Journal of Probability and Statistics The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . All rights reserved. Approximating Explicitly the Mean-Reverting CEV Process Mon, 23 Nov 2015 11:09:25 +0000 We are interested in the numerical solution of mean-reverting CEV processes that appear in financial mathematics models and are described as nonnegative solutions of certain stochastic differential equations with sublinear diffusion coefficients of the form where . Our goal is to construct explicit numerical schemes that preserve positivity. We prove convergence of the proposed SD scheme with rate depending on the parameter . Furthermore, we verify our findings through numerical experiments and compare with other positivity preserving schemes. Finally, we show how to treat the two-dimensional stochastic volatility model with instantaneous variance process given by the above mean-reverting CEV process. N. Halidias and I. S. Stamatiou Copyright © 2015 N. Halidias and I. S. Stamatiou. All rights reserved. Convex and Radially Concave Contoured Distributions Mon, 23 Nov 2015 08:02:58 +0000 Integral representations of the locally defined star-generalized surface content measures on star spheres are derived for boundary spheres of balls being convex or radially concave with respect to a fan in . As a result, the general geometric measure representation of star-shaped probability distributions and the general stochastic representation of the corresponding random vectors allow additional specific interpretations in the two mentioned cases. Applications to estimating and testing hypotheses on scaling parameters are presented, and two-dimensional sample clouds are simulated. Wolf-Dieter Richter Copyright © 2015 Wolf-Dieter Richter. All rights reserved. On Association Measures for Continuous Variables and Correction for Chance Wed, 18 Nov 2015 07:01:16 +0000 This paper studies correction for chance for association measures for continuous variables. The set of linear transformations of Pearson’s product-moment correlation is used as the domain of the correction for chance function. Examples of measures in this set are Tucker’s congruence coefficient, Jobson’s coefficient, and Pearson’s correlation. An equivalence relation on the set of linear transformations is defined. The fixed points of the correction for chance function are characterized. It is shown that each linear transformation is mapped to the fixed point in its equivalence class. Matthijs J. Warrens Copyright © 2015 Matthijs J. Warrens. All rights reserved. Statistical Tests for the Reciprocal of a Normal Mean with a Known Coefficient of Variation Wed, 11 Nov 2015 09:38:51 +0000 An asymptotic test and an approximate test for the reciprocal of a normal mean with a known coefficient of variation were proposed in this paper. The asymptotic test was based on the expectation and variance of the estimator of the reciprocal of a normal mean. The approximate test used the approximate expectation and variance of the estimator by Taylor series expansion. A Monte Carlo simulation study was conducted to compare the performance of the two statistical tests. Simulation results showed that the two proposed tests performed well in terms of empirical type I errors and power. Nevertheless, the approximate test was easier to compute than the asymptotic test. Wararit Panichkitkosolkul Copyright © 2015 Wararit Panichkitkosolkul. All rights reserved. Confidence Interval Estimation of an ROC Curve: An Application of Generalized Half Normal and Weibull Distributions Sun, 08 Nov 2015 12:49:16 +0000 In the recent past, the work in the area of ROC analysis gained attention in explaining the accuracy of a test and identification of the optimal threshold. Such types of ROC models are referred to as bidistributional ROC models, for example Binormal, Bi-Exponential, Bi-Logistic and so forth. However, in practical situations, we come across data which are skewed in nature with extended tails. Then to address this issue, the accuracy of a test is to be explained by involving the scale and shape parameters. Hence, the present paper focuses on proposing an ROC model which takes into account two generalized distributions which helps in explaining the accuracy of a test. Further, confidence intervals are constructed for the proposed curve; that is, coordinates of the curve (FPR, TPR) and accuracy measure, Area Under the Curve (AUC), which helps in explaining the variability of the curve and provides the sensitivity at a particular value of specificity and vice versa. The proposed methodology is supported by a real data set and simulation studies. S. Balaswamy and R. Vishnu Vardhan Copyright © 2015 S. Balaswamy and R. Vishnu Vardhan. All rights reserved. Estimation of Population Mean in Chain Ratio-Type Estimator under Systematic Sampling Tue, 03 Nov 2015 07:56:27 +0000 A chain ratio-type estimator is proposed for the estimation of finite population mean under systematic sampling scheme using two auxiliary variables. The mean square error of the proposed estimator is derived up to the first order of approximation and is compared with other relevant existing estimators. To illustrate the performances of the different estimators in comparison with the usual simple estimator, we have taken a real data set from the literature of survey sampling. Mursala Khan and Rajesh Singh Copyright © 2015 Mursala Khan and Rajesh Singh. All rights reserved. Some Characterization Results on Dynamic Cumulative Residual Tsallis Entropy Thu, 29 Oct 2015 11:13:12 +0000 We propose a generalized cumulative residual information measure based on Tsallis entropy and its dynamic version. We study the characterizations of the proposed information measure and define new classes of life distributions based on this measure. Some applications are provided in relation to weighted and equilibrium probability models. Finally the empirical cumulative Tsallis entropy is proposed to estimate the new information measure. Madan Mohan Sati and Nitin Gupta Copyright © 2015 Madan Mohan Sati and Nitin Gupta. All rights reserved. An Production Inventory Controlled Self-Service Queuing System Thu, 29 Oct 2015 08:59:46 +0000 We consider a multiserver Markovian queuing system where each server provides service only to one customer. Arrival of customers is according to a Poisson process and whenever a customer leaves the system after getting service, that server is also removed from the system. Here the servers are considered as a standard production inventory. Behavior of this system is studied using a three-dimensional QBD process. The condition for checking ergodicity and the steady state solutions are obtained using matrix analytic method. Unlike classical queuing models, the number of servers varies in this model according to an inventory policy. Anoop N. Nair and M. J. Jacob Copyright © 2015 Anoop N. Nair and M. J. Jacob. All rights reserved. Generalized Inferences about the Mean Vector of Several Multivariate Gaussian Processes Thu, 29 Oct 2015 08:39:15 +0000 We consider in this paper the problem of comparing the means of several multivariate Gaussian processes. It is assumed that the means depend linearly on an unknown vector parameter and that nuisance parameters appear in the covariance matrices. More precisely, we deal with the problem of testing hypotheses, as well as obtaining confidence regions for . Both methods will be based on the concepts of generalized value and generalized confidence region adapted to our context. Pilar Ibarrola and Ricardo Vélez Copyright © 2015 Pilar Ibarrola and Ricardo Vélez. All rights reserved. An Ambit Stochastic Approach to Pricing Electricity Forward Contracts: The Case of the German Energy Market Tue, 27 Oct 2015 12:57:51 +0000 We propose an ambit stochastic model to study the electricity forward prices. We provide a detailed analysis of the probabilistic properties of such model, discussing the related martingale conditions and deriving concrete implementation of it for the related underlying spot price. The latter is obtained from the forward model through a limiting argument. Furthermore, we show, also providing a concrete example, that a proper specification of these models is able to effectively forecast prices of forward contracts written on the European Energy Exchange (EEX) AG, or German Energy Exchange, market. Luca Di Persio and Isacco Perin Copyright © 2015 Luca Di Persio and Isacco Perin. 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.