Journal of Probability and Statistics The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . All rights reserved. Comparison of the Frequentist MATA Confidence Interval with Bayesian Model-Averaged Confidence Intervals Thu, 08 Oct 2015 12:05:45 +0000 Model averaging is a technique used to account for model uncertainty, in both Bayesian and frequentist multimodel inferences. In this paper, we compare the performance of model-averaged Bayesian credible intervals and frequentist confidence intervals. Frequentist intervals are constructed according to the model-averaged tail area (MATA) methodology. Differences between the Bayesian and frequentist methods are illustrated through an example involving cloud seeding. The coverage performance and interval width of each technique are then studied using simulation. A frequentist MATA interval performs best in the normal linear setting, while Bayesian credible intervals yield the best coverage performance in a lognormal setting. The use of a data-dependent prior probability for models improved the coverage of the model-averaged Bayesian interval, relative to that using uniform model prior probabilities. Data-dependent model prior probabilities are philosophically controversial in Bayesian statistics, and our results suggest that their use is beneficial when model averaging. Daniel Turek Copyright © 2015 Daniel Turek. All rights reserved. Measurement of Interobserver Disagreement: Correction of Cohen’s Kappa for Negative Values Wed, 30 Sep 2015 14:27:19 +0000 As measures of interobserver agreement for both nominal and ordinal categories, Cohen’s kappa coefficients appear to be the most widely used with simple and meaningful interpretations. However, for negative coefficient values when (the probability of) observed disagreement exceeds chance-expected disagreement, no fixed lower bounds exist for the kappa coefficients and their interpretations are no longer meaningful and may be entirely misleading. In this paper, alternative measures of disagreement (or negative agreement) are proposed as simple corrections or modifications of Cohen’s kappa coefficients. The new coefficients have a fixed lower bound of −1 that can be attained irrespective of the marginal distributions. A coefficient is formulated for the case when the classification categories are nominal and a weighted coefficient is proposed for ordinal categories. Besides coefficients for the overall disagreement across categories, disagreement coefficients for individual categories are presented. Statistical inference procedures are developed and numerical examples are provided. Tarald O. Kvålseth Copyright © 2015 Tarald O. Kvålseth. All rights reserved. Generalized Information for the -Order Normal Distribution Wed, 30 Sep 2015 11:33:26 +0000 This paper investigates a generalization of Fisher’s entropy type information measure under the multivariate -order normal distribution, related to his measure, as well as its corresponding Shannon entropy. Certain boundaries of this information measure are also proved and discussed. Thomas L. Toulias Copyright © 2015 Thomas L. Toulias. All rights reserved. Residual and Past Entropy for Concomitants of Ordered Random Variables of Morgenstern Family Sun, 27 Sep 2015 14:15:51 +0000 For a system, which is observed at time t, the residual and past entropies measure the uncertainty about the remaining and the past life of the distribution, respectively. In this paper, we have presented the residual and past entropy of Morgenstern family based on the concomitants of the different types of generalized order statistics (gos) and give the linear transformation of such model. Characterization results for these dynamic entropies for concomitants of ordered random variables have been considered. M. M. Mohie EL-Din, M. M. Amein, Nahed S. A. Ali, and M. S. Mohamed Copyright © 2015 M. M. Mohie EL-Din et al. All rights reserved. Robust Bayesian Regularized Estimation Based on Regression Model Sun, 20 Sep 2015 09:41:14 +0000 The distribution is a useful extension of the normal distribution, which can be used for statistical modeling of data sets with heavy tails, and provides robust estimation. In this paper, in view of the advantages of Bayesian analysis, we propose a new robust coefficient estimation and variable selection method based on Bayesian adaptive Lasso regression. A Gibbs sampler is developed based on the Bayesian hierarchical model framework, where we treat the distribution as a mixture of normal and gamma distributions and put different penalization parameters for different regression coefficients. We also consider the Bayesian regression with adaptive group Lasso and obtain the Gibbs sampler from the posterior distributions. Both simulation studies and real data example show that our method performs well compared with other existing methods when the error distribution has heavy tails and/or outliers. Zean Li and Weihua Zhao Copyright © 2015 Zean Li and Weihua Zhao. All rights reserved. Success Run Waiting Times and Fuss-Catalan Numbers Mon, 14 Sep 2015 13:32:19 +0000 We present power series expressions for all the roots of the auxiliary equation of the recurrence relation for the distribution of the waiting time for the first run of consecutive successes in a sequence of independent Bernoulli trials, that is, the geometric distribution of order . We show that the series coefficients are Fuss-Catalan numbers and write the roots in terms of the generating function of the Fuss-Catalan numbers. Our main result is a new exact expression for the distribution, which is more concise than previously published formulas. Our work extends the analysis by Feller, who gave asymptotic results. We obtain quantitative improvements of the error estimates obtained by Feller. S. J. Dilworth and S. R. Mane Copyright © 2015 S. J. Dilworth and S. R. Mane. All rights reserved. On the Computation of the Survival Probability of Brownian Motion with Drift in a Closed Time Interval When the Absorbing Boundary Is a Step Function Wed, 09 Sep 2015 07:40:00 +0000 This paper provides explicit formulae for the probability that an arithmetic or a geometric Brownian motion will not cross an absorbing boundary defined as a step function during a finite time interval. Various combinations of downward and upward steps are handled. Numerical computation of the survival probability is done quasi-instantaneously and with utmost precision. The sensitivity of the survival probability to the number and the ordering of the steps in the boundary is analyzed. Tristan Guillaume Copyright © 2015 Tristan Guillaume. All rights reserved. The Concepts of Pseudo Compound Poisson and Partition Representations in Discrete Probability Mon, 31 Aug 2015 13:28:04 +0000 The mathematical/statistical concepts of pseudo compound Poisson and partition representations in discrete probability are reviewed and clarified. A combinatorial interpretation of the convolution of geometric distributions in terms of a variant of Newton’s identities is obtained. The practical use of the twofold convolution leads to an improved goodness-of-fit for a data set from automobile insurance that was up to now not fitted satisfactorily. Werner Hürlimann Copyright © 2015 Werner Hürlimann. All rights reserved. The Type I Generalized Half-Logistic Distribution Based on Upper Record Values Thu, 13 Aug 2015 13:25:55 +0000 We consider the type I generalized half-logistic distribution and derive some new explicit expressions and recurrence relations for marginal and joint moment generating functions of upper record values. Here we show the computations for the first four moments and their variances. Next we show that results for record values of this distribution can be derived from our results as special cases. We obtain the characterization result of this distribution on using the recurrence relation for single moment and conditional expectation of upper record values. We obtain the maximum likelihood estimators of upper record values and their confidence intervals. Also, we compute the maximum likelihood estimates of the parameters of upper record values and their confidence intervals. At last, we present one real case data study to emphasize the results of this paper. Devendra Kumar, Neetu Jain, and Shivani Gupta Copyright © 2015 Devendra Kumar et al. All rights reserved. Kim and Omberg Revisited: The Duality Approach Wed, 12 Aug 2015 08:22:30 +0000 We give an alternative duality-based proof to the solution of the expected utility maximization problem analyzed by Kim and Omberg. In so doing, we also provide an example of incomplete-market optimal investment problem for which the duality approach is conducive to an explicit solution. Anna Battauz, Marzia De Donno, and Alessandro Sbuelz Copyright © 2015 Anna Battauz et al. All rights reserved. Optimal Bandwidth Selection for Kernel Density Functionals Estimation Thu, 06 Aug 2015 06:30:21 +0000 The choice of bandwidth is crucial to the kernel density estimation (KDE) and kernel based regression. Various bandwidth selection methods for KDE and local least square regression have been developed in the past decade. It has been known that scale and location parameters are proportional to density functionals with appropriate choice of and furthermore equality of scale and location tests can be transformed to comparisons of the density functionals among populations. can be estimated nonparametrically via kernel density functionals estimation (KDFE). However, the optimal bandwidth selection for KDFE of has not been examined. We propose a method to select the optimal bandwidth for the KDFE. The idea underlying this method is to search for the optimal bandwidth by minimizing the mean square error (MSE) of the KDFE. Two main practical bandwidth selection techniques for the KDFE of are provided: Normal scale bandwidth selection (namely, “Rule of Thumb”) and direct plug-in bandwidth selection. Simulation studies display that our proposed bandwidth selection methods are superior to existing density estimation bandwidth selection methods in estimating density functionals. Su Chen Copyright © 2015 Su Chen. All rights reserved. Generalized Residual Entropy and Upper Record Values Sun, 02 Aug 2015 13:31:48 +0000 In this communication, we deal with a generalized residual entropy of record values and weighted distributions. Some results on monotone behaviour of generalized residual entropy in record values are obtained. Upper and lower bounds are presented. Further, based on this measure, we study some comparison results between a random variable and its weighted version. Finally, we describe some estimation techniques to estimate the generalized residual entropy of a lifetime distribution. Suchandan Kayal Copyright © 2015 Suchandan Kayal. All rights reserved. Recurrence Relation and Accurate Value on Inverse Moment of Discrete Distributions Sun, 05 Jul 2015 07:43:52 +0000 Properties of the generalized hypergeometric series functions are employed to get the recurrence relation for inverse moments and inverse factorial moments of some discrete distributions. Meanwhile, with the existence of the recurrence relations, the accurate value for inverse moment of discrete distributions can thus be obtained. ChunYuan Wang and Wuyungaowa Copyright © 2015 ChunYuan Wang and Wuyungaowa. All rights reserved. Moderate and Large Deviations for the Smoothed Estimate of Sample Quantiles Thu, 11 Jun 2015 16:49:31 +0000 We derive the moderate and large deviations principle for the smoothed sample quantile from a sequence of independent and identically distributed samples of size . Xiaoxia He, Xi Liu, and Chun Yao Copyright © 2015 Xiaoxia He et al. All rights reserved. Generalized Fractional Integral Inequalities for Continuous Random Variables Thu, 01 Jan 2015 09:55:51 +0000 Some generalized integral inequalities are established for the fractional expectation and the fractional variance for continuous random variables. Special cases of integral inequalities in this paper are studied by Barnett et al. and Dahmani. Abdullah Akkurt, Zeynep Kaçar, and Hüseyin Yildirim Copyright © 2015 Abdullah Akkurt et al. All rights reserved. New Indices for Refining Multiple Choice Questions Tue, 23 Dec 2014 13:52:34 +0000 Multiple choice questions (MCQs) are one of the most popular tools to evaluate learning and knowledge in higher education. Nowadays, there are a few indices to measure reliability and validity of these questions, for instance, to check the difficulty of a particular question (item) or the ability to discriminate from less to more knowledge. In this work two new indices have been constructed: (i) the no answer index measures the relationship between the number of errors and the number of no answers; (ii) the homogeneity index measures homogeneity of the wrong responses (distractors). The indices are based on the lack-of-fit statistic, whose distribution is approximated by a chi-square distribution for a large number of errors. An algorithm combining several traditional and new indices has been developed to refine continuously a database of MCQs. The final objective of this work is the classification of MCQs from a large database of items in order to produce an automated-supervised system of generating tests with specific characteristics, such as more or less difficulty or capacity of discriminating knowledge of the topic. Mariano Amo-Salas, María del Mar Arroyo-Jimenez, David Bustos-Escribano, Eva Fairén-Jiménez, and Jesús López-Fidalgo Copyright © 2014 Mariano Amo-Salas et al. All rights reserved. Defining Sample Quantiles by the True Rank Probability Mon, 08 Dec 2014 09:25:58 +0000 Many definitions exist for sample quantiles and are included in statistical software. The need to adopt a standard definition of sample quantiles has been recognized and different definitions have been compared in terms of satisfying some desirable properties, but no consensus has been found. We outline here that comparisons of the sample quantile definitions are irrelevant because the probabilities associated with order-ranked sample values are known exactly. Accordingly, the standard definition for sample quantiles should be based on the true rank probabilities. We show that this allows more accurate inference of the tails of the distribution, and thus improves estimation of the probability of extreme events. Lasse Makkonen and Matti Pajari Copyright © 2014 Lasse Makkonen and Matti Pajari. All rights reserved. Bayesian Inference of a Multivariate Regression Model Mon, 24 Nov 2014 00:00:00 +0000 We explore Bayesian inference of a multivariate linear regression model with use of a flexible prior for the covariance structure. The commonly adopted Bayesian setup involves the conjugate prior, multivariate normal distribution for the regression coefficients and inverse Wishart specification for the covariance matrix. Here we depart from this approach and propose a novel Bayesian estimator for the covariance. A multivariate normal prior for the unique elements of the matrix logarithm of the covariance matrix is considered. Such structure allows for a richer class of prior distributions for the covariance, with respect to strength of beliefs in prior location hyperparameters, as well as the added ability, to model potential correlation amongst the covariance structure. The posterior moments of all relevant parameters of interest are calculated based upon numerical results via a Markov chain Monte Carlo procedure. The Metropolis-Hastings-within-Gibbs algorithm is invoked to account for the construction of a proposal density that closely matches the shape of the target posterior distribution. As an application of the proposed technique, we investigate a multiple regression based upon the 1980 High School and Beyond Survey. Marick S. Sinay and John S. J. Hsu Copyright © 2014 Marick S. Sinay and John S. J. Hsu. All rights reserved. Parameter Estimation of Population Pharmacokinetic Models with Stochastic Differential Equations: Implementation of an Estimation Algorithm Mon, 10 Nov 2014 10:28:23 +0000 Population pharmacokinetic (PPK) models play a pivotal role in quantitative pharmacology study, which are classically analyzed by nonlinear mixed-effects models based on ordinary differential equations. This paper describes the implementation of SDEs in population pharmacokinetic models, where parameters are estimated by a novel approximation of likelihood function. This approximation is constructed by combining the MCMC method used in nonlinear mixed-effects modeling with the extended Kalman filter used in SDE models. The analysis and simulation results show that the performance of the approximation of likelihood function for mixed-effects SDEs model and analysis of population pharmacokinetic data is reliable. The results suggest that the proposed method is feasible for the analysis of population pharmacokinetic data. Fang-Rong Yan, Ping Zhang, Jun-Lin Liu, Yu-Xi Tao, Xiao Lin, Tao Lu, and Jin-Guan Lin Copyright © 2014 Fang-Rong Yan et al. All rights reserved. On Volatility Swaps for Stock Market Forecast: Application Example CAC 40 French Index Sun, 09 Nov 2014 06:48:47 +0000 This paper focuses on the pricing of variance and volatility swaps under Heston model (1993). To this end, we apply this model to the empirical financial data: CAC 40 French Index. More precisely, we make an application example for stock market forecast: CAC 40 French Index to price swap on the volatility using GARCH(1,1) model. Halim Zeghdoudi, Abdellah Lallouche, and Mohamed Riad Remita Copyright © 2014 Halim Zeghdoudi et al. All rights reserved. New Approach for Finding Basic Performance Measures of Single Server Queue Mon, 20 Oct 2014 13:39:20 +0000 Consider the single server queue in which the system capacity is infinite and the customers are served on a first come, first served basis. Suppose the probability density function and the cumulative distribution function of the interarrival time are such that the rate tends to a constant as , and the rate computed from the distribution of the service time tends to another constant. When the queue is in a stationary state, we derive a set of equations for the probabilities of the queue length and the states of the arrival and service processes. Solving the equations, we obtain approximate results for the stationary probabilities which can be used to obtain the stationary queue length distribution and waiting time distribution of a customer who arrives when the queue is in the stationary state. Siew Khew Koh, Ah Hin Pooi, and Yi Fei Tan Copyright © 2014 Siew Khew Koh et al. All rights reserved. On Improving Ratio/Product Estimator by Ratio/Product-cum-Mean-per-Unit Estimator Targeting More Efficient Use of Auxiliary Information Tue, 23 Sep 2014 08:37:59 +0000 To achieve a more efficient use of auxiliary information we propose single-parameter ratio/product-cum-mean-per-unit estimators for a finite population mean in a simple random sample without replacement when the magnitude of the correlation coefficient is not very high (less than or equal to 0.7). The first order large sample approximation to the bias and the mean square error of our proposed estimators are obtained. We use simulation to compare our estimators with the well-known sample mean, ratio, and product estimators, as well as the classical linear regression estimator for efficient use of auxiliary information. The results are conforming to our motivating aim behind our proposition. Angela Shirley, Ashok Sahai, and Isaac Dialsingh Copyright © 2014 Angela Shirley et al. All rights reserved. Subgeometric Ergodicity Analysis of Continuous-Time Markov Chains under Random-Time State-Dependent Lyapunov Drift Conditions Sun, 31 Aug 2014 11:20:04 +0000 We investigate random-time state-dependent Foster-Lyapunov analysis on subgeometric rate ergodicity of continuous-time Markov chains (CTMCs). We are mainly concerned with making use of the available results on deterministic state-dependent drift conditions for CTMCs and on random-time state-dependent drift conditions for discrete-time Markov chains and transferring them to CTMCs. Mokaedi V. Lekgari Copyright © 2014 Mokaedi V. Lekgari. All rights reserved. Subgeometric Ergodicity under Random-Time State-Dependent Drift Conditions Thu, 24 Jul 2014 00:00:00 +0000 Motivated by possible applications of Lyapunov techniques in the stability of stochastic networks, subgeometric ergodicity of Markov chains is investigated. In a nutshell, in this study we take a look at -ergodic general Markov chains, subgeometrically ergodic at rate , when the random-time Foster-Lyapunov drift conditions on a set of stopping times are satisfied. Mokaedi V. Lekgari Copyright © 2014 Mokaedi V. Lekgari. All rights reserved. An Analysis of a Heuristic Procedure to Evaluate Tail (in)dependence Mon, 21 Jul 2014 08:45:54 +0000 Measuring tail dependence is an important issue in many applied sciences in order to quantify the risk of simultaneous extreme events. A usual measure is given by the tail dependence coefficient. The characteristics of events behave quite differently as these become more extreme, whereas we are in the class of asymptotic dependence or in the class of asymptotic independence. The literature has emphasized the asymptotic dependent class but wrongly infers that tail dependence will result in the overestimation of extreme value dependence and consequently of the risk. In this paper we analyze this issue through simulation based on a heuristic procedure. Marta Ferreira and Sérgio Silva Copyright © 2014 Marta Ferreira and Sérgio Silva. All rights reserved. On -Gamma and -Beta Distributions and Moment Generating Functions Tue, 15 Jul 2014 10:08:41 +0000 The main objective of the present paper is to define -gamma and -beta distributions and moments generating function for the said distributions in terms of a new parameter . Also, the authors prove some properties of these newly defined distributions. Gauhar Rahman, Shahid Mubeen, Abdur Rehman, and Mammona Naz Copyright © 2014 Gauhar Rahman et al. All rights reserved. Two Bootstrap Strategies for a -Problem up to Location-Scale with Dependent Samples Sun, 13 Jul 2014 11:53:53 +0000 This paper extends the work of Quessy and Éthier (2012) who considered tests for the -sample problem with dependent samples. Here, the marginal distributions are allowed, under , to differ according to their mean and their variance; in other words, one focuses on the shape of the distributions. Although easily stated, this problem nevertheless requires a careful treatment for the computation of valid values. To this end, two bootstrap strategies based on the multiplier central limit theorem are proposed, both exploiting a representation of the test statistics in terms of a Hadamard differentiable functional. This accounts for the fact that one works with empirically standardized data instead of the original observations. Simulations reported show the nice sample properties of the method based on Cramér-von Mises and characteristic function type statistics. The newly introduced tests are illustrated on the marginal distributions of the eight-dimensional Oil currency data set. Jean-François Quessy and François Éthier Copyright © 2014 Jean-François Quessy and François Éthier. All rights reserved. Risk Efficiencies of Empirical Bayes and Generalized Maximum Likelihood Estimates for Rayleigh Model under Censored Data Wed, 02 Jul 2014 11:26:37 +0000 The comparison of empirical Bayes and generalized maximum likelihood estimates of reliability performances is made in terms of risk efficiencies when the data are progressively Type II censored from Rayleigh distribution. The empirical Bayes estimates are obtained using an asymmetric loss function. The risk functions of the estimates and risk efficiencies are obtained under this loss function. A real data set is presented to illustrate the proposed comparison method, and the performance of the estimates is examined and compared in terms of risk efficiencies by means of Monte Carlo simulations. The simulation results indicate that the proposed empirical Bayes estimates are more preferable than the generalized maximum likelihood estimates. Dinesh Barot and Manhar Patel Copyright © 2014 Dinesh Barot and Manhar Patel. All rights reserved. Bandwidth Selection for Recursive Kernel Density Estimators Defined by Stochastic Approximation Method Mon, 02 Jun 2014 11:55:03 +0000 We propose an automatic selection of the bandwidth of the recursive kernel estimators of a probability density function defined by the stochastic approximation algorithm introduced by Mokkadem et al. (2009a). We showed that, using the selected bandwidth and the stepsize which minimize the MISE (mean integrated squared error) of the class of the recursive estimators defined in Mokkadem et al. (2009a), the recursive estimator will be better than the nonrecursive one for small sample setting in terms of estimation error and computational costs. We corroborated these theoretical results through simulation study. Yousri Slaoui Copyright © 2014 Yousri Slaoui. All rights reserved. Sum of Bernoulli Mixtures: Beyond Conditional Independence Mon, 02 Jun 2014 09:19:24 +0000 We consider the distribution of the sum of Bernoulli mixtures under a general dependence structure. The level of dependence is measured in terms of a limiting conditional correlation between two of the Bernoulli random variables. The conditioning event is that the mixing random variable is larger than a threshold and the limit is with respect to the threshold tending to one. The large-sample distribution of the empirical frequency and its use in approximating the risk measures, value at risk and conditional tail expectation, are presented for a new class of models which we call double mixtures. Several illustrative examples with a Beta mixing distribution, are given. As well, some data from the area of credit risk are fit with the models, and comparisons are made between the new models and also the classical Beta-binomial model. Taehan Bae and Ian Iscoe Copyright © 2014 Taehan Bae and Ian Iscoe. 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