Two ge neral classes of search designs for factor screening experiments with factors at three levels

In this paper, we are presenting general classes of factor screening designs for identifying a few important factors from a list of m (≥ 3) factors each at three levels. A design is a subset of 3^m possible runs. The problem of finding designs with small number of runs is considered here. A main effect plan requires at least (2m + 1) runs for estimating the general mean, linear and quadratic effects of m factors. An orthogonal main effect plan requires, in addition, the number of runs as a multiple of 9. For example, when m=5, a main effect plan requires at least 11 runs and an orthogonal main effect plan requires 18 runs. Two general factor screening designs presented here are nonorthogonal designs with (2m− 1) runs. These designs, called search designs permit us to search for and identify at most two important factors out of m factors under the search linear model introduced in Srivastava (1975). For example, when m=5, the two new plans given in this paper have 9 runs, which is a significant improvement over an orthogonal main effect plan with 18 runs in terms of the number of runs and an improvement over a main effect plan with at least 11 runs. We compare these designs, for 4≤m≤ 10, using arithmetic and geometric means of the determinants, traces, and maximum characteristic roots of certain matrices. Two designs D1 and D2 are identical for m=3 and this design is an optimal design in the class of all search designs under the six criteria discussed above. Designs D1 and D2 are also identical for m=4 under some row and column permutations. Consequently, D1 and D2 are equally good for searching and identifying one important factor out of m factors when m=4. The design D1 is marginally better than the design D2 for searching and identifying one important factor out of m factors when m=5, … , 10. The design D1 is marginally better than the D2 for searching and identifying two important factors out of m factors when m=5, 7, 9. The design D2 is somewhat better than the design D1 for m=6, 8. For m=10, D1 is marginally better than D2 w.r.t. the geometric mean and D2 is marginally better than D1 w.r.t. the arithmetic mean of the maximum characteristic roots.     

α-optimal sampling plans for lot-by-lot defects inspection

Investigations on acceptance sampling have attached rather few importance to defects inspection. For modern quality control, both the steadily increasing complexity of products and the need for differentiated cost calculation involve a clear demand for economic defects sampling in its practically most relevant form: lot-by-lot single sampling plans, where the OC (lot OC) is considered as a function of lot sizeN, sample sizen, acceptance numberc, number of defects in the lotK. Starting from an appropriate lot OC function, the present paper develops an economic cost and control model and an economic objective function for single defects sampling plans by adapting theα-optimal sampling scheme, introduced by E. von Collani for defectives inspection, to the purposes of defects inspection. A simple and accurate approximation ofα-optimal defects plans is derived by means of a Poisson approximation of the lot OC function.     

Optimum allocation in Bayesian stratified two-phase sampling when there arem-attributes

The problem of optimally allocating a sample amongk-strata at the second phase of a two-phase sampling procedure when the sampling is for proportion has been discussed byNewbold [1971] and then by the author [Khan, 1972].This paper deals with optimum allocation of the sample amongk-strata at the second phase of a two-phase sampling procedure, when the sampling is form-attributes. The problem is to estimate the proportion of each attribute in the population. Here (m–1) dimensional Dirichlet distribution is taken as the prior distribution.     

Mixed systems with minimal and maximal lifetime variances

We consider the mixed systems composed of a fixed number of components whose lifetimes are i.i.d. with a known distribution which has a positive and finite variance. We show that a certain of the k-out-of-n systems has the minimal lifetime variance, and the maximal one is attained by a mixture of series and parallel systems. The number of the k-out-of-n system, and the probability weights of the mixture depend on the first two moments of order statistics of the parent distribution of the component lifetimes. We also show methods of calculating extreme system lifetime variances under various restrictions on the system lifetime expectations, and vice versa.     

Estimation and testing of crossing‐points in fixed design regression

Many phenomena in the life sciences can be analyzed by using a fixed design regression model with a regression function m that exhibits a crossing‐point in the following sense: the regression function runs below or above its mean level, respectively, according as the input variable lies to the left or to the right of that crossing‐point, or vice versa. We propose a non‐parametric estimator and show weak and strong consistency as long as the crossing‐point is unique. It is defined as maximizing point arg max of a certain marked empirical process. For testing the hypothesis H₀ that the regression function m actually is constant (no crossing‐point), a decision rule is designed for the specific alternative H₁ that m possesses a crossing‐point. The pertaining test‐statistic is the ratio max/argmax of the maximum value and the maximizing point of the marked empirical process. Under the hypothesis the ratio converges in distribution to the corresponding ratio of a reflected Brownian bridge, for which we derive the distribution function. The test is consistent on the whole alternative and superior to the corresponding Kolmogorov–Smirnov test, which is based only on the maximal value max. Some practical examples of possible applications are given where a certain study about dental phobia is discussed in more detail.     

Multivariate t and beta distributions associated with the multivariate F distribution

Relationships between F, skew t and beta distributions in the univariate case are in this paper extended in a natural way to the multivariate case. The result is two new distributions: a multivariate t/skew t distribution (on ℜ^m) and a multivariate beta distribution (on (0,1)^m). A special case of the former distribution is a new multivariate symmetric t distribution. The new distributions have a natural relationship to the standard multivariate F distribution (on (ℜ⁺)^m) and many of their properties run in parallel. We look at: joint distributions, mathematically and graphically; marginal and conditional distributions; moments; correlations; local dependence; and some limiting cases. Received: March 2001     

Strong representation of the presmoothed quantile function estimator for censored data

We consider lifetime data subject to right random censorship. In this context, this paper deals with the topic of estimating the distribution function of the lifetime and the corresponding quantile function. As it has been shown that the classical Kaplan–Meier estimator of the distribution function can be improved by means of presmoothing ideas, we introduce a quantile function estimator via the presmoothed distribution function estimator studied by Cao et al. [Journal of Nonparametric statistics, Vol. 17 (2005) pp. 31–56.] The main result of this paper is an almost sure representation of this presmoothed estimator. As a consequence, its strong consistency and asymptotic normality are established. The performance of this new quantile estimator is analyzed in a simulation study and applied to a real data example.     

Kostenoptimale Prüfpläne für ein quantitatives Merkmal

Summary A sample inspection plan is said to be optimal in the sense of the minimax regret principle, if it minimizes the difference between the expected total costs and the unavoidable costs. The results of this article can be used to calculate such sample inspection plans for a quantitative quality control with one-sided tolerance limits and known or unknown variance of the test variate. As an example of practical importance the case of a normal variate with unknown variance is considered. Formulae are given to estimate the error that arises if the assumed distribution of the test variate differs from the actual distribution.     

Existence of balanced sampling plans avoiding cyclic distances

This article considers balanced sampling plans avoiding adjacent units. It is shown, using linear algebraic and design theoretic tools, that such plans exist for all sufficiently large populations. More generally, if any set of m circular distances is to be avoided in balanced sampling with subsets of size k, then such a plan exists, provided the population size N is at least mk ³. Research of the Peter J. Dukes is supported by NSERC.     

On bootstrapping periodic random arrays with increasing period

The aim of the paper is to determine when the periodic block bootstrap, procedure introduced by Chan et al. (Technometrics 46(2):215–224, 2004), can be applied to arrays of random variables. Formal consistency is obtained under α-mixing or m-dependence conditions together with the assumption that the length of the period tends to infinity. On the other hand, if the period is constant, inconsistency is shown. The performance of periodic block bootstrap is also compared in simulations with moving block bootstrap. It is suggested that for the case of long-period data the first method is more effective and much more stable with respect to the length of the block size.     

The Limit Distribution of the Bootstrap for the Unit Root Test Statistic when the Residuals are Dependent

In this article, the unit root test for the AR(1) model with dependent residuals is considered. We adopt a bootstrap procedure to bootstrap the residuals with bootstrap sample size m less than the size n of the original sample. Under the assumptions that m → ∞ and m/n → 0, the convergence in probability of the bootstrap distribution function is established. Research supported by National Natural Science Foundation of China (No. 10471126)     

On bootstrapping the mode in the nonparametric regression model with random design

In the nonparametric regression model with random design and based on i.i.d. pairs of observations (X _i, Y _i), where the regression function m is given by m(x)=?(Y _i|X _i=x), estimation of the location θ (mode) of a unique maximum of m by the location of a maximum of the Nadaraya-Watson kernel estimator for the curve m is considered. In order to obtain asymptotic confidence intervals for θ, the suitably normalized distribution of is bootstrapped in two ways: we present a paired bootstrap (PB) where resampling is done from the empirical distribution of the pairs of observations and a smoothed paired bootstrap (SPB) where the bootstrap variables are generated from a smooth bivariate density based on the pairs of observations. While the PB requires only relatively small computational effort when carried out in practice, it is shown to work only in the case of vanishing asymptotic bias, i.e. of “undersmoothing” when compared to optimal smoothing for mode estimation. On the other hand, the SPB, although causing more intricate computations, is able to capture the correct amount of bias if the pilot estimator for m oversmoothes. Received: May 2000     

Bestimmung kostenoptimaler Parameter für den kontinuierlichen Stichprobenplan vonDodge

Summary Dodge's sampling plan for a continuous production process is determined by two constantsi andk. Under some assumptions about the statistical properties of the production process the average loss due to the costs for inspection and for replacing defective units by good ones is described by a loss function. It is shown that there are always constantsi andk which minimize the loss function in a special sense. Tables of these optimum parameters are computed.     

Inequalities compared

We know that the partial means m_rof a sequence of i.i.d. standardized random variables tend to 0 with probability 1. If we want P{m_k≥εfor some k ≥r}≤δ for given positive ε and δ, how large should we take r? Several (strong) inequalities for the distribution of partial sums providing an answer to this question can be found in the literature (Hájek -Rényi Robbins , Khan ). Furthermore there exist wellknown (weak) inequalities (Bienaymé -Chebyshev , Bernstein , Okamoto ) that give us values of rfor which P{m_r≥ε}≤δ. We compare these inequalities and illustrate them with numerical results for a fixed choice ofε and δ. After a general survey and introduction in section 1, the normal and the binomial distribution are considered in more detail in the sections 2 and 3, while in section 4 it is shown that the strong inequality essentially due to Robbins can give an inferior result for particular distributions.     

Robust explicit estimators of Weibull parameters

The Weibull distribution plays a central role in modeling duration data. Its maximum likelihood estimator is very sensitive to outliers. We propose three robust and explicit Weibull parameter estimators: the quantile least squares, the repeated median and the median/Q _n estimator. We derive their breakdown point, influence function, asymptotic variance and study their finite sample properties in a Monte Carlo study. The methods are illustrated on real lifetime data affected by a recording error.     

Planning step-stress life test with progressively type I group-censored exponential data

Accelerated life testing of products is used to get information quickly on their lifetime distributions. This paper discusses a k -stage step-stress accelerated life test under progressive type I censoring with grouped data. An exponential lifetime distribution with mean life that is a log-linear function of stress is considered. A cumulative exposure model is also assumed. We use the maximum likelihood method to obtain the estimators of the model parameters. The methods for obtaining the optimum test plan are investigated using the variance-optimality and D-optimality criteria. Some numerical studies are discussed to illustrate the proposed criteria.     

Limit theorems for the spacings of weak records

Let W(1), W(2), . . . be weak record values obtained from a sample of independent variables with common discrete distribution. In the present paper, we derive weak and strong limit theorems for the spacings W(n + m) − W(n), m ≥ 1, n → ∞.     

D-optimal weighing designs for six objects

For all integers m≥6, we determine the maximum value of det X ^T X, where X is an m×6 (0, 1)-matrix, and exhibit (D-optimal) matrices X for which the maximum occurs. For D-optimal matrices X, the uniqueness of the Gram matrix X ^T X is discussed. Received: May 2000     

Compensatory inter vivos gifts

Parents' transfer motives are important for understanding, e.g., macroeconomics, income (re)distribution, savings, and public finance. Using data from six biennial waves of the Health and Retirement Study 1992–2002, we estimate censored regression models with nested multilevel error components. First, we interpret our findings that inter vivos transfers from parents to children are gifts, rather than temporary help to overcome liquidity constraints. Second, inter vivos gifts are compensatory in the sense that lifetime poorer children will receive higher transfers than their lifetime richer siblings. Third, inter vivos gifts do not, however, make up the entire difference in lifetime incomes. Copyright © 2009 John Wiley & Sons, Ltd.     

Increasing sample sizes do not always increase the power of UMPU-tests for 2×2 tables

We consider the uniformly most powerful unbiased (UMPU) one-sided test for the comparison of two proportions based on sample sizes m and n, i.e., the randomized version of Fisher's exact one-sided test. It will be shown that the power function of the one-sided UMPU-test based on sample sizes m and n can coincide with the power function of the UMPU-test based on sample sizes m+1 and n for certain levels on the entire parameter space. A characterization of all such cases with identical power functions is derived. Finally, this characterization is closely related to number theoretical problems concerning Fermat-like binomial equations. Some consequences for Fisher's original exact test will be discussed, too.