Asymptotic Relative Efficiencies of the Score and Robust Tests in Genetic Association Studies

Introduction:The score statisticZ(θ)and the maximin efficient robust test statisticZMERTare commonly used in genetic association study, but according to our knowledge there is no formal comparison of them.Methods:In this report, we compare the asymptotic behavior ofZ(θ)andZMERT, by computing their Asymptotic Relative Efficiencies (AREs) relative to each other. Four commonly used ARE measures, the Pitman ARE, Chernoff ARE, Hodges-Lehmann ARE and the Bahadur ARE are considered. Some modifications of these methods are made to simplify the computations. We found that the Chernoff, Hodges-Lehmann and Bahadur AREs are suitable for our setting.Results and Conclusion:Based on our study, the efficiencies of the two test statistic varies for different criterion used, and for different parameter values under the same criterion, so each test has its advantages and dis-advantages according to the criterion used and the parameters involved, which are described in the context. Numerical examples are given to illustrate the use of the two statistics in genetic association study.

Kumaraswamy Distribution and Random Extrema

Objective:We provide a new stochastic representation for a Kumaraswamy random variable with arbitrary non-negative parameters. The representation is in terms of maxima and minima of independent distributed standard uniform components and extends a similar representation for integer-valued parameters.Result:The result is further extended for generalized classes of distributions obtained from a “base” distribution functionFviz.G(x) =H(F(x)), whereHis the CDF of Kumaraswamy distribution.

Consistency of the Kaplan-Meier Estimator of the Survival Function in Competiting Risks

Introduction:In this article, we only focus on the probability distributions of the breakdown time whose causes are known, and we consider a partition of the observations into subgroups according to each of the causes as defined in Njamen and Ngatchou [1]. By adapting the stochastic processes developed by Aalen [2, 3], we derive a Kaplan-Meier [4] nonparametric estimator for the survival function in competiting risks.Result & Discussion:In a region where there is at least one observation, we prove on one hand that this new nonparametric estimator is unbiased in competiting risk and on the other hand, using the Lenglart inequality, we establish its uniform consistency in competiting risks.

Clustering Directions Based on the Estimation of a Mixture of Von Mises-Fisher Distributions

Background:In the statistical analysis of directional data, the von Mises-Fisher distribution plays an important role to model unit vectors. The estimation of the parameters of a mixture of von Mises-Fisher distributions can be done through the Estimation-Maximization algorithm.Objective:In this paper we propose a dynamic clusters type algorithm based on the estimation of the parameters of a mixture of von Mises-Fisher distributions for clustering directions, and we compare this algorithm with the Estimation-Maximization algorithm. We also define the between-groups and within-groups variability measures to compare the solutions obtained with the algorithms through these measures.Results:The comparison of the clusters obtained with both algorithms is provided for a simulation study based on samples generated from a mixture of two Fisher distributions and for an illustrative example with spherical data.

Bayesian Inference for Three Bivariate Beta Binomial Models

Background:This paper considers three two-dimensional beta binomial models previously introduced in the literature. These were proposed as candidate models for modelling forms of correlated and overdispersed bivariate count data. However, the first model has a complicated form of bivariate probability mass function involving a generalized hypergeometric function and the remaining two do not have closed forms of probability mass functions and are not amenable to analysis using maximum likelihood. This limited their applicability.Objective:In this paper, we will discuss how the Bayesian analyses of these models may go forward using Markov chain Monte Carlo and data augmentation.Results:An illustrative example having to do with student achievement in two related university courses is included. Posterior and posterior predictive inferences and predictive information criteria are discussed.

An Entropy Rate Theorem for a Hidden Inhomogeneous Markov Chain

Objective: The main object of our study is to extend some entropy rate theorems to a Hidden Inhomogeneous Markov Chain (HIMC) and establish an entropy rate theorem under some mild conditions. Introduction: A hidden inhomogeneous Markov chain contains two different stochastic processes; one is an inhomogeneous Markov chain whose states are hidden and the other is a stochastic process whose states are observable. Materials and Methods: The proof of theorem requires some ergodic properties of an inhomogeneous Markov chain, and the flexible application of the properties of norm and the bounded conditions of series are also indispensable. Results: This paper presents an entropy rate theorem for an HIMC under some mild conditions and two corollaries for a hidden Markov chain and an inhomogeneous Markov chain. Conclusion: Under some mild conditions, the entropy rates of an inhomogeneous Markov chains, a hidden Markov chain and an HIMC are similar and easy to calculate.

Comparing Different Information Levels

Objective: Given a sequence of random variables X = X1, X2, . . .suppose the aim is to maximize one’s return by picking a ‘favorable’ Xi. Obviously, the expected payoff crucially depends on the information at hand. An optimally informed person knows all the values Xi = xi and thus receives E(sup Xi). Method: We will compare this return to the expected payoffs of a number of gamblers having less information, in particular supi(EXi), the value of the sequence to a person who only knows the random variables’ expected values. In general, there is a stochastic environment, (F.E. a class of random variables C), and several levels of information. Given some XϵC, an observer possessing information j obtains rj(X). We are going to study ‘information sets’ of the form. characterizing the advantage of k relative to j. Since such a set measures the additional payoff by virtue of increased information, its analysis yields a number of interesting results, in particular ‘prophet-type’ inequalities.

On Limited Length Binary Strings with an Application in Statistical Control

In a 0 - 1 sequence of Markov dependent trials we consider a statistic which counts strings of a limited length run of 0s between subsequent 1s. Its probability mass function is used to determine the chance that a stochastic process remains or not in statistical control. Illustrative numerics are presented.

A New Class of Distributions Based on Hurwitz Zeta Function with Applications for Risk Management

This paper constructs a new family of distributions, which is based on the Hurwitz zeta function, which includes novel distributions as well important known distributions such as the normal, gamma, Weibull, Maxwell-Boltzmann and the exponential power distributions. We provide the n-th moment, the Esscher transform and premium and the tail conditional moments for this family.

A New Acceptance Sampling Plans Based on Percentiles for Odds Exponential Log Logistic Distribution

In this paper, acceptance sampling plans are developed for the odds exponential log logistic distribution (OELLD) introduced by Rosaiah et al. [1] based on lifetime percentiles when the life test is truncated at a predetermined time. The minimum sample size necessary to ensure the specified lifetime percentile is obtained under a given customer’s risk. The operating characteristic values of the sampling plans as well as the producer’s risk are presented. One example with real data set is also given as an illustration.