Journal of Probability and Statistics The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . 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. Improved Inference for Moving Average Disturbances in Nonlinear Regression Models Thu, 13 Feb 2014 09:46:24 +0000 This paper proposes an improved likelihood-based method to test for first-order moving average in the disturbances of nonlinear regression models. The proposed method has a third-order distributional accuracy which makes it particularly attractive for inference in small sample sizes models. Compared to the commonly used first-order methods such as likelihood ratio and Wald tests which rely on large samples and asymptotic properties of the maximum likelihood estimation, the proposed method has remarkable accuracy. Monte Carlo simulations are provided to show how the proposed method outperforms the existing ones. Two empirical examples including a power regression model of aggregate consumption and a Gompertz growth model of mobile cellular usage in the US are presented to illustrate the implementation and usefulness of the proposed method in practice. Pierre Nguimkeu Copyright © 2014 Pierre Nguimkeu. All rights reserved. A General Result on the Mean Integrated Squared Error of the Hard Thresholding Wavelet Estimator under -Mixing Dependence Sun, 19 Jan 2014 09:18:45 +0000 We consider the estimation of an unknown function for weakly dependent data (-mixing) in a general setting. Our contribution is theoretical: we prove that a hard thresholding wavelet estimator attains a sharp rate of convergence under the mean integrated squared error (MISE) over Besov balls without imposing too restrictive assumptions on the model. Applications are given for two types of inverse problems: the deconvolution density estimation and the density estimation in a GARCH-type model, both improve existing results in this dependent context. Another application concerns the regression model with random design. Christophe Chesneau Copyright © 2014 Christophe Chesneau. All rights reserved. A Batch Arrival Single Server Queue with Server Providing General Service in Two Fluctuating Modes and Reneging during Vacation and Breakdowns Wed, 08 Jan 2014 14:49:16 +0000 We study the behavior of a batch arrival queuing system equipped with a single server providing general arbitrary service to customers with different service rates in two fluctuating modes of service. In addition, the server is subject to random breakdown. As soon as the server faces breakdown, the customer whose service is interrupted comes back to the head of the queue. As soon as repair process of the server is complete, the server immediately starts providing service in mode 1. Also customers waiting for service may renege (leave the queue) when there is breakdown or when server takes vacation. The system provides service with complete or reduced efficiency due to the fluctuating rates of service. We derive the steady state queue size distribution. Some special cases are discussed and numerical illustration is provided to see the effect and validity of the results. Monita Baruah, Kailash C. Madan, and Tillal Eldabi Copyright © 2014 Monita Baruah et al. All rights reserved. Parametric Regression Models Using Reversed Hazard Rates Mon, 06 Jan 2014 09:30:22 +0000 Proportional hazard regression models are widely used in survival analysis to understand and exploit the relationship between survival time and covariates. For left censored survival times, reversed hazard rate functions are more appropriate. In this paper, we develop a parametric proportional hazard rates model using an inverted Weibull distribution. The estimation and construction of confidence intervals for the parameters are discussed. We assess the performance of the proposed procedure based on a large number of Monte Carlo simulations. We illustrate the proposed method using a real case example. Asokan Mulayath Variyath and P. G. Sankaran Copyright © 2014 Asokan Mulayath Variyath and P. G. Sankaran. All rights reserved. The Beta Generalized Half-Normal Distribution: New Properties Tue, 31 Dec 2013 14:20:42 +0000 We study some mathematical properties of the beta generalized half-normal distribution recently proposed by Pescim et al. (2010). This model is quite flexible for analyzing positive real data since it contains as special models the half-normal, exponentiated half-normal, and generalized half-normal distributions. We provide a useful power series for the quantile function. Some new explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability, and entropy. We demonstrate that the density function of the beta generalized half-normal order statistics can be expressed as a mixture of generalized half-normal densities. We obtain two closed-form expressions for their moments and other statistical measures. The method of maximum likelihood is used to estimate the model parameters censored data. The beta generalized half-normal model is modified to cope with long-term survivors may be present in the data. The usefulness of this distribution is illustrated in the analysis of four real data sets. Gauss M. Cordeiro, Rodrigo R. Pescim, Edwin M. M. Ortega, and Clarice G. B. Demétrio Copyright © 2013 Gauss M. Cordeiro et al. All rights reserved. Block Empirical Likelihood for Semiparametric Varying-Coefficient Partially Linear Errors-in-Variables Models with Longitudinal Data Sun, 29 Dec 2013 14:44:41 +0000 Block empirical likelihood inference for semiparametric varying-coeffcient partially linear errors-in-variables models with longitudinal data is investigated. We apply the block empirical likelihood procedure to accommodate the within-group correlation of the longitudinal data. The block empirical log-likelihood ratio statistic for the parametric component is suggested. And the nonparametric version of Wilk’s theorem is derived under mild conditions. Simulations are carried out to access the performance of the proposed procedure. Yafeng Xia and Hu Da Copyright © 2013 Yafeng Xia and Hu Da. All rights reserved. Estimation of Parameters of Generalized Inverted Exponential Distribution for Progressive Type-II Censored Sample with Binomial Removals Sun, 29 Dec 2013 08:36:05 +0000 We obtained the maximum likelihood and Bayes estimators of the parameters of the generalized inverted exponential distribution in case of the progressive type-II censoring scheme with binomial removals. Bayesian estimation procedure has been discussed under the consideration of the square error and general entropy loss functions while the model parameters follow the gamma prior distributions. The performances of the maximum likelihood and Bayes estimators are compared in terms of their risks through the simulation study. Further, we have also derived the expression of the expected experiment time to get a progressively censored sample with binomial removals, consisting of specified number of observations from generalized inverted exponential distribution. An illustrative example based on a real data set has also been given. Sanjay Kumar Singh, Umesh Singh, and Manoj Kumar Copyright © 2013 Sanjay Kumar Singh et al. All rights reserved. The Central Limit Theorem for th-Order Nonhomogeneous Markov Information Source Wed, 11 Dec 2013 13:34:15 +0000 We prove a central limit theorem for th-order nonhomogeneous Markov information source by using the martingale central limit theorem under the condition of convergence of transition probability matrices for nonhomogeneous Markov chain in Cesàro sense. Huilin Huang Copyright © 2013 Huilin Huang. All rights reserved. Confidence Intervals for the Coefficient of Variation in a Normal Distribution with a Known Population Mean Thu, 21 Nov 2013 13:24:06 +0000 This paper presents three confidence intervals for the coefficient of variation in a normal distribution with a known population mean. One of the proposed confidence intervals is based on the normal approximation. The other proposed confidence intervals are the shortest-length confidence interval and the equal-tailed confidence interval. A Monte Carlo simulation study was conducted to compare the performance of the proposed confidence intervals with the existing confidence intervals. Simulation results have shown that all three proposed confidence intervals perform well in terms of coverage probability and expected length. Wararit Panichkitkosolkul Copyright © 2013 Wararit Panichkitkosolkul. All rights reserved. Gaussian Estimation of One-Factor Mean Reversion Processes Sun, 20 Oct 2013 15:30:11 +0000 We propose a new alternative method to estimate the parameters in one-factor mean reversion processes based on the maximum likelihood technique. This approach makes use of Euler-Maruyama scheme to approximate the continuous-time model and build a new process discretized. The closed formulas for the estimators are obtained. Using simulated data series, we compare the results obtained with the results published by other authors. Freddy H. Marín Sánchez and J. Sebastian Palacio Copyright © 2013 Freddy H. Marín Sánchez and J. Sebastian Palacio. All rights reserved. Parameter Estimation for Type III Discrete Weibull Distribution: A Comparative Study Sat, 28 Sep 2013 14:51:26 +0000 The type III discrete Weibull distribution can be used in reliability analysis for modeling failure data such as the number of shocks, cycles, or runs a component or a structure can overcome before failing. This paper describes three methods for estimating its parameters: two customary techniques and a technique particularly suitable for discrete distributions, which, in contrast to the two other techniques, provides analytical estimates, whose derivation is detailed here. The techniques’ peculiarities and practical limits are outlined. A Monte Carlo simulation study has been performed to assess the statistical performance of these methods for different parameter combinations and sample sizes and then give some indication for their mindful use. Two applications of real data are provided with the aim of showing how the type III discrete Weibull distribution can fit real data, even better than other popular discrete models, and how the inferential procedures work. A software implementation of the model is also provided. Alessandro Barbiero Copyright © 2013 Alessandro Barbiero. All rights reserved. Scale-Free Property for Degrees and Weights in a Preferential Attachment Random Graph Model Sun, 22 Sep 2013 15:26:27 +0000 A random graph evolution mechanism is defined. The evolution studied is a combination of the preferential attachment model and the interaction of four vertices. The asymptotic behaviour of the graph is described. It is proved that the graph exhibits a power law degree distribution; in other words, it is scale-free. It turns out that any exponent in can be achieved. The proofs are based on martingale methods. István Fazekas and Bettina Porvázsnyik Copyright © 2013 István Fazekas and Bettina Porvázsnyik. All rights reserved. Depth-Based Classification for Distributions with Nonconvex Support Sun, 22 Sep 2013 11:48:41 +0000 Halfspace depth became a popular nonparametric tool for statistical analysis of multivariate data during the last two decades. One of applications of data depth considered recently in literature is the classification problem. The data depth approach is used instead of the linear discriminant analysis mostly to avoid the parametric assumptions and to get better classifier for data whose distribution is not elliptically symmetric, for example, skewed data. In our paper, we suggest to use weighted version of halfspace depth rather than the halfspace depth itself in order to obtain lower misclassification rate in the case of “nonconvex” distributions. Simulations show that the results of depth-based classifiers are comparable with linear discriminant analysis for two normal populations, while for nonelliptic distributions the classifier based on weighted halfspace depth outperforms both linear discriminant analysis and classifier based on the usual (nonweighted) halfspace depth. Daniel Hlubinka and Ondrej Vencalek Copyright © 2013 Daniel Hlubinka and Ondrej Vencalek. All rights reserved. Bayesian and Non-Bayesian Inference for Survival Data Using Generalised Exponential Distribution Thu, 12 Sep 2013 10:10:04 +0000 A two-parameter lifetime distribution was introduced by Kundu and Gupta known as generalised exponential distribution. This distribution has been touted to be an alternative to the well-known 2-parameter Weibull and gamma distributions. We seek to determine the parameters and the survival function of this distribution. The survival function determines the probability that a unit under investigation will survive beyond a certain specified time, say, (). We have employed different data sets to estimate the parameters and see how well the distribution can be used to analyse survival data. A comparison is made about the estimators used in this study. Standard errors of the estimators are determined and used for the comparisons. A simulation study is also carried out, and the mean squared errors and absolute bias are obtained for the purpose of comparison. Chris Bambey Guure and Samuel Bosomprah Copyright © 2013 Chris Bambey Guure and Samuel Bosomprah. All rights reserved. Growth Estimators and Confidence Intervals for the Mean of Negative Binomial Random Variables with Unknown Dispersion Mon, 26 Aug 2013 17:31:49 +0000 The negative binomial distribution becomes highly skewed under extreme dispersion. Even at moderately large sample sizes, the sample mean exhibits a heavy right tail. The standard normal approximation often does not provide adequate inferences about the data's expected value in this setting. In previous work, we have examined alternative methods of generating confidence intervals for the expected value. These methods were based upon Gamma and Chi Square approximations or tail probability bounds such as Bernstein's inequality. We now propose growth estimators of the negative binomial mean. Under high dispersion, zero values are likely to be overrepresented in the data. A growth estimator constructs a normal-style confidence interval by effectively removing a small, predetermined number of zeros from the data. We propose growth estimators based upon multiplicative adjustments of the sample mean and direct removal of zeros from the sample. These methods do not require estimating the nuisance dispersion parameter. We will demonstrate that the growth estimators' confidence intervals provide improved coverage over a wide range of parameter values and asymptotically converge to the sample mean. Interestingly, the proposed methods succeed despite adding both bias and variance to the normal approximation. David Shilane and Derek Bean Copyright © 2013 David Shilane and Derek Bean. All rights reserved. On the Generalized Lognormal Distribution Mon, 29 Jul 2013 13:14:24 +0000 This paper introduces, investigates, and discusses the -order generalized lognormal distribution (-GLD). Under certain values of the extra shape parameter , the usual lognormal, log-Laplace, and log-uniform distribution, are obtained, as well as the degenerate Dirac distribution. The shape of all the members of the -GLD family is extensively discussed. The cumulative distribution function is evaluated through the generalized error function, while series expansion forms are derived. Moreover, the moments for the -GLD are also studied. Thomas L. Toulias and Christos P. Kitsos Copyright © 2013 Thomas L. Toulias and Christos P. Kitsos. All rights reserved. A Note on Functional Averages over Gaussian Ensembles Thu, 11 Jul 2013 14:19:24 +0000 We find a new formula for matrix averages over the Gaussian ensemble. Let be an Gaussian random matrix with complex, independent, and identically distributed entries of zero mean and unit variance. Given an positive definite matrix and a continuous function such that for every , we find a new formula for the expectation . Taking gives another formula for the capacity of the MIMO communication channel, and taking gives the MMSE achieved by a linear receiver. Gabriel H. Tucci and Maria V. Vega Copyright © 2013 Gabriel H. Tucci and Maria V. Vega. All rights reserved. Bayesian Estimation and Prediction for Flexible Weibull Model under Type-II Censoring Scheme Thu, 11 Jul 2013 13:45:42 +0000 We have developed the Bayesian estimation procedure for flexible Weibull distribution under Type-II censoring scheme assuming Jeffrey's scale invariant (noninformative) and Gamma (informative) priors for the model parameters. The interval estimation for the model parameters has been performed through normal approximation, bootstrap, and highest posterior density (HPD) procedures. Further, we have also derived the predictive posteriors and the corresponding predictive survival functions for the future observations based on Type-II censored data from the flexible Weibull distribution. Since the predictive posteriors are not in the closed form, we proposed to use the Monte Carlo Markov chain (MCMC) methods to approximate the posteriors of interest. The performance of the Bayes estimators has also been compared with the classical estimators of the model parameters through the Monte Carlo simulation study. A real data set representing the time between failures of secondary reactor pumps has been analysed for illustration purpose. Sanjay Kumar Singh, Umesh Singh, and Vikas Kumar Sharma Copyright © 2013 Sanjay Kumar Singh et al. All rights reserved. A Survey Design for a Sensitive Binary Variable Correlated with Another Nonsensitive Binary Variable Sun, 23 Jun 2013 12:33:50 +0000 Tian et al. (2007) introduced a so-called hidden sensitivity model for evaluating the association of two sensitive questions with binary outcomes. However, in practice, we sometimes need to assess the association between one sensitive binary variable (e.g., whether or not a drug user, the number of sex partner being ⩽1 or >1, and so on) and one nonsensitive binary variable (e.g., good or poor health status, with or without cervical cancer, and so on). To address this issue, by sufficiently utilizing the information contained in the non-sensitive binary variable, in this paper, we propose a new survey scheme, called combination questionnaire design/model, which consists of a main questionnaire and a supplemental questionnaire. The introduction of the supplemental questionnaire which is indeed a design of direct questioning can effectively reduce the noncompliance behavior since more respondents will not be faced with the sensitive question. Likelihood-based inferences including maximum likelihood estimates via the expectation-maximization algorithm, asymptotic confidence intervals, and bootstrap confidence intervals of parameters of interest are derived. A likelihood ratio test is provided to test the association between the two binary random variables. Bayesian inferences are also discussed. Simulation studies are performed, and a cervical cancer data set in Atlanta is used to illustrate the proposed methods. Jun-Wu Yu, Yang Lu, and Guo-Liang Tian Copyright © 2013 Jun-Wu Yu et al. All rights reserved. Bayesian Estimation of the Scale Parameter of Inverse Weibull Distribution under the Asymmetric Loss Functions Sun, 23 Jun 2013 10:38:18 +0000 This paper proposes different methods of estimating the scale parameter in the inverse Weibull distribution (IWD). Specifically, the maximum likelihood estimator of the scale parameter in IWD is introduced. We then derived the Bayes estimators for the scale parameter in IWD by considering quasi, gamma, and uniform priors distributions under the square error, entropy, and precautionary loss functions. Finally, the different proposed estimators have been compared by the extensive simulation studies in corresponding the mean square errors and the evolution of risk functions. Farhad Yahgmaei, Manoochehr Babanezhad, and Omid S. Moghadam Copyright © 2013 Farhad Yahgmaei et al. All rights reserved. Weighted Kappas for Tables Sun, 16 Jun 2013 14:36:39 +0000 Weighted kappa is a widely used statistic for summarizing inter-rater agreement on a categorical scale. For rating scales with three categories, there are seven versions of weighted kappa. It is shown analytically how these weighted kappas are related. Several conditional equalities and inequalities between the weighted kappas are derived. The analytical analysis indicates that the weighted kappas are measuring the same thing but to a different extent. One cannot, therefore, use the same magnitude guidelines for all weighted kappas. Matthijs J. Warrens Copyright © 2013 Matthijs J. Warrens. All rights reserved. On the Study of Transience and Recurrence of the Markov Chain Defined by Directed Weighted Circuits Associated with a Random Walk in Fixed Environment Thu, 06 Jun 2013 09:36:53 +0000 By using the cycle representation theory of Markov processes, we investigate proper criterions regarding transience and recurrence of the corresponding Markov chain represented uniquely by directed cycles (especially by directed circuits) and weights of a random walk with jumps in a fixed environment. Chrysoula Ganatsiou Copyright © 2013 Chrysoula Ganatsiou. All rights reserved. Recent Advances in Univariate and Multivariate Models Tue, 14 May 2013 13:28:23 +0000 Gauss M. Cordeiro, Artur Lemonte, Edwin Ortega, and Jose M. Sarabia Copyright © 2013 Gauss M. Cordeiro et al. All rights reserved. Testing for Main Random Effects in Two-Way Random and Mixed Effects Models: Modifying the Statistic Thu, 21 Mar 2013 17:37:30 +0000 A procedure for testing the significance of the main random effect is proposed under a model which does not require the traditional assumptions of symmetry, homoscedasticity, and normality for the error term and random effects. To accommodate this level of model generality, and also unbalanced designs, suitable adjustments to the F-test are made. The extensive simulations performed under the random effects model, and the unrestricted and restricted versions of the mixed effects model, indicate that the classical F procedure is extremely liberal under heteroscedasticity and unbalancedness. The proposed test procedure performs well in all settings and is comparable to the classical F-test when the classical assumptions are met. An analysis of a dataset from the Mussel Watch Project is presented. Trent Gaugler and Michael G. Akritas Copyright © 2013 Trent Gaugler and Michael G. Akritas. All rights reserved.