Some properties of extreme stable laws and related infinitely divisible random variables

In the course of studying the moment sequence {nⁿ:n=0,1,…}$\{n^n:n=0,1,\dots \}$, Eaton et al. [1971. On extreme stable laws and some applications. J. Appl. Probab. 8, 794–801] have shown that this sequence, which is, indeed, the moment sequence of a log-extreme stable law with characteristic exponent γ=1$\gamma =1$, corresponds to a scale mixture of exponential distributions and hence to a distribution with decreasing failure rate. Following essentially the approach of Shanbhag et al. [1977. Some further results in infinite divisibility. Math. Proc. Cambridge Philos. Soc. 82, 289–295] we show that, under certain conditions, log-extreme stable laws with characteristic exponent γ∈[1,2)$\gamma \in [1,2)$ are scale mixtures of exponential distributions and hence are infinitely divisible and have decreasing failure rates. In addition, we study the moment problem associated with the log-extreme stable laws with characteristic exponent γ∈(0,2]$\gamma \in (0,2]$ and throw further light on the existing literature on the subject. As a by-product, we show that generalized Poisson and generalized negative binomial distributions are mixed Poisson distributions. Finally, we address some relevant questions on structural aspects of infinitely divisible distributions, and make new observations, including in particular that certain results appearing in Steutel and van Harn [2004. Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker, New York] have links with the Wiener–Hopf factorization met in the theory of random walk.     

Mantel-Haenszel estimators for irregular sparse K2×J tables

This paper deals with sparse K2×J(J>2)$K2\times J(J>2)$ tables. Projection-method Mantel–Haenszel (MH) estimators of the common odds ratios have been proposed for K2×J$K2\times J$ tables, which include Greenland's generalized MH estimator as a special case. The method projects log-transformed MH estimators for all K2×2$K2\times 2$ subtables, which were called naive MH estimators, onto a linear space spanned by log odds ratios. However, for sparse tables it is often the case that naive MH estimators are unable to be computed. In this paper we introduce alternative naive MH estimators using a graph that represents K2×J$K2\times J$ tables, and apply the projection to these alternative estimators. The idea leads to infinitely many reasonable estimators and we propose a method to choose the optimal one by solving a quadratic optimization problem induced by the graph, where some graph-theoretic arguments play important roles to simplify the optimization problem. An illustration is given using data from a case–control study. A simulation study is also conducted, which indicates that the MH estimator tends to have a smaller mean squared error than the MH estimator previously suggested and the conditional maximum likelihood estimator for sparse tables.     

Comparing several treatments with a control

Consider the partially balanced one-way layout for comparing k   treatments _μi,${\mu }_i,$1?i?k,$1?i?k,$ with a control _μ0${\mu }_0$. We propose a new test which is similar to the test statistics of Marcus [1976. The powers of some tests of the equality of normal means against an ordered alternative. Biometrika 63, 177–183]. By simulation we find that the proposed test has a good power performance when compared with other tests. Moreover, it can produce confidence intervals for _μi-_μ0,1?i?k.${\mu }_i-{\mu }_0,1?i?k.$     

Confidence intervals in regression utilizing prior information

We consider a linear regression model with regression parameter β=(_β1,…,_βp)$\beta =({\beta }_1,\dots ,{\beta }_p)$ and independent and identically N(0,σ²)$N(0,{\sigma }^2)$ distributed errors. Suppose that the parameter of interest is θ=a^Tβ$\theta =a^T\beta$ where a$a$ is a specified vector. Define the parameter τ=c^Tβ-t$\tau =c^T\beta -t$ where the vector c$c$ and the number t$t$ are specified and a$a$ and c$c$ are linearly independent. Also suppose that we have uncertain prior information that τ=0$\tau =0$. We present a new frequentist 1-α$1-\alpha$ confidence interval for θ$\theta$ that utilizes this prior information. We require this confidence interval to (a) have endpoints that are continuous functions of the data and (b) coincide with the standard 1-α$1-\alpha$ confidence interval when the data strongly contradict this prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when τ=0$\tau =0$. This minimization leads to an interval that has the following desirable properties. This interval has expected length that (a) is relatively small when the prior information about τ$\tau$ is correct and (b) has a maximum value that is not too large. The following problem will be used to illustrate the application of this new confidence interval. Consider a 2×2$2\times 2$ factorial experiment with 20 replicates. Suppose that the parameter of interest θ$\theta$ is a specified simple   effect and that we have uncertain prior information that the two-factor interaction is zero. Our aim is to find a frequentist 0.95 confidence interval for θ$\theta$ that utilizes this prior information.     

Decomposable pseudodistances and applications in statistical estimation

The aim of this paper is to introduce new statistical criteria for estimation, suitable for inference in models with common continuous support. This proposal is in the direct line of a renewed interest for divergence based inference tools imbedding the most classical ones, such as maximum likelihood, Chi-square or Kullback–Leibler. General pseudodistances with decomposable structure are considered, they allowing defining minimum pseudodistance estimators, without using nonparametric density estimators. A special class of pseudodistances indexed by α>0$\alpha >0$, leading for α↓0$\alpha \downarrow 0$ to the Kullback–Leibler divergence, is presented in detail. Corresponding estimation criteria are developed and asymptotic properties are studied. The estimation method is then extended to regression models. Finally, some examples based on Monte Carlo simulations are discussed.     

A new estimation method for Weibull-type tails based on the mean excess function

Studying the right tail of a distribution, one can classify the distributions into three classes based on the extreme value index γ$\gamma$. The class γ>0$\gamma >0$ corresponds to Pareto-type or heavy tailed distributions, while γ<0$\gamma <0$ indicates that the underlying distribution has a finite endpoint. The Weibull-type distributions form an important subgroup within the Gumbel class with γ=0$\gamma =0$. The tail behaviour can then be specified using the Weibull tail index. Classical estimators of this index show severe bias. In this paper we present a new estimation approach based on the mean excess function, which exhibits improved bias and mean squared error. The asserted properties are supported by simulation experiments and asymptotic results. Illustrations with real life data sets are provided.     

Results for two-level fractional factorial designs of resolution IV or more

The main theorem of this paper shows that foldover designs are the only (regular or nonregular) two-level factorial designs of resolution IV (strength 3) or more for n   runs and n/3?m?n/2$n/3?m?n/2$ factors. This theorem is a generalization of a coding theory result of Davydov and Tombak [1990. Quasiperfect linear binary codes with distance 4 and complete caps in projective geometry. Problems Inform. Transmission 25, 265–275] which, under translation, effectively states that foldover (or even) designs are the only regular two-level factorial designs of resolution IV or more for n   runs and 5n/16?m?n/2$5n/16?m?n/2$ factors. This paper also contains other theorems including an alternative proof of Davydov and Tombak's result.     

On E(s)-optimal supersaturated designs

A popular measure to assess 2-level supersaturated designs is the E(s²)$E(s^2)$ criterion. In this paper, improved lower bounds on E(s²)$E(s^2)$ are obtained. The same improvement has recently been established by Ryan and Bulutoglu [2007. E(s²)$E(s^2)$-optimal supersaturated designs with good minimax properties. J. Statist. Plann. Inference 137, 2250–2262]. However, our analysis provides more details on precisely when an improvement is possible, which is lacking in Ryan and Bulutoglu [2007. E(s²)$E(s^2)$-optimal supersaturated designs with good minimax properties. J. Statist. Plann. Inference 137, 2250–2262]. The equivalence of the bounds obtained by Butler et al. [2001. A general method of constructing E(s²)$E(s^2)$-optimal supersaturated designs. J. Roy. Statist. Soc. B 63, 621–632] (in the cases where their result applies) and those obtained by Bulutoglu and Cheng [2004. Construction of E(s²)$E(s^2)$-optimal supersaturated designs. Ann. Statist. 32, 1662–1678] is established. We also give two simple methods of constructing E(s²)$E(s^2)$-optimal designs.     

Optimal and efficient semi-Latin squares

An (n×n)/k$(n\times n)/k$semi-Latin square is an n×n square array in which nk distinct symbols (representing treatments) are placed in such a way that there are exactly k   symbols in each cell (row–column intersection) and each symbol occurs once in each row and once in each column. Semi-Latin squares form a class of row–column designs generalising Latin squares, and have applications in areas including the design of agricultural experiments, consumer testing, and via their duals, human–machine interaction. In the present paper, new theoretical and computational methods are developed to determine optimal or efficient (n×n)/k$(n\times n)/k$ semi-Latin squares for values of n and k for which such semi-Latin squares were previously unknown. The concept of subsquares of uniform semi-Latin squares is studied, new applications of the DESIGN package for GAP are developed, and exact algebraic computational techniques for comparing efficiency measures of binary equireplicate block designs are described. Applications include the complete enumeration of the (4×4)/k$(4\times 4)/k$ semi-Latin squares for k=2,…,10$k=2,\dots ,10$, and the determination of those that are A-, D-, and E-optimal, the construction of efficient (6×6)/k$(6\times 6)/k$ semi-Latin squares for k=4,5,6$k=4,5,6$, and counterexamples to a long-standing conjecture of R.A. Bailey and to a similar conjecture of D. Bedford and R.M. Whitaker.     

Classification of orthogonal arrays by integer programming

The problem of classifying all isomorphism classes of OA(N,k,s,t)$\mathrm{OA}(N,k,s,t)$'s is shown to be equivalent to finding all isomorphism classes of non-negative integer solutions to a system of linear equations under the symmetry group of the system of equations. A branch-and-cut algorithm developed by Margot [2002. Pruning by isomorphism in branch-and-cut. Math. Programming Ser. A 94, 71–90; 2003a. Exploiting orbits in symmetric ILP. Math. Programming Ser. B 98, 3–21; 2003b. Small covering designs by branch-and-cut. Math. Programming Ser. B 94, 207–220; 2007. Symmetric ILP: coloring and small integers. Discrete Optim., 4, 40–62] for solving integer programming problems with large symmetry groups is used to find all non-isomorphic OA(24,7,2,2)$\mathrm{OA}(24,7,2,2)$'s, OA(24,k,2,3)$\mathrm{OA}(24,k,2,3)$'s for 6?k?11$6?k?11$, OA(32,k,2,3)$\mathrm{OA}(32,k,2,3)$'s for 6?k?11$6?k?11$, OA(40,k,2,3)$\mathrm{OA}(40,k,2,3)$'s for 6?k?10$6?k?10$, OA(48,k,2,3)$\mathrm{OA}(48,k,2,3)$'s for 6?k?8$6?k?8$, OA(56,k,2,3)$\mathrm{OA}(56,k,2,3)$'s, OA(80,k,2,4)$\mathrm{OA}(80,k,2,4)$'s, OA(112,k,2,4)$\mathrm{OA}(112,k,2,4)$'s, for k=6,7$k=6,7$, OA(64,k,2,4)$\mathrm{OA}(64,k,2,4)$'s, OA(96,k,2,4)$\mathrm{OA}(96,k,2,4)$'s for k=7,8$k=7,8$, and OA(144,k,2,4)$\mathrm{OA}(144,k,2,4)$'s for k=8,9$k=8,9$. Further applications to classifying covering arrays with the minimum number of runs and packing arrays with the maximum number of runs are presented.     

Nonparametric Bayes estimation in repair models

Consider a sequence of dependent random variables _X1,_X2,…,_Xn$X_1,X_2,\dots ,X_n$, where _X1$X_1$ has distribution F (or probability measure P  ), and the distribution of _Xi+1$X_{i+1}$ given _X1,…,_Xi$X_1,\dots ,X_i$ and other covariates and environmental factors depends on F   and the previous data, i=1,…,n-1$i=1,\dots ,n-1$. General repair models give rise to such random variables as the failure times of an item subject to repair. There exist nonparametric non-Bayes methods of estimating F in the literature, for instance, Whitaker and Samaniego [1989. Estimating the reliability of systems subject to imperfect repair. J. Amer. Statist. Assoc. 84, 301–309], Hollander et al. [1992. Nonparametric methods for imperfect repair models. Ann. Statist. 20, 879–896] and Dorado et al. [1997. Nonparametric estimation for a general repair model. Ann. Statist. 25, 1140–1160], etc. Typically these methods apply only to special repair models and also require repair data on N independent items until exactly only one item is left awaiting a “perfect repair”.     

A confidence interval for the median of a finite population under unequal probability sampling: A model-assisted approach

This paper presents a method for constructing confidence intervals for the median of a finite population under unequal probability sampling. The model-assisted approach makes use of the _L1$L_1$-norm to motivate the estimating function which is then used to develop a unified approach to inference which includes not only confidence intervals but hypothesis tests and point estimates. The approach relies on large sample theory to construct the confidence intervals. In cases when second-order inclusion probabilities are not available or easy to compute, the Hartley–Rao variance approximation is employed. Simulations show that the confidence intervals achieve the appropriate confidence level, whether or not the Hartley–Rao variance is employed.     

Smallest confidence intervals for one binomial proportion

We specify three classes of one-sided and two-sided $1-\alpha$ confidence intervals with certain monotonicity and symmetry on the confidence limits for the probability of success, the parameter in a binomial distribution. For each class of one-sided confidence intervals the smallest interval, in the sense of the set inclusion, is obtained based on the direct analysis of coverage probability functions. A simple sufficient and necessary condition for the existence of the smallest two-sided confidence interval is provided and the smallest interval is derived if it exists. Thus the proposed intervals are uniformly most accurate, and have the uniformly minimum expected length as well.     

A test for the parametric form of the variance function in a partial linear regression model

We consider the problem of testing for a parametric form of the variance function in a partial linear regression model. A new test is derived, which can detect local alternatives converging to the null hypothesis at a rate n^-1/2$n^{-1/2}$ and is based on a stochastic process of the integrated variance function. We establish weak convergence to a Gaussian process under the null hypothesis, fixed and local alternatives. In the special case of testing for homoscedasticity the limiting process is a scaled Brownian bridge. We also compare the finite sample properties with a test based on an L²$L^2$-distance, which was recently proposed by You and Chen [2005. Testing heteroscedasticity in partially linear regression models. Statist. Probab. Lett. 73, 61–70].     

Asymptotic properties of the EPMC for modified linear discriminant analysis when sample size and dimension are both large

We deal with the problem of classifying a new observation vector into one of two known multivariate normal distributions when the dimension p and training sample size N   are both large with p<N$p<N$. Modified linear discriminant analysis (MLDA) was suggested by Xu et al. [10]. Error rate of MLDA is smaller than the one of LDA. However, if p and N   are moderately large, error rate of MLDA is close to the one of LDA. These results are conditional ones, so we should investigate whether they hold unconditionally. In this paper, we give two types of asymptotic approximations of expected probability of misclassification (EPMC) for MLDA as n→∞$n\to \infty$ with p=O(n^δ)$p=O(n^{\delta })$, 0<δ<1$0<\delta <1$. The one of two is the same as the asymptotic approximation of LDA, and the other is corrected version of the approximation. Simulation reveals that the modified version of approximation has good accuracy for the case in which p and N are moderately large.