Likelihood-free parallel tempering

Approximate Bayesian Computational (ABC) methods, or likelihood-free methods, have appeared in the past fifteen years as useful methods to perform Bayesian analysis when the likelihood is analytically or computationally intractable. Several ABC methods have been proposed: MCMC methods have been developed by Marjoram et al. (2003) and by Bortot et al. (2007) for instance, and sequential methods have been proposed among others by Sisson et al. (2007), Beaumont et al. (2009) and Del Moral et al. (2012). Recently, sequential ABC methods have appeared as an alternative to ABC-PMC methods (see for instance McKinley et al., 2009; Sisson et al., 2007). In this paper a new algorithm combining population-based MCMC methods with ABC requirements is proposed, using an analogy with the parallel tempering algorithm (Geyer 1991). Performance is compared with existing ABC algorithms on simulations and on a real example.     

Empirical likelihood for varying-coefficient semiparametric mixed-effects errors-in-variables models with longitudinal data

In this paper, the empirical likelihood inferences for varying-coefficient semiparametric mixed-effects errors-in-variables models with longitudinal data are investigated. We construct the empirical log-likelihood ratio function for the fixed-effects parameters and the mean parameters of random-effects. The empirical log-likelihood ratio at the true parameters is proven to be asymptotically $\chi ^2_{q+r}$ , where $q$ and $r$ are dimensions of the fixed and random effects respectively, and the corresponding confidence regions for them are then constructed. We also obtain the maximum empirical likelihood estimator of the parameters of interest, and prove it is the asymptotically normal under some suitable conditions. A simulation study and a real data application are undertaken to assess the finite sample performance of the proposed method.     

On the joint distribution of success runs of several lengths in the sequence of MBT and its applications

Let \(X_1 ,X_2 ,\ldots ,X_n \) be a sequence of Markov Bernoulli trials (MBT) and \(\underline{X}_n =( {X_{n,k_1 } ,X_{n,k_2 } ,\ldots ,X_{n,k_r } })\) be a random vector where \(X_{n,k_i } \) represents the number of occurrences of success runs of length \(k_i \,( {i=1,2,\ldots ,r})\) . In this paper the joint distribution of \(\underline{X}_n \) in the sequence of \(n\) MBT is studied using method of conditional probability generating functions. Five different counting schemes of runs namely non-overlapping runs, runs of length at least \(k\) , overlapping runs, runs of exact length \(k\) and \(\ell \) -overlapping runs (i.e. \(\ell \) -overlapping counting scheme), \(0\le \ell are considered. The pgf of joint distribution of \(\underline{X}_n \) is obtained in terms of matrix polynomial and an algorithm is developed to get exact probability distribution. Numerical results are included to demonstrate the computational flexibility of the developed results. Various applications of the joint distribution of \(\underline{X}_n \) such as in evaluation of the reliability of \(( {n,f,k})\!\!:\!\!G\) and \(\!:\!\!G\) system, in evaluation of quantities related to start-up demonstration tests, acceptance sampling plans are also discussed.     

On exact and optimal single-sampling plans by variables

We deal with sampling by variables with two-way protection in the case of a $N\>(\mu ,\sigma ^2)$ distributed characteristic with unknown $\sigma $ . The LR sampling plan proposed by Lieberman and Resnikoff (JASA 50: 457 ${-}$ 516, 1955) and the BSK sampling plan proposed by Bruhn-Suhr and Krumbholz (Stat. Papers 31: 195–207, 1990) are based on the UMVU and the plug-in estimator, respectively. For given $p_1$ (AQL), $p_2$ (RQL) and $\alpha ,\beta $ (type I and II errors) we present an algorithm allowing to determine the optimal LR and BSK plans having minimal sample size among all plans satisfying the corresponding two-point condition on the OC. An R (R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org/ 2012) package, ExLiebeRes‘ (Krumbholz and Steuer ExLiebeRes: calculating exact LR- and BSK-plans, R-package version 0.9.9. http://exlieberes.r-forge.r-project.org 2012) implementing that algorithm is provided to the public.     

Classical and Bayesian estimation of P(X<Y) using upper record values from Kumaraswamy’s distribution

In this paper, maximum likelihood and Bayesian approaches have been used to obtain the estimation of \(P(X<Y)\) based on a set of upper record values from Kumaraswamy distribution. The existence and uniqueness of the maximum likelihood estimates of the Kumaraswamy distribution parameters are obtained. Confidence intervals, exact and approximate, as well as Bayesian credible intervals are constructed. Bayes estimators have been developed under symmetric (squared error) and asymmetric (LINEX) loss functions using the conjugate and non informative prior distributions. The approximation forms of Lindley (Trabajos de Estadistica 3:281–288, 1980) and Tierney and Kadane (J Am Stat Assoc 81:82–86, 1986) are used for the Bayesian cases. Monte Carlo simulations are performed to compare the different proposed methods.     

Conditional MSE-based discrimination of the sample mean and the post-stratification estimator in population sampling

Whenever a random sample is drawn from a stratified population, the post-stratification estimator $\tilde X$ usually is preferred to the sample mean $\tilde X$ , when the population mean is to be estimated. This is due to the fact that the variance of $\tilde X$ is asymptotically smaller than that of $\tilde X$ , while both estimators are asymptotically unbiased. However, this only holds looking at post-stratification unconditionally, when strata sample sizes are random. Conditioned on the realized sample sizes, the MSE of $\tilde X$ can be higher than that of $\tilde X$ which means that $\tilde X$ should be preferred to $\tilde X$ , even if it is biased. The conditional MSE difference of $\tilde X$ and $\tilde X$ is estimated, and using this estimation and its variance a heuristic test based on the Vysochanskiî-Petunin inequality is derived.     

Life behavior of \delta -shock models for uniformly distributed interarrival times

In this paper we study the life behavior of \(\delta \) -shock models when the shocks occur according to a renewal process whose interarrival distribution is uniform. In particular, we obtain the first two moments of the corresponding lifetime random variables for general interarrival distribution, and survival functions when the interarrival distribution is uniform.     

Comparative Studies on Frailties in Survival Analysis

A mixed survival function is derived through proportional hazards model assuming Gamma or Inverse Gaussian frailty under Weibull baseline via Bayesian approach, and the maximum likelihood estimation (MLE) is proposed to estimate the parameters in the population. Through intensive simulations designed to generate right-censored data from the proposed survival function, we compare the MLE in this article with Cox estimates (in many different versions) for the covariate coefficient. Finally, the proposed survival model is applied to fit Leukemia data from Cox and Oakes (1992 Cox , D. R. , Oakes , D. ( 1992 ). Analysis of Survival Data . London : Chapman & Hall . [Google Scholar]).     

A discrete version of the half-normal distribution and its generalization with applications

A new discrete distribution depending on two parameters $\alpha >-1$ and $\sigma >0$ is obtained by discretizing the generalized normal distribution proposed in García et al. (Comput Stat and Data Anal 54:2021–2034, 2010), which was derived from the normal distribution by using the Marshall and Olkin (Biometrika 84(3):641–652, 1997) scheme. The particular case $\alpha =1$ leads us to the discrete half-normal distribution which is different from the discrete half-normal distribution proposed previously in the statistical literature. This distribution is unimodal, overdispersed (the responses show a mean sample greater than the variance) and with an increasing failure rate. We revise its properties and the question of parameter estimation. Expected frequencies were calculated for two overdispersed and underdispersed (the responses show a variance greater than the mean) examples, and the distribution was found to provide a very satisfactory fit.     

A unified method for constructing expectation tolerance intervals

Given a random sample of size \(n\) with mean \(\overline{X} \) and standard deviation \(s\) from a symmetric distribution \(F(x; \mu , \sigma ) = F_{0} (( x- \mu ) / \sigma ) \) with \(F_0\) known, and \(X \sim F(x;\; \mu , \sigma )\) independent of the sample, we show how to construct an expansion \( a_n^{\prime } = \sum _{i=0}^\infty \ c_i \ n^{-i} \) such that \(\overline{X} - s a_n^{\prime } < X < \overline{X} + s a_n^{\prime } \) with a given probability \(\beta \) . The practical value of this result is illustrated by simulation and using a real data set.     

Equalities between OLSE,BLUE and BLUP in the linear model

We consider equalities between the ordinary least squares estimator ( $\mathrm {OLSE} $ ), the best linear unbiased estimator ( $\mathrm {BLUE} $ ) and the best linear unbiased predictor ( $\mathrm {BLUP} $ ) in the general linear model $\{ \mathbf y , \mathbf X \varvec{\beta }, \mathbf V \}$ extended with the new unobservable future value $ \mathbf y _{*}$ of the response whose expectation is $ \mathbf X _{*}\varvec{\beta }$ . Our aim is to provide some new insight and new proofs for the equalities under consideration. We also collect together various expressions, without rank assumptions, for the $\mathrm {BLUP} $ and provide new results giving upper bounds for the Euclidean norm of the difference between the $\mathrm {BLUP} ( \mathbf y _{*})$ and $\mathrm {BLUE} ( \mathbf X _{*}\varvec{\beta })$ and between the $\mathrm {BLUP} ( \mathbf y _{*})$ and $\mathrm {OLSE} ( \mathbf X _{*}\varvec{\beta })$ . A remark is made on the application to small area estimation.     

On the central limit theorem along subsequences of sums of i.i.d. random variables

Let \(\mathbb{N } = \{1, 2, 3, \ldots \}\) . Let \(\{X, X_{n}; n \in \mathbb N \}\) be a sequence of i.i.d. random variables, and let \(S_{n} = \sum _{i=1}^{n}X_{i}, n \in \mathbb N \) . Then \( S_{n}/\sqrt{n} \Rightarrow N(0, \sigma ^{2})\) for some \(\sigma ^{2} < \infty \) whenever, for a subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/\sqrt{n_{k}} \Rightarrow N(0, \sigma ^{2})\) . Motivated by this result, we study the central limit theorem along subsequences of sums of i.i.d. random variables when \(\{\sqrt{n}; n \in \mathbb N \}\) is replaced by \(\{\sqrt{na_{n}};n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) . We show that, for given positive nondecreasing sequence \(\{a_{n}; n \in \mathbb N \}\) with \(\lim _{n \rightarrow \infty } a_{n} = \infty \) and \(\lim _{n \rightarrow \infty } a_{n+1}/a_{n} = 1\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \) , there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/\sqrt{n_{k}a_{n_{k}}} \Rightarrow N(0, 1)\) . In particular, for given \(0 < p < 2\) and given nondecreasing function \(h(\cdot ): (0, \infty ) \rightarrow (0, \infty )\) with \(\lim _{x \rightarrow \infty } h(x) = \infty \) , there exists a sequence \(\{X, X_{n}; n \in \mathbb N \}\) of symmetric i.i.d. random variables such that \(\mathbb E h(|X|) = \infty \) and, for some subsequence \(\{n_{k}; k \in \mathbb N \}\) of \(\mathbb N \) , \( S_{n_{k}}/n_{k}^{1/p} \Rightarrow N(0, 1)\) .     

EWMA charts: ARL considerations in case of changes in location and scale

Widely spread tools within the area of Statistical Process Control are control charts of various designs. Control chart applications are used to keep process parameters (e.g., mean \(\mu \) , standard deviation \(\sigma \) or percent defective \(p\) ) under surveillance so that a certain level of process quality can be assured. Well-established schemes such as exponentially weighted moving average charts (EWMA), cumulative sum charts or the classical Shewhart charts are frequently treated in theory and practice. Since Shewhart introduced a \(p\) chart (for attribute data), the question of controlling the percent defective was rarely a subject of an analysis, while several extensions were made using more advanced schemes (e.g., EWMA) to monitor effects on parameter deteriorations. Here, performance comparisons between a newly designed EWMA \(p\) control chart for application to continuous types of data, \(p=f(\mu ,\sigma )\) , and popular EWMA designs ( \(\bar{X}\) , \(\bar{X}\) - \(S^2\) ) are presented. Thus, isolines of the average run length are introduced for each scheme taking both changes in mean and standard deviation into account. Adequate extensions of the classical EWMA designs are used to make these specific comparisons feasible. The results presented are computed by using numerical methods.     

Some Results on Dynamic Generalized Survival Entropy

Recently, Zografos and Nadarajah (2005 Zografos, K., Nadarajah, S. (2005). Survival exponential entropies. IEEE Trans. Inform. Theor. 51:1239–1246.[Crossref], [Web of Science ®] , [Google Scholar]) proposed two measures of uncertainty based on the survival function, called the survival exponential entropy and the generalized survival exponential entropy. In this article, we explore properties of the generalized survival entropy and the dynamic version of it. We study conditions under which the generalized survival entropy of first order statistic can uniquely determines the parent distribution. The exponential, Pareto, and finite range distributions, which are commonly used in reliability, have been characterized using this generalized measure. Another measure of entropy is also introduced in analogy with cumulative entropy which has been proposed by Di Crescenzo and Longobardi (2009) and some properties of it are given.     

Bayesian variable selection for correlated covariates via colored cliques

We propose a Bayesian method to select groups of correlated explanatory variables in a linear regression framework. We do this by introducing in the prior distribution assigned to the regression coefficients a random matrix $G$ that encodes the group structure. The groups can thus be inferred by sampling from the posterior distribution of $G$ . We then give a graph-theoretic interpretation of this random matrix $G$ as the adjacency matrix of cliques. We discuss the extension of the groups from cliques to more general random graphs, so that the proposed approach can be viewed as a method to find networks of correlated covariates that are associated with the response.     

On dynamic weighted survival entropy of order α

Recently, Abbasnejad et al. (2010 Abbasnejad, M., Arghami, N.R., Morgenthaler, S., Mohtashami Borzadaran, G.R. (2010). On the dynamic survival entropy. Stat. Probab. Lett. 80:1962–1971.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a measure of uncertainty based on survival function, called the survival entropy of order α. A dynamic form of the survival entropy of order α is also proposed by them. In this paper, we derive the weighted form of these measures. The properties of the new measures are also discussed.     

A Note on Influence Assessment in Score Tests

Recently, [1] Ebrahimi, N. 1996. How to measure uncertainty about residual life time. Sankhya Ser. A, 58: 48–57.  [Google Scholar] proposed a dynamic measure based on differential entropy applied to the residual lifetime. This measure has been used for the classification and ordering of survival functions. More recently, [2] Ebrahimi, N. 1997. Testing whether lifetime distribution is decreasing uncertainty. Journal of Statistical Planning and Inference, 64: 9–19. [Crossref], [Web of Science ®] , [Google Scholar] has considered the problem of testing the monotonicity of this measure. We propose and study several kernel type estimators of the entropy of residual life through the estimation of f(x) log f(x). These estimators can be applied to the classification and comparison of lifetime distribution.     

Consistency of a nonparametric conditional mode estimator for random fields

Given a stationary multidimensional spatial process $\left\{ Z_{\mathbf{i}}=\left( X_{\mathbf{i}},\ Y_{\mathbf{i}}\right) \in \mathbb R ^d\right. \left. \times \mathbb R ,\mathbf{i}\in \mathbb Z ^{N}\right\} $ , we investigate a kernel estimate of the spatial conditional mode function of the response variable $Y_{\mathbf{i}}$ given the explicative variable $X_{\mathbf{i}}$ . Consistency in $L^p$ norm and strong convergence of the kernel estimate are obtained when the sample considered is a $\alpha $ -mixing sequence. An application to real data is given in order to illustrate the behavior of our methodology.     

Shock models with renewal failure rate properties

For the counting process N={N(t), t≥0} and the probability that a device survives the first k shocks \(\bar P_k \) , the probability that the device survives beyond t that is \(\bar H(t) = \sum\limits_{k = 0}^\omega {P(N(t) = k)} \bar P_k \) is considered. The survival \(\bar H(t)\) is proved to have the new better (worse) than used renewal failure rate and the new better (worse) than average failure rate properties under, some conditions on N and \((\bar P_k )_{k = \rho }^\omega \) . In particular we study the survival probability when N is a nonhomogeneous Poisson process or birth process. Acumulative damage model and Laplace transform characterization for properties are investigated. Further the generating functions for these renewal failure rates properties are given.     

Principal points for an allometric extension model

A set of \(n\) -principal points of a \(p\) -dimensional distribution is an optimal \(n\) -point-approximation of the distribution in terms of a squared error loss. It is in general difficult to derive an explicit expression of principal points. Hence, we may have to search the whole space \(R^p\) for \(n\) -principal points. Many efforts have been devoted to establish results that specify a linear subspace in which principal points lie. However, the previous studies focused on elliptically symmetric distributions and location mixtures of spherically symmetric distributions, which may not be suitable to many practical situations. In this paper, we deal with a mixture of elliptically symmetric distributions that form an allometric extension model, which has been widely used in the context of principal component analysis. We give conditions under which principal points lie in the linear subspace spanned by the first several principal components.