Journal of Probability and Statistics The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . 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. All rights reserved. Ratio Estimator in Adaptive Cluster Sampling without Replacement of Networks Tue, 27 May 2014 06:18:30 +0000 In this paper, we study the estimators of the population total in adaptive cluster sampling by using the information of the auxiliary variable. The numerical examples showed that the ratio estimator in adaptive cluster sampling without replacement of networks is more efficient than the ratio estimators in adaptive cluster sampling without replacement of units. Nipaporn Chutiman and Monchaya Chiangpradit Copyright © 2014 Nipaporn Chutiman and Monchaya Chiangpradit. All rights reserved. A New Lifetime Distribution and Its Power Transformation Sun, 18 May 2014 00:00:00 +0000 New one-parameter and two-parameter distributions are introduced in this paper. The failure rate of the one-parameter distribution is unimodal (upside-down bathtub), while the failure rate of the two-parameter distribution can be decreasing, increasing, unimodal, increasing-decreasing-increasing, or decreasing-increasing-decreasing, depending on the values of its two parameters. The two-parameter distribution is derived from the one-parameter distribution by using a power transformation. We discuss some properties of these two distributions, such as the behavior of the failure rate function, the probability density function, the moments, skewness, and kurtosis, and limiting distributions of order statistics. Maximum likelihood estimation for the two-parameter model using complete samples is investigated. Different algorithms for generating random samples from the two new models are given. Applications to real data are discussed and compared with the fit attained by some one- and two-parameter distributions. Finally, a simulation study is carried out to investigate the mean square error of the maximum likelihood estimators, the coverage probability, and the width of the confidence intervals of the unknown parameters. Ammar M. Sarhan, Lotfi Tadj, and David C. Hamilton Copyright © 2014 Ammar M. Sarhan et al. All rights reserved. Point-Symmetric Multivariate Density Function and Its Decomposition Tue, 13 May 2014 09:21:25 +0000 For a -variate density function, the present paper defines the point-symmetry, quasi-point-symmetry of order (), and the marginal point-symmetry of order and gives the theorem that the density function is -variate point-symmetric if and only if it is quasi-point-symmetric and marginal point-symmetric of order . The theorem is illustrated for the multivariate normal density function. Kiyotaka Iki and Sadao Tomizawa Copyright © 2014 Kiyotaka Iki and Sadao Tomizawa. All rights reserved. A Study of Probability Models in Monitoring Environmental Pollution in Nigeria Mon, 05 May 2014 13:52:47 +0000 In Lagos State, Nigeria, pollutant emissions were monitored across the state to detect any significant change which may cause harm to human health and the environment at large. In this research, three theoretical distributions, Weibull, lognormal, and gamma distributions, were examined on the carbon monoxide observations to determine the best fit. The characteristics of the pollutant observation were established and the probabilities of exceeding the Lagos State Environmental Protection Agency (LASEPA) and the Federal Environmental Protection Agency (FEPA) acceptable limits have been successfully predicted. Increase in the use of vehicles and increase in the establishment of industries have been found not to contribute significantly to the high level of carbon monoxide concentration in Lagos State for the period studied. P. E. Oguntunde, O. A. Odetunmibi, and A. O. Adejumo Copyright © 2014 P. E. Oguntunde et al. All rights reserved. Estimating the Reliability Function for a Family of Exponentiated Distributions Tue, 29 Apr 2014 12:19:05 +0000 A family of exponentiated distributions is proposed. The problems of estimating the reliability function are considered. Uniformly minimum variance unbiased estimators and maximum likelihood estimators are derived. A comparative study of the two methods of estimation is done. Simulation study is preformed. Ajit Chaturvedi and Anupam Pathak Copyright © 2014 Ajit Chaturvedi and Anupam Pathak. All rights reserved. Direct Determination of Smoothing Parameter for Penalized Spline Regression Tue, 22 Apr 2014 08:49:19 +0000 Penalized spline estimator is one of the useful smoothing methods. To construct the estimator, having goodness of fit and smoothness, the smoothing parameter should be appropriately selected. The purpose of this paper is to select the smoothing parameter using the asymptotic property of the penalized splines. The new smoothing parameter selection method is established in the context of minimization asymptotic form of MISE of the penalized splines. The mathematical and the numerical properties of the proposed method are studied. First we organize the new method in univariate regression model. Next we extend to the additive models. A simulation study to confirm the efficiency of the proposed method is addressed. Takuma Yoshida Copyright © 2014 Takuma Yoshida. All rights reserved. Convergence in Distribution of Some Self-Interacting Diffusions Tue, 15 Apr 2014 16:24:39 +0000 The present paper is concerned with some self-interacting diffusions living on . These diffusions are solutions to stochastic differential equations: , where is the empirical mean of the process , is an asymptotically strictly convex potential, and is a given positive function. We study the asymptotic behaviour of for three different families of functions . If with small enough, then the process converges in distribution towards the global minima of , whereas if or if , then converges in distribution if and only if. Aline Kurtzmann Copyright © 2014 Aline Kurtzmann. All rights reserved. Increased Statistical Efficiency in a Lognormal Mean Model Mon, 14 Apr 2014 08:01:30 +0000 Within the context of clinical and other scientific research, a substantial need exists for an accurate determination of the point estimate in a lognormal mean model, given that highly skewed data are often present. As such, logarithmic transformations are often advocated to achieve the assumptions of parametric statistical inference. Despite this, existing approaches that utilize only a sample’s mean and variance may not necessarily yield the most efficient estimator. The current investigation developed and tested an improved efficient point estimator for a lognormal mean by capturing more complete information via the sample’s coefficient of variation. Results of an empirical simulation study across varying sample sizes and population standard deviations indicated relative improvements in efficiency of up to 129.47 percent compared to the usual maximum likelihood estimator and up to 21.33 absolute percentage points above the efficient estimator presented by Shen and colleagues (2006). The relative efficiency of the proposed estimator increased particularly as a function of decreasing sample size and increasing population standard deviation. Grant H. Skrepnek and Ashok Sahai Copyright © 2014 Grant H. Skrepnek and Ashok Sahai. All rights reserved. The Exponentiated Half-Logistic Family of Distributions: Properties and Applications Thu, 13 Mar 2014 12:13:19 +0000 We study some mathematical properties of a new generator of continuous distributions with two extra parameters called the exponentiated half-logistic family. We present some special models. We investigate the shapes of the density and hazard rate function. We derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions, probability weighted moments, Bonferroni and Lorenz curves, Shannon and Rényi entropies, and order statistics, which hold for any baseline model. We introduce two bivariate extensions of this family. We discuss the estimation of the model parameters by maximum likelihood and demonstrate the potentiality of the new family by means of two real data sets. Gauss M. Cordeiro, Morad Alizadeh, and Edwin M. M. Ortega Copyright © 2014 Gauss M. Cordeiro et al. All rights reserved. A Software Reliability Model Using Quantile Function Tue, 11 Mar 2014 15:45:56 +0000 We study a class of software reliability models using quantile function. Various distributional properties of the class of distributions are studied. We also discuss the reliability characteristics of the class of distributions. Inference procedures on parameters of the model based on L-moments are studied. We apply the proposed model to a real data set. Bijamma Thomas, Midhu Narayanan Nellikkattu, and Sankaran Godan Paduthol Copyright © 2014 Bijamma Thomas et al. All rights reserved. An Improved Class of Chain Ratio-Product Type Estimators in Two-Phase Sampling Using Two Auxiliary Variables Thu, 06 Mar 2014 16:04:38 +0000 This paper presents a technique for estimating finite population mean of the study variable in the presence of two auxiliary variables using two-phase sampling scheme when the regression line does not pass through the neighborhood of the origin. The properties of the proposed class of estimators are studied under large sample approximation. In addition, bias and efficiency comparisons are carried out to study the performances of the proposed class of estimators over the existing estimators. It has also been shown that the proposed technique has greater applicability in survey research. An empirical study is carried out to demonstrate the performance of the proposed estimators. Gajendra K. Vishwakarma and Manish Kumar Copyright © 2014 Gajendra K. Vishwakarma and Manish Kumar. All rights reserved. A Study on the Chain Ratio-Type Estimator of Finite Population Variance Mon, 24 Feb 2014 06:53:50 +0000 We suggest an estimator using two auxiliary variables for the estimation of the unknown population variance. The bias and the mean square error of the proposed estimator are obtained to the first order of approximations. In addition, the problem is extended to two-phase sampling scheme. After theoretical comparisons, as an illustration, a numerical comparison is carried out to examine the performance of the suggested estimator with several estimators. Yunusa Olufadi and Cem Kadilar Copyright © 2014 Yunusa Olufadi and Cem Kadilar. All rights reserved.