Empirical Bayes testing for guarantee lifetime: Non identical components case

This article considers an empirical Bayes testing problem for the guarantee lifetime in the two-parameter exponential distributions with non identical components. We study a method of constructing empirical Bayes tests under a class of unknown prior distributions for the sequence of the component testing problems. The asymptotic optimality of the sequence of empirical Bayes tests is studied. Under certain regularity conditions on the prior distributions, it is shown that the sequence of the constructed empirical Bayes tests is asymptotically optimal, and the associated sequence of regrets converges to zero at a rate O(n^{? 1 + 1/[2(r + α) + 1]}) for some integer r ? 0 and 0 ? α ? 1 depending on the unknown prior distributions, where n is the number of past data available when the (n + 1)st component testing problem is considered.     

Some results on constructing two-level block designs with general minimum lower order confounding

Block designs are widely used in experimental situations where the experimental units are heterogeneous. The blocked general minimum lower order confounding (B-GMC) criterion is suitable for selecting optimal block designs when the experimenters have some prior information on the importance of ordering of the treatment factors. This paper constructs B-GMC 2^{n ? m}: 2^r designs with 5 × 2^l/16 + 1 ? n ? (N ? 2^l) < 2^{l ? 1} for l(r + 1 ? l ? n ? m ? 1), where 2^{n ? m}: 2^r denotes a two-level regular block design with N = 2^{n ? m} runs, n treatment factors, and 2^r blocks. With suitable choice of the blocking factors, each B-GMC block design has a common specific structure. Some examples illustrate the simple and effective construction method.     

Maximizing survival time in a random walk on an interval

A gambler buys N tokens that enable him to play N rounds of the following game. A symmetric random walk on a discrete interval { ? r, …, r} starts from the point 0. The gambler knows only the number of steps made so far, but is unaware of the current position of the walk. Once the walk hits one of the barriers ? r or r for the first time in the current round, the round ends with no payoff. The gambler can start a new round by inserting a new token, if there are any tokens left. The gambler can end the game at any time getting the payoff equal to the number of steps made in the current round. We find the optimal stopping strategy for this game and calculate the expected payoff once the optimal strategy is applied.     

Equineighboured balanced nested row-column designs

This paper presents equineighboured balanced nested row-column designs for v treatments arranged in b blocks each comprising pq units further grouped into p rows and q columns. These are two-dimensional designs with the property that all pairs of treatments are neighbours equally frequently at all levels in both rows and columns. Methods of construction are given both for designs based on Latin squares and those where pq≤v. Cyclic equineighboured designs are defined and tabulated for 5≤v≤10, p≤3, q≤5, where r=bpq/v is the number of replications of each treatment.     

Robust model-based sampling designs

We investigate methods for the design of sample surveys, and address the traditional resistance of survey samplers to the use of model-based methods by incorporating model robustness at the design stage. The designs are intended to be sufficiently flexible and robust that resulting estimates, based on the designer’s best guess at an appropriate model, remain reasonably accurate in a neighbourhood of this central model. Thus, consider a finite population of N units in which a survey variable Y is related to a q dimensional auxiliary variable x. We assume that the values of x are known for all N population units, and that we will select a sample of n≤N population units and then observe the n corresponding values of Y. The objective is to predict the population total $T=\sum_{i=1}^{N}Y_{i}$ . The design problem which we consider is to specify a selection rule, using only the values of the auxiliary variable, to select the n units for the sample so that the predictor has optimal robustness properties. We suppose that T will be predicted by methods based on a linear relationship between Y—possibly transformed—and given functions of x. We maximise the mean squared error of the prediction of T over realistic neighbourhoods of the fitted linear relationship, and of the assumed variance and correlation structures. This maximised mean squared error is then minimised over the class of possible samples, yielding an optimally robust (‘minimax’) design. To carry out the minimisation step we introduce a genetic algorithm and discuss its tuning for maximal efficiency.     

A modified cluster sampling method for estimating the bacterial density in Xanthan gum

Suppose particles are randomly distributed in a certain medium, powder or liquid, which is conceptually divided into N cells. Let p_i denote the probability that a particle falls in the ith cell and Y_i denote the number of particles in the ith cell. Assume that the joint probability function of the Y_i follows a multinomial distribution with cell probabilities p_i respectively. Take n (≤N) cells at random without replacement and put each of the cells separately through a mixing mechanism of dilution and swirl. These n cells constitute the first stage samples and the number of particles in these cells are not observable. Now conceptually divide each of n cells into M subcells of equal size and let X_ij denote the number of particles in the jth subcell of the ith cell selected in the first stage; i=1,2,…,N and j=1,2,…,M. Consequently assume that the conditional joint probability function of the X_ij given Y_i=y_i follows a multinomial distribution with equal cell probabilities. Now take m (≤M) subcells at random from each of the cells selected in the first stage sample. Assume that the numbers of particles in M×N subcells are observable. The properties of the estimator of the particle density per sample unit are investigated under the modified two-stage cluster sampling method. A laboratory experiment for Xanthan Gum Products is analyzed in order to examine the appropriateness of the model assumed in this paper.     

Empirical nonparametric control charts for high-quality processes

For attribute data with (very) small failure rates often control charts are used which decide whether to stop or to continue each time r failures have occurred, for some r?1. Because of the small probabilities involved, such charts are very sensitive to estimation effects. This is true in particular if the underlying failure rate varies and hence the distributions involved are not geometric. Such a situation calls for a nonparametric approach, but this may require far more Phase I observations than are typically available in practice. In the present paper it is shown how this obstacle can be effectively overcome by looking not at the sum but rather at the maximum of each group of size r.     

On the distribution of crossings in a generalized random walk

This paper deals with the derivation of the probability distribution of the rank order statistic N¹_μ,n(r), the number of crossings of heightr(≥0) in a generalized random walk with steps 1 and ?μ by using the modified Dwass technique.     

Optimal selection and ordering of columns in supersaturated designs

Two methods to select columns for assigning factors to work on supersaturated designs are proposed. The focus of interest is the degree of non-orthogonality between the selected columns. One method is the exhaustive enumeration of selections of p columns from all k columns to find the exact optimality, while the other is intended to find an approximate solution by applying techniques used in the corresponding analysis, aiming for ease of use as well as a reduction in the large computing time required for large k with the first method. Numerical illustrations for several typical design matrices reveal that the resulting “approximately” optimal assignments of factors to their columns are exactly optimal for any p. Ordering the columns in E(s²)-optimal designs results in promising new findings including a large number of E(s²)-optimal designs.     

A-optimal design measures for one-way layouts with additive regression

For linear models with one discrete factor and additive general regression term the problem of characterizing A-optimal design measures for inference on (i) treatment effects, (ii) the regression parameters and (iii) all parameters will be considered. In any of these problems product designs can be found which are optimal among all designs, and equal weigth 1/J may be given to each of the J levels of the discrete factor. For problem (i) and (ii) the allocation of the continuous factors for the regression term should follow a suitable optimal design for the corresponding pure regression model, whereas for problem (iii) this would not give an A-optimal product design. For this problem an equivalence theorem for A-optimal product designs will be given. An example will illustrate these results. Finally, by analyzing a model with two discrete factors it will be shown that for enlarged models the best product designs may not be A-optimal.     

Optimal designs for covariates’ models

A linear model with one treatment at V levels and first order regression on K continuous covariates with values on a K-cube is considered. The D-criterion is used to judge the ‘goodness’ of any design for estimating the parameters of this model. Since this criterion is based on the determinant of the information matrix M(d) of a design d, upper bounds for |M(d)| yield lower bounds for the D-efficiency of any design d in estimating the vector of parameters in the model. We consider here only classes of designs d for which the number N of observations to be taken is a multiple of V, that is, there exists R≥2 such that N=V×R.Under these conditions, we determine the maximum of |M(d)|, and conditions under which the maximum is attained. These conditions include R being even, each treatment level being observed the same number of times, that is, R times, and N being a multiple of four. For the other cases of congruence of N (modulo 4) we further determine upper bounds on |M (d)| for equireplicated designs, i.e. for designs with equal number of observations per treatment level. These upper bounds are shown to depend also on the congruence of V (modulo 4). For some triples (N,V,K), the upper bounds determined are shown to be attained.Construction methods yielding families of designs which attain the upper bounds of |M(d)| are presented, for each of the sixteen cases of congruence of N and V.We also determine the upper bound for D-optimal designs for estimating only the treatment parameters, when first order regression on one continuous covariate is present.     

A sampling plan for one-shot systems considering destructive inspection

Abstract

The paper is concerned with an acceptance sampling problem under destructive inspections for one-shot systems. The systems may fail at random times while they are operating (as the systems are considered to be operating when storage begins), and these failures can only be identified by inspection. Thus, n samples are randomly selected from N one-shot systems for periodic destructive inspection. After storage time T, the N systems are replaced if the number of working systems is less than a pre-specified threshold k. The primary purpose of this study is to determine the optimal number of samples n*, extracted from the N for destructive detection and the optimal acceptance number k*, in the sample under the constraint of the system interval availability, to minimize the expected cost rate. Numerical experiments are studied to investigate the effect of the parameters in sampling inspection on the optimal solutions.     

Nested Hadamard difference sets

A Hadamard difference set (HDS) has the parameters (4N², 2N² − N, N² − N). In the abelian case it is equivalent to a perfect binary array, which is a multidimensional matrix with elements ±1 such that all out-of-phase periodic autocorrelation coefficients are zero. We show that if a group of the form H × Z²_p^r contains a (hp^2r, √hp^r(2√hp^r − 1), √hp^r(√hp^r − 1)) HDS (HDS), p a prime not dividing |H| = h and p^j ≡ −1 (mod exp(H)) for some j, then H × Z²_p^t has a (hp^2t, √hp^t(2√hp^t − 1), √hp^t(√hp^t − 1)) HDS for every 0⩽t⩽r. Thus, if these families do not exist, we simply need to show that H × Z²_p does not support a HDS. We give two examples of families that are ruled out by this procedure.     

Bayes A-optimal designs for comparing test treatments with a control

The problem of comparing v test treatments simultaneously with a control treatment when k, v ⩾ 3 is considered. Following the work of Majumdar (1992), we use exact design theory to derive Bayes A-optimal block designs and optimal Г-minimax designs for a more general prior assumption for the one-way elimination of heterogeneity model. Examples of robust optimal designs, highly efficient designs, and the comparisons of the approximate optimal designs that are derived by our methods and by some other existing rounding-off schemes when using Owen's procedure are also provided.     

Bayesian analysis of a multiple-recapture model

In the Bayesian analysis of a multiple-recapture census, different diffuse prior distributions can lead to markedly different inferences about the population size N. Through consideration of the Fisher information matrix it is shown that the number of captures in each sample typically provides little information about N. This suggests that if there is no prior information about capture probabilities, then knowledge of just the sample sizes and not the number of recaptures should leave the distribution of Nunchanged. A prior model that has this property is identified and the posterior distribution is examined. In particular, asymptotic estimates of the posterior mean and variance are derived. Differences between Bayesian and classical point and interval estimators are illustrated through examples.     

Selection rules based on divergences

This paper deals with a special adaptive estimation problem, namely how can one select for each set of i.i.d. data X ₁, …, X _n the better of two given estimates of the data-generating probability density. Such a problem was studied by Devroye and Lugosi [Combinatorial Methods in Density Estimation, Springer, Berlin, 2001] who proposed a feasible suboptimal selection (called the Scheffé selection) as an alternative to the optimal but nonfeasible selection which minimizes the L₁-error. In many typical situations, the L₁-error of the Scheffé selection was shown to tend to zero for n→∞ as fast as the L₁-error of the optimal estimate. This asymptotic result was based on an inequality between the total variation errors of the Scheffé and optimal selections. The present paper extends this inequality to the class of φ-divergence errors containing the L₁-error as a special case. The first extension compares the φ-divergence errors of the mentioned Scheffé and optimal selections of Devroye and Lugosi. The second extension deals with a class of generalized Scheffé selections adapted to the φ-divergence error criteria and reducing to the classical Scheffé selection for the L₁-criterion. It compares the φ-divergence errors of these feasible selections and the optimal nonfeasible selections minimizing the φ-divergence errors. Both extensions are motivated and illustrated by examples.     

On limiting distribution laws of statistics analogous to pearson's chi-square

Two test-statistics analogous to Pearson's chi-square test function - given in (1.6) and (1.7) - are investigated. These statistics utilize, apart from the number of sample elements lying in the respective intervals of the partition, their positions within the intervals too. It is shown that the test-statistics are asymptotically distributed - as the sample size N tends to infinity - according to the x ²distribution with parameter r, i.e. the number of intervals chosen. The limiting distribution of the test statistics under the null-hypothesis when N tends to the infinity and r =O(N ^α) (0<α<1), further the consistency of the tests based on these statistics is considered. Some remarks are made concerning the efficiency of the corresponding goodness of fit tests also; the authors intend to return to a more detailed treatment of the efficiency later.     

Classification of difference matrices over cyclic groups

The existence of difference matrices over small cyclic groups is investigated in this computer-aided work. The maximum values of the parameters for which difference matrices exist as well as the number of inequivalent difference matrices in each case is determined up to the computational limit. Several new difference matrices have been found in this manner. The maximum number of rows is 9 for an r ×15 difference matrix over Z₃, 8 for an r ×15 difference matrix over Z₅, and 6 for an r ×12 difference matrix over Z₆; the number of inequivalent matrices with these parameters is 5, 2, and 7, respectively.     

On a problem of counting by weighing

Suppose it is desired to obtain a large number N_s of items for which individual counting is impractical, but one can demand a batch to weigh at least w units so that the number of items N in the batch may be close to the desired number N_s. If the items have mean weight ωTH, it is reasonable to have w equal to ωTHN_s when ωTH is known. When ωTH is unknown, one can take a sample of size n, not bigger than N_s, estimate ωTH by a good estimator ω_n, and set w equal to ω_nN_s. Let R_n = Kp²N²_s/n + K_sn be a measure of loss, where K_e and K_s are the coefficients representing the cost of the error in estimation and the cost of the sampling respectively, and p is the coefficient of variation for the weight of the items. If one determines the sample size to be the integer closest to pCN_s when p is known, where C is (K_e/K_s)^1/2, then R_n will be minimized. If p is unknown, a simple sequential procedure is proposed for which the average sample number is shown to be asymptotically equal to the optimal fixed sample size. When the weights are assumed to have a gamma distribution given ω and ω has a prior inverted gamma distribution, the optimal sample size can be found to be the nonnegative integer closest to pCN_s + p²A(pC – 1), where A is a known constant given in the prior distribution.