Order statistics from trivariate normal and -distributions in terms of generalized skew-normal and skew- distributions

We consider here a generalization of the skew-normal distribution, GSN(λ₁,λ₂,ρ), defined through a standard bivariate normal distribution with correlation ρ, which is a special case of the unified multivariate skew-normal distribution studied recently by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574]. We then present some simple and useful properties of this distribution and also derive its moment generating function in an explicit form. Next, we show that distributions of order statistics from the trivariate normal distribution are mixtures of these generalized skew-normal distributions; thence, using the established properties of the generalized skew-normal distribution, we derive the moment generating functions of order statistics, and also present expressions for means and variances of these order statistics.Next, we introduce a generalized skew-t_ν distribution, which is a special case of the unified multivariate skew-elliptical distribution presented by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574] and is in fact a three-parameter generalization of Azzalini and Capitanio's [2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J. Roy. Statist. Soc. Ser. B 65, 367–389] univariate skew-t_ν form. We then use the relationship between the generalized skew-normal and skew-t_ν distributions to discuss some properties of generalized skew-t_ν as well as distributions of order statistics from bivariate and trivariate t_ν distributions. We show that these distributions of order statistics are indeed mixtures of generalized skew-t_ν distributions, and then use this property to derive explicit expressions for means and variances of these order statistics.     

De-aliasing effects using semifoldover techniques

Semifoldover designs, obtained by semifolding a regular two-level factorial design, have been discussed recently in the literature. In this article, with the use of indicator functions, we investigate various semifoldover designs that are obtained from a general two-level factorial design. We discuss when a main factor or a two-factor interaction can be de-aliased from their aliased two-factor interactions, and extend some of the existing results from regular designs to non-regular designs. Finally, we present some examples to illustrate the results developed here.     

Nonparametric control charts based on runs and Wilcoxon-type rank-sum statistics

In this article, we introduce three new distribution-free Shewhart-type control charts that exploit run and Wilcoxon-type rank-sum statistics to detect possible shifts of a monitored process. Exact formulae for the alarm rate, the run length distribution, and the average run length (ARL) are all derived. A key advantage of these charts is that, due to their nonparametric nature, the false alarm rate (FAR) and in-control run length distribution is the same for all continuous process distributions. Tables are provided for the implementation of the charts for some typical FAR values. Furthermore, a numerical study carried out reveals that the new charts are quite flexible and efficient in detecting shifts to Lehmann-type out-of-control situations.     

Semiparametric estimation for count data through weighted distributions

This paper is concerned with semiparametric discrete kernel estimators when the unknown count distribution can be considered to have a general weighted Poisson form. The estimator is constructed by multiplying the Poisson estimate with a nonparametric discrete kernel-type estimate of the Poisson weight function. Comparisons are then carried out with the ordinary discrete kernel probability mass function estimators. The Poisson weight function is thus a local multiplicative correction factor, and is considered as the uniform measure to detect departures from the equidispersed Poisson distribution. In this way, the effects of dispersion and zero-proportion with respect to the standard Poisson distribution are also minimized. This method of estimation is also applied to the weighted binomial form for the count distribution having a finite support. The proposed estimators, in addition to being simple, easy-to-implement and effective, also outperform the competing nonparametric and parametric estimators in finite-sample situations. Two examples illustrate this new semiparametric estimation.     

Iterative bias reduction: a comparative study

Multivariate nonparametric smoothers, such as kernel based smoothers and thin plate splines smoothers, are adversely impacted by the sparseness of data in high dimension, also known as the curse of dimensionality. Adaptive smoothers, that can exploit the underlying smoothness of the regression function, may partially mitigate this effect. This paper presents a comparative simulation study of a novel adaptive smoother (IBR) with competing multivariate smoothers available as package or function within the R language and environment for statistical computing. Comparison between the methods are made on simulated datasets of moderate size, from 50 to 200 observations, with two, five or 10 potential explanatory variables, and on a real dataset. The results show that the good asymptotic properties of IBR are complemented by a very good behavior on moderate sized datasets, results which are similar to those obtained with Duchon low rank splines.     

Nonparametric regression estimates with censored data based on block thresholding method

Here we consider wavelet-based identification and estimation of a censored nonparametric regression model via block thresholding methods and investigate their asymptotic convergence rates. We show that these estimators, based on block thresholding of empirical wavelet coefficients, achieve optimal convergence rates over a large range of Besov function classes, and in particular enjoy those rates without the extraneous logarithmic penalties that are usually suffered by term-by-term thresholding methods. This work is extension of results in Li et al. (2008). The performance of proposed estimator is investigated by a numerical study.     

On the recoding of continuous and bounded indexes to a binomial form: an application to quality-of-life scores

The statistical analysis of patient-reported outcomes (PROs) as endpoints has shown to be of great practical relevance. The resulting scores or indexes from the questionnaires used to measure PROs could be treated as continuous or ordinal. The goal of this study is to propose and evaluate a recoding process of the scores, so that they can be treated as binomial outcomes and, therefore, analyzed using logistic regression with random effects. The general methodology of recoding is based on the observable values of the scores. In order to obtain an optimal recoding, the evaluation of the recoding method is tested for different values of the parameters of the binomial distribution and different probability distributions of the random effects. We illustrate, evaluate and validate the proposed method of recoding with the Short Form-36 (SF-36) Survey and real data. The optimal recoding approach is very useful and flexible. Moreover, it has a natural interpretation, not only for ordinal scores, but also for questionnaires with many dimensions and different profiles, where a common method of analysis is desired, such as the SF-36.     

Progressive type II censored order statistics from exponential distributions

In the model of progressive type II censoring, point and interval estimation as well as relations for single and product moments are considered. Based on two-parameter exponential distributions, maximum likelihood estimators (MLEs), uniformly minimum variance unbiased estimators (UMVUEs) and best linear unbiased estimators (BLUEs) are derived for both location and scale parameters. Some properties of these estimators are shown. Moreover, results for single and product moments of progressive type II censored order statistics are presented to obtain recurrence relations from exponential and truncated exponential distributions. These relations may then be used to compute all the means, variances and covariances of progressive type II censored order statistics based on exponential distributions for arbitrary censoring schemes. The presented recurrence relations simplify those given by Aggarwala and Balakrishnan (1996)     

INFLUENCE FUNCTIONS IN FUNCTIONAL MEASUREMENT ERROR MODELS WITH REPLICATED DATA

We consider the construction and properties of influence functions in the context of functional measurement error models with replicated data. In these models estimates of the parameters can be affected both by the individual observations and the means of replicated observations. We show that influence function of the means of replicates on the estimate of regression coefficients can be only derived under the assumption that the variances of the errors are known, while one for the individual observations can be only derived simultaneously with their influence function on the estimators of the variances of the errors.