Journal of Probability and Statistics The latest articles from Hindawi Publishing Corporation © 2016 , Hindawi Publishing Corporation . All rights reserved. Odds Ratios Estimation of Rare Event in Binomial Distribution Wed, 19 Oct 2016 09:20:32 +0000 We introduce the new estimator of odds ratios in rare events using Empirical Bayes method in two independent binomial distributions. We compare the proposed estimates of odds ratios with two estimators, modified maximum likelihood estimator (MMLE) and modified median unbiased estimator (MMUE), using the Estimated Relative Error (ERE) as a criterion of comparison. It is found that the new estimator is more efficient when compared to the other methods. Kobkun Raweesawat, Yupaporn Areepong, Katechan Jampachaisri, and Saowanit Sukparungsee Copyright © 2016 Kobkun Raweesawat et al. All rights reserved. Bayesian Estimation in Delta and Nabla Discrete Fractional Weibull Distributions Mon, 26 Sep 2016 11:28:08 +0000 We use discrete fractional calculus for showing the existence of delta and nabla discrete distributions and then apply time scales for definitions of delta and nabla discrete fractional Weibull distributions. Also, we study the Bayesian estimation of the functions of parameters of these distributions. M. Ganji and F. Gharari Copyright © 2016 M. Ganji and F. Gharari. All rights reserved. Parameter Estimation in Mean Reversion Processes with Deterministic Long-Term Trend Mon, 22 Aug 2016 16:50:05 +0000 This paper describes a procedure based on maximum likelihood technique in two phases for estimating the parameters in mean reversion processes when the long-term trend is defined by a continued deterministic function. Closed formulas for the estimators that depend on observations of discrete paths and an estimation of the expected value of the process are obtained in the first phase. In the second phase, a reestimation scheme is proposed when a priori knowledge exists of the long-term trend. Some experimental results using simulated data sets are graphically illustrated. Freddy H. Marín Sánchez and Verónica M. Gallego Copyright © 2016 Freddy H. Marín Sánchez and Verónica M. Gallego. All rights reserved. A Generalized Class of Exponential Type Estimators for Population Mean under Systematic Sampling Using Two Auxiliary Variables Wed, 17 Aug 2016 09:49:32 +0000 We have proposed a generalized class of exponential type estimators for population mean under the framework of systematic sampling using the knowledge of two auxiliary variables. The expressions for the mean square error of the proposed class of estimators have been corrected up to first order of approximation. Comparisons of the efficiency of the proposed class of estimators under the optimal conditions with the other existing estimators have been presented through a real secondary data. The statistical study provides strong evidence that the proposed class of estimators in survey estimation procedure results in substantial efficiency improvements over the other existing estimation approaches. Mursala Khan Copyright © 2016 Mursala Khan. All rights reserved. A Mixture of Generalized Tukey’s Distributions Mon, 15 Aug 2016 11:08:57 +0000 Mixtures of symmetric distributions, in particular normal mixtures as a tool in statistical modeling, have been widely studied. In recent years, mixtures of asymmetric distributions have emerged as a top contender for analyzing statistical data. Tukey’s family of generalized distributions depend on the parameters, namely, , which controls the skewness. This paper presents the probability density function (pdf) associated with a mixture of Tukey’s family of generalized distributions. The mixture of this class of skewed distributions is a generalization of Tukey’s family of distributions. In this paper, we calculate a closed form expression for the density and distribution of the mixture of two Tukey’s families of generalized distributions, which allows us to easily compute probabilities, moments, and related measures. This class of distributions contains the mixture of Log-symmetric distributions as a special case. José Alfredo Jiménez and Viswanathan Arunachalam Copyright © 2016 José Alfredo Jiménez and Viswanathan Arunachalam. All rights reserved. Cesàro Summable Sequence Spaces over the Non-Newtonian Complex Field Mon, 09 May 2016 08:48:18 +0000 The spaces , , and can be considered the sets of all sequences that are strongly summable to zero, strongly summable, and bounded, by the Cesàro method of order with index . Here we define the sets of sequences which are related to strong Cesàro summability over the non-Newtonian complex field by using two generator functions. Also we determine the -duals of the new spaces and characterize matrix transformations on them into the sets of -bounded, -convergent, and -null sequences of non-Newtonian complex numbers. Uğur Kadak Copyright © 2016 Uğur Kadak. All rights reserved. Variable Selection and Parameter Estimation with the Atan Regularization Method Wed, 16 Mar 2016 12:50:18 +0000 Variable selection is fundamental to high-dimensional statistical modeling. Many variable selection techniques may be implemented by penalized least squares using various penalty functions. In this paper, an arctangent type penalty which very closely resembles penalty is proposed; we call it Atan penalty. The Atan-penalized least squares procedure is shown to consistently select the correct model and is asymptotically normal, provided the number of variables grows slower than the number of observations. The Atan procedure is efficiently implemented using an iteratively reweighted Lasso algorithm. Simulation results and data example show that the Atan procedure with BIC-type criterion performs very well in a variety of settings. Yanxin Wang and Li Zhu Copyright © 2016 Yanxin Wang and Li Zhu. All rights reserved. On a Power Transformation of Half-Logistic Distribution Mon, 07 Mar 2016 12:19:01 +0000 A new continuous distribution on the positive real line is constructed from half-logistic distribution, using a transformation and its analytical characteristics are studied. Some characterization results are derived. Classical procedures for the estimation of parameters of the new distribution are discussed and a comparative study is done through numerical examples. Further, different families of continuous distributions on the positive real line are generated using this distribution. Application is discussed with the help of real-life data sets. S. D. Krishnarani Copyright © 2016 S. D. Krishnarani. All rights reserved. Properties of Matrix Variate Confluent Hypergeometric Function Distribution Mon, 08 Feb 2016 08:34:53 +0000 We study matrix variate confluent hypergeometric function kind 1 distribution which is a generalization of the matrix variate gamma distribution. We give several properties of this distribution. We also derive density functions of , , and , where independent random matrices and follow confluent hypergeometric function kind 1 and gamma distributions, respectively. Arjun K. Gupta, Daya K. Nagar, and Luz Estela Sánchez Copyright © 2016 Arjun K. Gupta et al. All rights reserved. General Results for the Transmuted Family of Distributions and New Models Sun, 31 Jan 2016 11:20:12 +0000 The transmuted family of distributions has been receiving increased attention over the last few years. For a baseline G distribution, we derive a simple representation for the transmuted-G family density function as a linear mixture of the G and exponentiated-G densities. We investigate the asymptotes and shapes and obtain explicit expressions for the ordinary and incomplete moments, quantile and generating functions, mean deviations, Rényi and Shannon entropies, and order statistics and their moments. We estimate the model parameters of the family by the method of maximum likelihood. We prove empirically the flexibility of the proposed model by means of an application to a real data set. Marcelo Bourguignon, Indranil Ghosh, and Gauss M. Cordeiro Copyright © 2016 Marcelo Bourguignon et al. All rights reserved. Classical and Bayesian Approach in Estimation of Scale Parameter of Nakagami Distribution Sun, 17 Jan 2016 11:51:03 +0000 Nakagami distribution is considered. The classical maximum likelihood estimator has been obtained. Bayesian method of estimation is employed in order to estimate the scale parameter of Nakagami distribution by using Jeffreys’, Extension of Jeffreys’, and Quasi priors under three different loss functions. Also the simulation study is conducted in R software. Kaisar Ahmad, S. P. Ahmad, and A. Ahmed Copyright © 2016 Kaisar Ahmad et al. All rights reserved. Applications of Fuss-Catalan Numbers to Success Runs of Bernoulli Trials Tue, 12 Jan 2016 12:16:43 +0000 In a recent paper, the authors derived the exact solution for the probability mass function of the geometric distribution of order , expressing the roots of the associated auxiliary equation in terms of generating functions for Fuss-Catalan numbers. This paper applies the above formalism for the Fuss-Catalan numbers to treat additional problems pertaining to occurrences of success runs. New exact analytical expressions for the probability mass function and probability generating function and so forth are derived. First, we treat sequences of Bernoulli trials with occurrences of success runs of length with -overlapping. The case , where there must be a gap of at least trials between success runs, is also studied. Next we treat the distribution of the waiting time for the nonoverlapping appearance of a pair of successes separated by at most failures (). S. J. Dilworth and S. R. Mane Copyright © 2016 S. J. Dilworth and S. R. Mane. All rights reserved. Scan Statistics for Detecting High-Variance Clusters Tue, 05 Jan 2016 07:59:40 +0000 Scan statistics are mostly used to detect spatial areas or time intervals in which the mean level of a given variable is more important. Sometimes, when this variable is continuous, there is an interest in looking for clusters in which its variability is more important. In this paper, two scan statistics are introduced for identifying clusters of values exhibiting higher variance. Like many classical scan statistics, the first one relies on a generalized likelihood ratio test whereas the second one is based on ratios of empirical variances. These methods are useful to identify spatial areas or time intervals in which the variability of a given variable is more important. In an application of the new methods, I look for geographical clusters of high-variability income in France and then for residuals exhibiting higher variance in a linear regression context. Lionel Cucala Copyright © 2016 Lionel Cucala. All rights reserved. Robust Stability Best Subset Selection for Autocorrelated Data Based on Robust Location and Dispersion Estimator Thu, 31 Dec 2015 13:08:48 +0000 Stability selection (multisplit) approach is a variable selection procedure which relies on multisplit data to overcome the shortcomings that may occur to single-split data. Unfortunately, this procedure yields very poor results in the presence of outliers and other contamination in the original data. The problem becomes more complicated when the regression residuals are serially correlated. This paper presents a new robust stability selection procedure to remedy the combined problem of autocorrelation and outliers. We demonstrate the good performance of our proposed robust selection method using real air quality data and simulation study. Hassan S. Uraibi, Habshah Midi, and Sohel Rana Copyright © 2015 Hassan S. Uraibi et al. All rights reserved. Confidence Region Approach for Assessing Bioequivalence and Biosimilarity Accounting for Heterogeneity of Variability Sun, 27 Dec 2015 12:04:29 +0000 For approval of generic drugs, the FDA requires that evidence of bioequivalence in average bioequivalence in terms of drug absorption be provided through the conduct of a bioequivalence study. A test product is said to be average bioequivalent to a reference (innovative) product if the 90% confidence interval of the ratio of means (after log-transformation) is totally within (80%, 125%). This approach is considered a one-parameter approach, which does not account for possible heterogeneity of variability between drug products. In this paper, we study a two-parameter approach (i.e., confidence region approach) for assessing bioequivalence, which can also be applied to assessing biosimilarity of biosimilar products. The proposed confidence region approach is compared with the traditional one-parameter approach both theoretically and numerically (i.e., simulation study) for finite sample performance. Jianghao Li and Shein-Chung Chow Copyright © 2015 Jianghao Li and Shein-Chung Chow. All rights reserved. Estimating Parameters of a Probabilistic Heterogeneous Block Model via the EM Algorithm Thu, 24 Dec 2015 10:00:52 +0000 We introduce a semiparametric block model for graphs, where the within- and between-cluster edge probabilities are not constants within the blocks but are described by logistic type models, reminiscent of the 50-year-old Rasch model and the newly introduced - models. Our purpose is to give a partition of the vertices of an observed graph so that the induced subgraphs and bipartite graphs obey these models, where their strongly interlaced parameters give multiscale evaluation of the vertices at the same time. In this way, a profoundly heterogeneous version of the stochastic block model is built via mixtures of the above submodels, while the parameters are estimated with a special EM iteration. Marianna Bolla and Ahmed Elbanna Copyright © 2015 Marianna Bolla and Ahmed Elbanna. All rights reserved. Posterior Analysis of State Space Model with Spherical Symmetricity Mon, 07 Dec 2015 13:10:09 +0000 The present work investigates state space model with nonnormal disturbances when the deviation from normality has been observed only with respect to kurtosis and the distribution of disturbances continues to follow a symmetric family of distributions. Spherically symmetric distribution is used to approximate behavior of symmetric nonnormal disturbances for discrete time series. The conditional posterior densities of the involved parameters are derived, which are further utilized in Gibbs sampler scheme for estimating the marginal posterior densities. The state space model with disturbances following multivariate- distribution, which is a particular case of spherically symmetric distribution, is discussed. Ranjita Pandey Copyright © 2015 Ranjita Pandey. All rights reserved. Polynomial Chaos Expansion Approach to Interest Rate Models Thu, 03 Dec 2015 14:05:47 +0000 The Polynomial Chaos Expansion (PCE) technique allows us to recover a finite second-order random variable exploiting suitable linear combinations of orthogonal polynomials which are functions of a given stochastic quantity , hence acting as a kind of random basis. The PCE methodology has been developed as a mathematically rigorous Uncertainty Quantification (UQ) method which aims at providing reliable numerical estimates for some uncertain physical quantities defining the dynamic of certain engineering models and their related simulations. In the present paper, we use the PCE approach in order to analyze some equity and interest rate models. In particular, we take into consideration those models which are based on, for example, the Geometric Brownian Motion, the Vasicek model, and the CIR model. We present theoretical as well as related concrete numerical approximation results considering, without loss of generality, the one-dimensional case. We also provide both an efficiency study and an accuracy study of our approach by comparing its outputs with the ones obtained adopting the Monte Carlo approach, both in its standard and its enhanced version. Luca Di Persio, Gregorio Pellegrini, and Michele Bonollo Copyright © 2015 Luca Di Persio et al. All rights reserved. Portfolio Theory for -Symmetric and Pseudoisotropic Distributions: -Fund Separation and the CAPM Tue, 01 Dec 2015 16:24:14 +0000 The shifted pseudoisotropic multivariate distributions are shown to satisfy Ross’ stochastic dominance criterion for two-fund monetary separation in the case with risk-free investment opportunity and furthermore to admit the Capital Asset Pricing Model under an embedding in condition if , with the betas given in an explicit form. For the -symmetric subclass, the market without risk-free investment opportunity admits -fund separation if , , generalizing the classical elliptical case , and we also give the precise number of funds needed, from which it follows that we cannot, except degenerate cases, have a CAPM without risk-free opportunity. For the symmetric stable subclass, the index of stability is only of secondary interest, and several common restrictions in terms of that index can be weakened by replacing it by the (no smaller) indices of symmetry/of embedding. Finally, dynamic models with intermediate consumption inherit the separation properties of the static models. Nils Chr. Framstad Copyright © 2015 Nils Chr. Framstad. 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.