A Lower Bound to the Measure of Optimality for Main Effect Plans in the Symmetric Factorial Experiments

A design d is called D-optimal if it maximizes det(M _d ), and is called MS-optimal if it maximizes tr(M _d ) and minimizes tr[(M _d )²] among those which maximize tr(M _d ), where M _d stands for the information matrix produced from d under a given model. In this article, we establish a lower bound for tr[(M _d )²] with respect to a main effects model, where d is an s-level symmetric orthogonal array of strength at least one. Non isomorphic two level MS-optimal orthogonal arrays of strength one with N = 10, 14, and 18 runs, non isomorphic three level MS-optimal orthogonal arrays of strength one with N = 6, 12, and 15 runs and non isomorphic four level MS-optimal orthogonal arrays of strength one with N = 12 runs are also presented.     

On tail parameter estimation in certain point process models

Let μ(ds, dx) denote Poisson random measure with intensity dsG(dx) on (0, ∞) × (0, ∞), for a measure G(dx) with tails varying regularly at ∞. We deal with estimation of index of regular variation α and weight parameter ξ if the point process is observed in certain windows K_n = [0, T_n] × [Y_n, ∞), where Y_n → ∞ as n → ∞. In particular, we look at asymptotic behaviour of the Hill estimator for α. In certain submodels, better estimators are available; they converge at higher speed and have a strong optimality property. This is deduced from the parametric case G(dx) = ξαx^−α−1 dx via a neighbourhood argument in terms of Hellinger distances.     

A Strong Invariance Theorem of the Tail Empirical Copula Processes

We study the behavior of bivariate empirical copula process 𝔾 _n (·, ·) on pavements [0, k _n /n]² of [0, 1]², where k _n is a sequence of positive constants fulfilling some conditions. We provide a upper bound for the strong approximation of 𝔾 _n (·, ·) by a Gaussian process when k _n /n↘γ as n → ∞, where 0 ≤ γ ≤1.     

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.     

Optimum invariant tests for random manova models

Consider the canonical-form MANOVA setup with X: n × p = (+ E, X_i n_i × p, i = 1, 2, 3, M_i: n_i × p, i = 1, 2, n₁ + n₂ + n₃) p, where E is a normally distributed error matrix with mean zero and dispersion I_n (> 0 (positive definite). Assume (in contrast with the usual case) that M₁i is normal with mean zero and dispersion I_n1) and M₂2 is either fixed or random normal with mean zero and different dispersion matrix I_n2 (being unknown. It is also assumed that M₁ E, and M₂ (if random) are all independent. For testing H₀) = 0 versus H₁: (> 0, it is shown that when either n₂ = 0 or M₂ is fixed if n₂ > 0, the trace test of Pillai (1955) is uniformly most powerful invariant (UMPI) if min(n₁, p)= 1 and locally best invariant (LBI) if min(n₁ p) > 1 underthe action of the full linear group Gl (p). When p > 1, the LBI test is also derived under a somewhat smaller group G_T(p) of p × p lower triangular matrices with positive diagonal elements. However, such results do not hold if n₂ > 0 and M₂ is random. The null, nonnull, and optimality robustness of Pillai's trace test under Gl(p) for suitable deviations from normality is pointed out.     

Prediction based on linear combinations of order statistics and bivariate concomitants in the case of multivariate elliptical distributions

In this paper, by considering a (3n+1) -dimensional random vector (X₀, X^T, Y^T, Z^T)^T having a multivariate elliptical distribution, we derive the exact joint distribution of (X₀, a^TX_(n), b^TY_[n], c^TZ_[n])^T, where a, b, c∈?ⁿ, X_(n)=(X₍₁₎, …, X_(n))^T, X₍₁₎<···<X_(n), is the vector of order statistics arising from X, and Y_[n]=(Y_[1], …, Y_[n])^T and Z_[n]=(Z_[1], …, Z_[n])^T denote the vectors of concomitants corresponding to X_(n) ((Y_[r], Z_[r])^T, for r=1, …, n, is the vector of bivariate concomitants corresponding to X_(r)). We then present an alternate approach for the derivation of the exact joint distribution of (X₀, X_(r), Y_[r], Z_[r])^T, for r=1, …, n. We show that these joint distributions can be expressed as mixtures of four-variate unified skew-elliptical distributions and these mixture forms facilitate the prediction of X_(r), say, based on the concomitants Y_[r] and Z_[r]. Finally, we illustrate the usefulness of our results by a real data.     

Improved minimax estimation of a normal precision matrix

Let S_{p × p} have a Wishart distribution with parameter matrix Σ and n degrees of freedom. We consider here the problem of estimating the precision matrix Σ^?1 under the loss functions L₁(σ) tr (σ) - log |σ| and L₂(σ) = tr (σ). James-Stein-type estimators have been derived for an arbitrary p. We also obtain an orthogonal invariant and a diagonal invariant minimax estimator under both loss functions. A Monte-Carlo simulation study indicates that the risk improvement of the orthogonal invariant estimators over the James-Stein type estimators, the Haff (1979) estimator, and the “testimator” given by Sinha and Ghosh (1987) is substantial.     

Admissible minimax model selection rule

This paper considers the problem of choosing one between the simple model N(0, I_d) and the full model N(0 I_d) based on the observation X from N(θ I_d) where X, θεR^d, 0 is the null vector in R^d and I_d is the d×d identity matrix. It is shown that the selection rule which chooses the full model if |x| > a_o , for some a₀ > 0 and the simple model otherwise is an admissible minimax model selection rule relative to a loss function which takes into account both inaccuracy and complexity.     

Monotonic minimax estimators of a 2×2 covariance matrix

Let S : 2 × 2 have a nonsingular Wishart distribution with unknown matrix σ and n degrees of freedom. For estimating σ two families of mimmax estimators, with respect to the entropy loss, are presented. These estimators are of the form σ(S) = Rø(L)R^t where R is orthogonal, L and Φ are diagonal, and RLR^T = S. Conditions under which the components of Φ and L follow the same order relation [i.e., writing Φ = diag(Φ₁,Φ₂) and L = diag(l₁,/₂) with l₁ ≥ l₂, we have Φ₁ ≥ Φ₂] are established. Comparisons with Stein's estimators and other orthogonally invariant estimators are discussed.     

THE CONVERGENCE RATE OF SEQUENTIAL FIXED-WIDTH CONFIDENCE INTERVALS FOR REGULAR FUNCTIONALS

Let X₁X₂,.be i.i.d. random variables and let U_n= (n r)^-1S?^(n,r) h (X_i1,., X_ir,) be a U-statistic with EU_n= v, v unknown. Assume that g(X₁) =E[h(X₁,.,X_r) - v |X₁]has a strictly positive variance s?². Further, let a be such that φ(a) - φ(-a) =α for fixed α, 0 < α < 1, where φ is the standard normal d.f., and let S²_n be the Jackknife estimator of n Var U_n. Consider the stopping times N(d)= min {n: S²_n: + n^-1≤²a^-2},d > 0, and a confidence interval for v of length 2d,of the form I_n,d= [U_n,-d, Un + d]. We assume that Var U_n is unknown, and hence, no fixed sample size method is available for finding a confidence interval for v of prescribed width 2d and prescribed coverage probability α Turning to a sequential procedure, let I_N(d),d be a sequence of sequential confidence intervals for v. The asymptotic consistency of this procedure, i.e. lim_{d → 0}P(v ∈ I_N(d),d)=α follows from Sproule (1969). In this paper, the rate at which |P(v ∈ I_N(d),d) converges to α is investigated. We obtain that |P(v ∈ I_N(d),d) - α| = 0 (d^{1/2-(1+k)/2(1+m)}), d → 0, where K = max {0,4 - m}, under the condition that E|h(X₁, X_r)|^m < ∞m > 2. This improves and extends recent results of Ghosh & DasGupta (1980) and Mukhopadhyay (1981).     

Tree Structured QBD Markov Chains and Tree‐Like QBD Processes

Abstract

In this paper, we show that an arbitrary tree structured quasi‐birth–death (QBD) Markov chain can be embedded in a tree‐like QBD process with a special structure. Moreover, we present an algebraic proof that applying the natural fixed point iteration (FPI) to the nonlinear matrix equation V = B + ∑ _s=1 ^d U _s (I ? V)^?1 D _s that solves the tree‐like QBD process, is equivalent to the more complicated iterative algorithm presented by Yeung and Alfa (1996).     

Estimated Generalized Least Squares for a Heteroscedastic Regression Model

This paper considers the general linear regression model yc = X₁β+u_t under the heteroscedastic structure E(u_t) = 0, E(u²) =σ²- (X_tβ)², E(u_t u_s) = 0, tæs, t, s= 1, T. It is shown that any estimated GLS estimator for β is asymptotically equivalent to the GLS estimator under some regularity conditions. A three-step GLS estimator, which calls upon the assumption E(u_t²) =s?²(X,β)² for the estimation of the disturbance covariance matrix, is considered.     

Maximum Likelihood Estimation in the Bivariate Binomial (0,1) Distribution:Application TO 2×2 Tables

We consider a 2×2 contingency table, with dichotomized qualitative characters (A,A) and (B,B), as a sample of size n drawn from a bivariate binomial (0,1) distribution. Maximum likelihood estimates p?₁p?₂ and p? are derived for the parameters of the two marginals p₁p₂ and the coefficient of correlation. It is found that p? is identical to Pearson's (1904)?=(χ²/n)½, where ?² is Pearson's usual chi-square for the 2×2 table. The asymptotic variance-covariance matrix of p?_lp?₂and p is also derived.     

On the construction of orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set

Complete sets of orthogonal F-squares of order n = s^p, where g is a prime or prime power and p is a positive integer have been constructed by Hedayat, Raghavarao, and Seiden (1975). Federer (1977) has constructed complete sets of orthogonal F-squares of order n = 4t, where t is a positive integer. We give a general procedure for constructing orthogonal F-squares of order n from an orthogonal array (n, k, s, 2) and an OL(s, t) set, where n is not necessarily a prime or prime power. In particular, we show how to construct sets of orthogonal F-squares of order n = 2s^p, where s is a prime or prime power and p is a positive integer. These sets are shown to be near complete and approach complete sets as s and/or p become large. We have also shown how to construct orthogonal arrays by these methods. In addition, the best upper bound on the number t of orthogonal F(n, λ₁), F(n, λ₂), …, F(n, λ₁) squares is given.     

Empirical Bayes Testing for the Mean Lifetime of Exponential Distributions: Unequal Sample Sizes Case

This paper deals with an empirical Bayes testing problem for the mean lifetimes of exponential distributions with unequal sample sizes. We study a method to construct empirical Bayes tests {δ* _nl + 1,n }^∞ _n = 1 for the sequence of the testing problems. The asymptotic optimality of {δ* _nl + 1,n }^∞ _n = 1 is studied. It is shown that the sequence of empirical Bayes tests {δ* _nl + 1,n }^∞ _n = 1 is asymptotically optimal, and its associated sequence of regrets converges to zero at a rate (ln n)^4M?1/n, where M is an upper bound of sample sizes.     

Bayes and Empirical Bayes Two-Stage Procedures for Sampling Inspection

A batch of M items is inspected for defectives. Suppose there are d defective items in the batch. Let d ₀ be a given standard used to evaluate the quality of the population where 0 < d ₀ < M. The problem of testing H ₀: d < d ₀ versus H ₁: d ≥ d ₀ is considered. It is assumed that past observations are available when the current testing problem is considered. Accordingly, the empirical Bayes approach is employed. By using information obtained from the past data, an empirical Bayes two-stage testing procedure is developed. The associated asymptotic optimality is investigated. It is proved that the rate of convergence of the empirical Bayes two-stage testing procedure is of order O (exp(? c? n)), for some constant c? > 0, where n is the number of past observations at hand.     

SOME LAW OF THE ITERATED LOGARITHM TYPE RESULTS FOR THE EMPIRICAL PROCESS

We consider Z^±_n= sup_{0< t ≤ 1/2}2 U^±_n (t)/(t(1- t))^1/2, where + and -denote the positive and negative parts respectively of the sample paths of the empirical process U_n. U^±_n and U_n are seen to behave rather differently, which is tied to the asymmetry of the binomial distribution, or to the asymmetry of the distribution of small order statistics. Csáki (1975) showed that log Z^±_n/log₂n is the appropriate normalization for a law of the iterated logarithm (LIL) for Z^±_n we show that Z^-_n/(2 log₂n)^1/2 is the appropriate normalization for Z^-_n. Csörgö & Révész (1975) posed the question: if we replace the sup over (0,1/2) above, by -the sup over [a_n, 1-a_n] where a_n→0, how fast can a_n→0 and still have |Z_n|/(2 log₂n)^1/2 maintain a finite lim sup a.s.? This question is answered herein. The techniques developed are then used in Section 4 to give an interesting new proof of the upper class half of a result of Chung (1949) for |U_n(t)|. The proofs draw heavily on James (1975); two basic inequalities of that paper are strengthened to their potential, and are felt to be of independent interest.     

Algorithms for the construction of optimizing distributions

We will consider the following problem.Maximise Φ(p)over P={p=(p₁,P₂,…,p_j):P_j≧0,∑p_j=1}". We require to calcute an optimizing distribution. Examples arise in optimal regression design,maximum likelihood estimation and stratified sazmpling problems. A class of multiplicative algorithms, indexed by functions which depend on the derivatives of Φ(·)is considered for solving this problem.Iterations are of the form:p_j ^(r+1)αp_j ^(r)f(x_j ^(r)), where x_{j (r)}=d_j ^(r) or F_j ^(r)and d_j ^(r)=?Φ/?p_j While F_j ^(r)=D_j ^(r)?∑p_i ^(r)d_i ^(r) (a directional derivative)at p=p^(r)f(·)satisfies some suitable properties and may depend on one or more free parameters. These iterations neatly submit to the constraints ofv the problem. Some results will be reported and extensions to problems dependin on two or more distributions and to problems with additional constraints will be considered.     

Minimal orthogonal main-effect plans and partial replication

This work extends Jacroux's results (Jacroux, Technometrics 34 (1992) 92–96) involving minimal orthogonal main-effect plans (OMEP's). Tables of correct minimal numbers of runs of OMEP's are also provided. In addition, a general method is proposed to construct a class of minimal OMEP's. Moreover, a construction method is provided to obtain partially replicated OMEP's (PROMEP's). Our results provide more duplicate points for some plans than Jacroux's results on PROMEP's (Jacroux Technometrics 35 (1993) 32–36). In addition, a method is derived to construct s₁ × (s₁ − 1)²∥s²₁ minimal PROMEP's with maximal duplicate points.     

Embedding of orthogonal arrays of strength two and deficiency greater than two

Let x ≥ 0 and n ≥ 2 be integers. Suppose there exists an orthogonal array $\text{A(n, q, μ}{}^1\text{)}$ of strength 2 in n symbols with q rows and $\text{n}{}^2\text{μ}{}^1$ columns where $\text{q = q}{}^1\text{? d, q}{}^1\text{= n}{}^2\text{x + n + 1}$, $\text{μ}{}^1\text{= (n ?}\text{1}\text{)x +}\text{1}$ and d is a positive integer. Then d is called the deficiency of the orthogonal array. The question of embedding such an array into a complete array $\text{A(n, q}{}^1\text{, μ}{}^1\text{)}$ is considered for the case d ≥ 3. It is shown that for d = 3 such an embedding is always possible if n ≥ 2(d ? 1)²(2d² ? 2d + 1). Partial results are indicated if d ≥ 4 for the embedding of a related design in a corresponding balanced incomplete block design.