

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.     

Reliability analysis of systems with components having two dependent subcomponents

In this article, a system that consists of n independent components each having two dependent subcomponents (A_i, B_i), i = 1, …, n is considered. The system is assumed to compose of components that have two correlated subcomponents (A_i, B_i), and functions iff both systems of subcomponents A₁, A₂, …, A_n and B₁, B₂, …, B_n work under certain structural rules. The expressions for reliability and mean time to failure of such systems are obtained. A sufficient condition to compare two systems of bivariate components in terms of stochastic ordering is also presented.     

Exact distributions associated with an i×j×k contingency table

In this paper an exact distributional framework is developed for analysing an IxJxK contingency table. It is shown that for the case of hypotheses H₀:P_ijk=P_i..P._j./K and H₀:P_ijk =P_i..P._j.P.._k the exact distributional results do not follow as simple extensions of the corresponding results obtained for an I×J table under the hypothesis of independence. From the factorial moment generating functions, expressions for the covariance matrices in terms of the Kronecker products of matrices, are presented. These expressions give indications whether or not Pearson's chi-square statistic should be corrected by the factor (n?1)/n or not. Marginal and conditional distributions are considered briefly and important differences with regard to the resuits for marginal and conditional distributions for an IxJ table are mentioned.     

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.     

Bayesian analysis of growth curves with AR(1) dependence

In this paper we consider Bayesian analysis of the generalized growth curve model when the covariance matrix Σ = σ²C where C = (ϱ^{i − j}), σ² > 0 and −1 < ϱ < 1 are unknown. We consider both parameter estimation and prediction of future values. Results are illustrated with real and simulated data.     

A Variation on the Coupon Collecting Problem

The classical coupon collector's problem is considered, where each new coupon collected is type i with probability p_i , ∑ ⁿ _{i = 1} p_i = 1. Suppose coupons are collected in a sequence of independent trials. An expression is developed for the probability that all coupon types i, i ≠ j, have been collected prior to collecting r ? 1 coupons of type j in the sequence of trials. Given two different coupon subsets A, B of {1, 2, …, n}, the foregoing is then generalized to an expression for the probability that s ? 1 copies of A appear in the sequence of trials before r ? 1 copies of B. Some computational considerations are discussed.