Asymptotics for Euclidean functionals of mixing processes

We extend previously known asymptotic results in the framework of subadditive Euclidean functionals and the related superadditive functionals. Our results generalize the standard independent identically distributed assumption, vastly expanding the scope of the existing theory. As an application of our results, a limit theorem for the average nearest neighbor distance is established, extending earlier results in the literature.     

Aspects of the mean residual life order for weighted distributions

In this paper, we first provide conditions for preservation of the mean residual life (mrl) order under weighting. Then we apply the obtained results to establish our results about preservation of the decreasing mrl class by weighted distributions. In addition, we present some results for comparing the original random variable to its weighted version in terms of the mrl order. Also, some examples are given to illustrate the results.     

General frailty model and stochastic orderings

In this paper, we propose a general frailty model and develop its properties including some results for stochastic comparisons. More specifically, our main results lie in seeing how the well known stochastic orderings between distributions of two frailties translate into the orderings between the corresponding survival functions. These results are used to obtain the properties of the classical multiplicative frailty model and the additive frailty model. Several of the results, in the literature, are obtained as special cases.     

On the strong convergence of weighted sums of widely dependent random variables

ABSTRACT

For widely dependent random variables, we present some results on the strong convergence of weighted sums, including results on almost surely (a.s.) and complete convergence. To this end, we verified some Borel–Cantelli lemmas of the widely dependent random variables. The above-mentioned random variables contain common negatively dependent random variables, some positively dependent random variables, and some others; therefore, the obtained results extend and improve some existing results.     

Further results for parallel systems with two heterogeneous exponential components

In this paper, we investigate ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components with respect to likelihood ratio and hazard rate orders. Two sufficient conditions are provided for likelihood ratio and hazard rate orders to hold between the lifetimes of two parallel systems, respectively. Moreover, we extend the results from exponential case to the proportional hazard rate models. The results established here strength some of the results known in the literature. Finally, some numerical examples are given to illustrate the theoretical results derived here as well.     

The efficacy of the correlation dimension technique in detecting determinism in small samples

This paper presents the results of a simulation study investigating the efficacy of the correlation dimension technique in detecting the presence of deterministic structure in small data sets. The results show that for some very simple structures a mere 100 data points suffice; but for some slightly more complicated structures, which arguably are still simple relative to real-world structures, even 2500 data points are inadequate. These results suggest that substantial caution be used in interpreting negative results on empirical data.     

Non-Gaussian Conditional Linear AR(1) Models

This paper gives a general formulation of a non-Gaussian conditional linear AR(1) model subsuming most of the non-Gaussian AR(1) models that have appeared in the literature. It derives some general results giving properties for the stationary process mean, variance and correlation structure, and conditions for stationarity. These results highlight similarities with and differences from the Gaussian AR(1) model, and unify many separate results appearing in the literature. Examples illustrate the wide range of properties that can appear under the conditional linear autoregressive assumption. These results are used in analysing three real datasets, illustrating general methods of estimation, model diagnostics and model selection. In particular, the theoretical results can be used to develop diagnostics for deciding if a time series can be modelled by some linear autoregressive model, and for selecting among several candidate models.     

Least Squares Programs-A Look at the Square Root Procedure

Various algorithms are in use on computers to solve least squares problems. Apparently very few programs use the square root procedure. In this paper, results from programs based on the square root procedure are compared with results based on some other algorithms. Remarkably good results are obtained using the square root procedure.     

Some Uniform Limit Results in Additive Regression Model

In the setting of additive regression model components estimation, we establish some uniform limit results in probability. Our results allow to give an asymptotic simultaneous 100% confidence band for these components. These results are stated in the framework of i.i.d random vectors when the marginal integration estimation method is used.     

Selecting significant effects in factorial designs: Lenth’s method versus the Box-Meyer approach

The Lenth method is conceptually simple and probably the most common approach to analyzing the significance of the effects in factorial designs. Here, we compare it with a Bayesian approach proposed by Box and Meyer and which does not appear in the usual software packages. The comparison is made by simulating the results of 4, 8 and 16 run designs in a set of scenarios that mirror practical situations and analyzing the results provided by both methods. Although the results depend on the number of runs and the scenario considered, the use of the Box and Meyer method generally produces better results.     

Optimality of Equal vs. Unequal Cluster Sizes in Multilevel Intervention Studies: A Monte Carlo Study for Small Sample Sizes

Optimality of equal versus unequal cluster sizes in the context of multilevel intervention studies is examined. A Monte Carlo study is done to examine to what degree asymptotic results on the optimality hold for realistic sample sizes and for different estimation methods. The relative D-criterion, comparing equal versus unequal cluster sizes, almost always exceeded 85%, implying that loss of information due to unequal cluster sizes can be compensated for by increasing the number of clusters by 18%. The simulation results are in line with asymptotic results, showing that, for realistic sample sizes and various estimation methods, the asymptotic results can be used in planning multilevel intervention studies.     

The effects of covariate adjustment in generalized linear models

Results from classical linear regression regarding the effects of covariate adjustment, with respect to the issues of confounding, the precision with which an exposure effect can be estimated, and the efficiency of hypothesis tests for no treatment effect in randomized experiments, are often assumed to apply more generally to other types of regression models. In this paper results pertaining to several generalized linear models involving a dichotomous response variable are given, demonstrating that with respect to the issues of confounding and precision, for models having a linear or log link function the results of classical linear regression do generally apply, whereas for other models, including those having a logit, probit, log-log, complementary log-log, or generalized logistic link function, the results of classical linear regression do not always apply. It is also shown, however, that for any link function, covariate adjustment results in improved efficiency of hypothesis tests for no treatment effect in randomized experiments, and hence that the classical linear regression results regarding efficiency do apply for all models having a dichotomous response variable.     

Referee recognition

The paper provides theoretical and experimental results concerning the distribution of the sample lead. They apply to sample lead of one party over another in political opinion polls, one product over another in market research surveys, share of market of one company over another in industrial studies, one class over another in social investigations, one programme over another in TV viewing surveys, etc. The theoretical results include the derivation of the moment and cumulant generating functions. it is shown formally that the sampling distribution approaches normality as sample size increases. Experimental results provide guidelines for the choice of sample size to ensure adequacy of the normal approximation. Calculations based on one of the Central Limit Theorems support the results of the simulation.     

Closed-form evaluation of integrals involving the gamma function

In this article, we obtain simplified as well as original closed-form expressions for integrals involving the gamma function. These explicit results are found neither in the literature nor using symbolic computation software. Subsequent results follow, giving rise to explicit expressions for integrals involving the error function, with application in order statistics. In particular, these results can be used within the framework of reliability theory.     

Heteroscedastic Regression Models and Applications to Off-line Quality Control

We discuss in the present paper the analysis of heteroscedastic regression models and their applications to off-line quality control problems. It is well known that the method of pseudo-likelihood is usually preferred to full maximum likelihood since estimators of the parameters in the regression function obtained are more robust to misspecification of the variance function. Despite its popularity, however, existing theoretical results are difficult to apply and are of limited use in many applications. Using more recent results in estimating equations, we obtain an efficient algorithm for computing the pseudo-likelihood estimator with desirable convergence properties and also derive simple, explicit and easy to apply asymptotic results. These results are used to look in detail at variance minimization in off-line quality control, yielding techniques of inferences for the optimized design parameter. In application of some existing approaches to off-line quality control, such as the dual response methodology, rigorous statistical inference techniques are scarce and difficult to obtain. An example of off-line quality control is presented to discuss the practical aspects involved in the application of the results obtained and to address issues such as data transformation, model building and the optimization of design parameters. The analysis shows very encouraging results, and is seen to be able to unveil some important information not found in previous analyses.     

Analysis of an European union election using principal component analysis

While studying the results from one European Parliament election, the question of principal component analysis (PCA) suitability for this kind of data was raised. Since multiparty data should be seen as compositional data (CD), the application of PCA is inadvisable and may conduct to ineligible results. This work points out the limitations of PCA to CD and presents a practical application to the results from the European Parliament election in 2004. We present a comparative study between the results of PCA, Crude PCA and Logcontrast PCA (Aitchison in: Biometrika 70:57–61, 1983; Kucera, Malmgren in: Marine Micropaleontology 34:117–120, 1998). As a conclusion of this study, and concerning the mentioned data set, the approach which produced clearer results was the Logcontrast PCA. Moreover, Crude PCA conducted to misleading results since nonlinear relations were presented between variables and the linear PCA proved, once again, to be inappropriate to analyse data which can be seen as CD.     

Modified chain sampling plans for lot inspection by variables and attributes

The purpose of acceptance sampling is to develop decision rules to accept or reject production lots based on sample data. When testing is destructive or expensive, dependent sampling procedures cumulate results from several preceding lots. This chaining of past lot results reduces the required size of the samples. A large part of these procedures only chain past lot results when defects are found in the current sample. However, such selective use of past lot results only achieves a limited reduction of sample sizes. In this article, a modified approach for chaining past lot results is proposed that is less selective in its use of quality history and, as a result, requires a smaller sample size than the one required for commonly used dependent sampling procedures, such as multiple dependent sampling plans and chain sampling plans of Dodge. The proposed plans are applicable for inspection by attributes and inspection by variables. Several properties of their operating characteristic-curves are derived, and search procedures are given to select such modified chain sampling plans by using the two-point method.     

Comparisons of coherent systems with non-identically distributed components

The signature-based mixture representations for coherent systems are a good way to obtain distribution-free comparisons of systems. Unfortunately, these representations only hold for systems whose component lifetimes are independent and identically distributed (IID) or exchangeable (i.e., their joint distribution is invariant under permutations). In this paper we obtain comparison results for generalized mixtures, that is, for reliability functions that can be written as linear combinations of some baseline reliability functions with positive and negative coefficients. These results are based on some concepts in Graph Theory. We apply these results to obtain new comparison results for coherent systems without the IID or exchangeability assumptions by using their generalized mixture representations based on the minimal path sets.     

Inference for Quantiles of a Finite Population: Asymptotic versus Resampling Results

The aim of the paper is to study the problem of estimating the quantile function of a finite population. Attention is first focused on point estimation, and asymptotic results are obtained. Confidence intervals are then constructed, based on both the following: (i) asymptotic results and (ii) a resampling technique based on rescaling the ‘usual’ bootstrap. A simulation study to compare asymptotic and resampling‐based results, as well as an application to a real population, is finally performed.     

Karlin's corollary: a topological approach to pitman's measure

In this article, many of the known univariate results about Pitman's Measure of Closeness (PMC) are synthesized through a topological approach. The proofs of many known results are simplified and clarified. The approach extends some previous results established under other restrictions. Connections between PMC and Bayesian estimation are discussed but the inherent interpretations differ. A discourse on this connection can be found in the article of Ghosh and Sen (1991). A transitiveness property for ordered estimators is established and a counter example is given for unordered ones. These results help distinguish between the Bayesian and classical interpretations of Pitman's measure.