Journal of Probability and Statistics The latest articles from Hindawi © 2017 , Hindawi Limited . All rights reserved. Maximum Likelihood and Bayes Estimation in Randomly Censored Geometric Distribution Tue, 21 Feb 2017 06:37:41 +0000 In this article, we study the geometric distribution under randomly censored data. Maximum likelihood estimators and confidence intervals based on Fisher information matrix are derived for the unknown parameters with randomly censored data. Bayes estimators are also developed using beta priors under generalized entropy and LINEX loss functions. Also, Bayesian credible and highest posterior density (HPD) credible intervals are obtained for the parameters. Expected time on test and reliability characteristics are also analyzed in this article. To compare various estimates developed in the article, a Monte Carlo simulation study is carried out. Finally, for illustration purpose, a randomly censored real data set is discussed. Hare Krishna and Neha Goel Copyright © 2017 Hare Krishna and Neha Goel. All rights reserved. Modified Slash Lindley Distribution Sun, 19 Feb 2017 00:00:00 +0000 In this paper we introduce a new distribution, called the modified slash Lindley distribution, which can be seen as an extension of the Lindley distribution. We show that this new distribution provides more flexibility in terms of kurtosis and skewness than the Lindley distribution. We derive moments and some basic properties for the new distribution. Moment estimators and maximum likelihood estimators are calculated using numerical procedures. We carry out a simulation study for the maximum likelihood estimators. A fit of the proposed model indicates good performance when compared with other less flexible models. Jimmy Reyes, Osvaldo Venegas, and Héctor W. Gómez Copyright © 2017 Jimmy Reyes et al. All rights reserved. Polyhedral Star-Shaped Distributions Tue, 14 Feb 2017 00:00:00 +0000 A new method of probabilistic modelling of polyhedrally contoured sample clouds is presented and applied to statistical reasoning for a real dataset. Various representations of the new class of polyhedral star-shaped distributions are derived and basic properties of the moments as well as characteristic and moment generating functions of these distributions are studied. Along with location-scale transformations, estimating and hypothesis testing are dealt with. Wolf-Dieter Richter and Kay Schicker Copyright © 2017 Wolf-Dieter Richter and Kay Schicker. All rights reserved. Estimation of the Parameters of a Chirp Type Model with Stationary Residuals Thu, 09 Feb 2017 10:02:15 +0000 Let be the observations from a chirp type statistical model , , where is a stationary noise. We consider a method of estimation of parameters, , , , , and , (where is the variance of ’s) which is basically an approximate least-squares method. The main advantage of the proposed approach is that no assumptions are required. We make use of the three theorems which were established associated with the kernel and then use them to prove, under certain conditions, the consistency of the estimators. K. Perera Copyright © 2017 K. Perera. All rights reserved. Gram-Charlier Processes and Applications to Option Pricing Wed, 08 Feb 2017 00:00:00 +0000 A Gram-Charlier distribution has a density that is a polynomial times a normal density. For option pricing this retains the tractability of the normal distribution while allowing nonzero skewness and excess kurtosis. Properties of the Gram-Charlier distributions are derived, leading to the definition of a process with independent Gram-Charlier increments, as well as formulas for option prices and their sensitivities. A procedure for simulating Gram-Charlier distributions and processes is given. Numerical illustrations show the effect of skewness and kurtosis on option prices. Jean-Pierre Chateau and Daniel Dufresne Copyright © 2017 Jean-Pierre Chateau and Daniel Dufresne. All rights reserved. Upper Bound of the Generalized Value for the Population Variances of Lognormal Distributions with Known Coefficients of Variation Mon, 16 Jan 2017 06:40:07 +0000 This paper presents an upper bound for each of the generalized values for testing the one population variance, the difference between two population variances, and the ratio of population variances for lognormal distribution when coefficients of variation are known. For each of the proposed generalized values, we derive a closed form expression of the upper bound of the generalized value. Numerical computations illustrate the theoretical results. Rada Somkhuean, Sa-aat Niwitpong, and Suparat Niwitpong Copyright © 2017 Rada Somkhuean et al. All rights reserved. Stochastic Models for the Infectivity Function in an Infinite Population of Susceptible Individuals Wed, 11 Jan 2017 11:22:46 +0000 Two stochastic models to study the course of the transient behaviour of the total infectivity present in an infinite population of susceptible individuals are developed. The conditional intensity function of the contagion comprises two components: one is due to the external sources only and the other is the contribution of each of the infected persons which is nonstationary in nature. The statistical characteristics of the number of infected individuals at any time are explicitly obtained. Estimation of the model parameters is also indicated. Viswanathan Arunachalam and Liliana Blanco Copyright © 2017 Viswanathan Arunachalam and Liliana Blanco. All rights reserved. Numerical Reconstruction of the Covariance Matrix of a Spherically Truncated Multinormal Distribution Tue, 10 Jan 2017 10:54:26 +0000 We relate the matrix of the second moments of a spherically truncated normal multivariate to its full covariance matrix and present an algorithm to invert the relation and reconstruct from . While the eigenvectors of are left invariant by the truncation, its eigenvalues are nonuniformly damped. We show that the eigenvalues of can be reconstructed from their truncated counterparts via a fixed point iteration, whose convergence we prove analytically. The procedure requires the computation of multidimensional Gaussian integrals over an Euclidean ball, for which we extend a numerical technique, originally proposed by Ruben in 1962, based on a series expansion in chi-square distributions. In order to study the feasibility of our approach, we examine the convergence rate of some iterative schemes on suitably chosen ensembles of Wishart matrices. We finally discuss the practical difficulties arising in sample space and outline a regularization of the problem based on perturbation theory. Filippo Palombi, Simona Toti, and Romina Filippini Copyright © 2017 Filippo Palombi et al. All rights reserved. Sample Dependence in the Maximum Entropy Solution to the Generalized Moment Problem Wed, 14 Dec 2016 06:03:58 +0000 The method of maximum entropy is quite a powerful tool to solve the generalized moment problem, which consists in determining the probability density of a random variable from the knowledge of the expected values of a few functions of the variable. In actual practice, such expected values are determined from empirical samples, leaving open the question of the dependence of the solution upon the sample. It is the purpose of this note to take a few steps towards the analysis of such dependence. Henryk Gzyl Copyright © 2016 Henryk Gzyl. All rights reserved. Exact Interval Inference for the Two-Parameter Rayleigh Distribution Based on the Upper Record Values Mon, 12 Dec 2016 14:10:46 +0000 The maximum likelihood method is the most widely used estimation method. On the other hand, it can produce substantial bias, and an approximate confidence interval based on the maximum likelihood estimator cannot be valid when the sample size is small. Because the sizes of the record values are considerably smaller than the original sequence observed in the majority of cases, a method appropriate for this situation is required for precise inference. This paper provides the exact confidence intervals for unknown parameters and exact predictive intervals for the future upper record values by providing some pivotal quantities in the two-parameter Rayleigh distribution based on the upper record values. Finally, the validity of the proposed inference methods was examined from Monte Carlo simulations and real data. Jung-In Seo, Jae-Woo Jeon, and Suk-Bok Kang Copyright © 2016 Jung-In Seo et al. All rights reserved. Estimating the Proportion of True Null Hypotheses in Multiple Testing Problems Thu, 08 Dec 2016 12:41:43 +0000 The problem of estimating the proportion, , of the true null hypotheses in a multiple testing problem is important in cases where large scale parallel hypotheses tests are performed independently. While the problem is a quantity of interest in its own right in applications, the estimate of can be used for assessing or controlling an overall false discovery rate. In this article, we develop an innovative nonparametric maximum likelihood approach to estimate . The nonparametric likelihood is proposed to be restricted to multinomial models and an EM algorithm is also developed to approximate the estimate of . Simulation studies show that the proposed method outperforms other existing methods. Using experimental microarray datasets, we demonstrate that the new method provides satisfactory estimate in practice. Oluyemi Oyeniran and Hanfeng Chen Copyright © 2016 Oluyemi Oyeniran and Hanfeng Chen. All rights reserved. Multivariate Macdonald Distribution and Its Properties Tue, 22 Nov 2016 14:14:43 +0000 We give multivariate generalization of Macdonald distribution and study several of its properties. We also define the multivariate Macdonald-gamma distribution and derive a number of results pertaining to it. Daya K. Nagar, Edwin Zarrazola, and Luz Estela Sánchez Copyright © 2016 Daya K. Nagar et al. All rights reserved. Exploratory Methods for the Study of Incomplete and Intersecting Shape Boundaries from Landmark Data Mon, 21 Nov 2016 13:50:43 +0000 Structured spatial point patterns appear in many applications within the natural sciences. The points often record the location of key features, called landmarks, on continuous object boundaries, such as anatomical features on a human face. In other situations, the points may simply be arbitrarily spaced marks along a smooth curve, such as on handwritten numbers. This paper proposes novel exploratory methods for the identification of structure within point datasets. In particular, points are linked together to form curves which estimate the original shape from which the points are the only recorded information. Nonparametric regression methods are applied to polar coordinate variables obtained from the point locations and periodic modelling allows closed curves to be fitted even when data are available on only part of the boundary. Further, the model allows discontinuities to be identified to describe rapid changes in the curves. These generalizations are particularly important when the points represent shapes which are occluded or are intersecting. A range of real-data examples is used to motivate the modelling and to illustrate the flexibility of the approach. The method successfully identifies the underlying structure and its output could also be used as the basis for further analysis. Fathi M. O. Hamed and Robert G. Aykroyd Copyright © 2016 Fathi M. O. Hamed and Robert G. Aykroyd. All rights reserved. Accuracy of Approximation for Discrete Distributions Tue, 08 Nov 2016 09:22:15 +0000 The paper is a contribution to the problem of estimating the deviation of two discrete probability distributions in terms of the supremum distance between their generating functions over the interval . Deviation can be measured by the difference of the th terms or by total variation distance. Our new bounds have better order of magnitude than those proved previously, and they are even sharp in certain cases. Tamás F. Móri Copyright © 2016 Tamás F. Móri. 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.