نشریه اندیشه آماری
سال بیست و دوم شماره 1 (پیاپی 43، بهار و تابستان 1396)

ýGiven the importance of Markov chains in information theoryý, ýthe definition of conditional probability for these random processes can also be defined in terms of mutual informationý. ýIn this paperý, ýthe relationship between the concept of sufficiency and Markov chains from the perspective of information theory and the relationship between probabilistic sufficiency and algorithmic sufficiency is determinedý.

ýIn this articleý, ýfirst of allý, ýthe Kumaraswamy distribution is introducedý. ýThený, ýthe joint and marginal distributions of W = X1/X2 and T = X1/X1 where X1 and X2 are independent Kumaraswamy random variablesý, ýare obtained and the moments of these random variables are computedý.
ýThe distribution of random variables W and T can be used in reliability studies and statistical models such as stress-strengthý.

ýIn this paperý, ýa new distribution of the three-parameter lifetime model called the Marshall-Olkin Gompertz is proposed on the basis of the Gompertz distributioný. ýIt is a generalization of the Gompertz distribution having decreasing failure rate and can also be increasing and bathtub-shaped depending on its parametersý. ýThe probability density functioný, ýcumulative distribution functioný, ýhazard rate function and some mathematical properties of this model such asý, ýcentral momentsý, ýmoments of order statisticsý, ýRenyi and Shannon entropies and quantile function are derivedý. ýIn additioný, ýthe maximum likelihood of its parameters method is estimated and this new distribution compared with some Gompertz distribution generalizations by means of a set of real dataý.

ýIn this paperý, ýan M/M/1 queue with instantaneous Bernoulli feedback is studied in the event of server failureý, ýthe catastrophe occurs and after repairý, ýit starts to work againý. ýThe transient response for the probability function of the system size is presentedý. ýThe steady state analysis of system size probabilities and some performance measures of system are providedý. ýThen the results are used to consider the performance of an ATMý. ýThen to observe and optimize the performace of the ATMý, ýwe illustrate the effects of changing parameters on system performance measuresý. ýAt lastý, ýwe simulate the system by using the R applicationý. ýThen we compare its results with expected resultsý.

ýIn this paper some properties of logisticsý - ýx family are discussed and a member of the familyý, ýthe logistic–normal distributioný, ýis studied in detailý. ýAverage deviationsý, ýrisk function and fashion for logistic–normal distribution is obtainedý. ýThe method of maximum likelihood estimation is proposed for estimating the parameters of the logistic–normal distribution and a data set is used to show applications of logistic–normal distributioný.

ýGeneralized Lambda Distribution is an extension of Tukey's lambda distributioný, ýthat is very flexible in modeling information and statistical dataý. ýIn this paperý, ýWe introduced two parameterization of this distributioný. ýThen We estimate parameters by moment matchingý, ýpercentileý, ýstarship and maximum likelihood methods and compare two parameterization and parameter estimation methods with Kolmogorov-Smirnov testý.

ýNormal distribution is widely used in many applicationsý. ýThe problem of testing whether observations come from a normal distribution has been studied extensively by many researchersý. ýOur main goal in this article is to present a simple test procedure for testing multivariate ýnormalityýý.

ýIn this paper we propose a utility function and obtain the Bayese stimate and the optimum sample size under this utility functioný. ýThis utility function is designed especially to obtain the Bayes estimate when the posterior follows a gamma distributioný. ýWe consider a Normal with known meaný, ýa Paretoý, ýan Exponential and a Poisson distribution for an optimum sample size under the proposed utility functioný, ýso that minimizes the cost of samplingý. ýIn this processý, ýwe use Lindley cost function in order to minimize the costý. ýHereý, ýbecause of the complicated form of computationý, ýwe are unable to solve it analytically and use the mumerical methids to get the optimum sample size.

ýAnalysis of large geostatistical data setsý, ýusuallyý, ýentail the expensive matrix computationsý. ýThis problem creates challenges in implementing statistical inferences of traditional Bayesian modelsý. ýIn addition,researchers often face with multiple spatial data sets with complex spatial dependence structures that their analysis is difficultý. ýThis is a problem for MCMC sampling algorithms that are commonly used in Bayesian analysis of spatial modelsý, ýcausing serious problems such as slowing down and chain integrationý. ýTo escape from such computational problemsý, ýwe use low-rank modelsý, ýto analyze Gaussian geostatistical dataý. ýThis models improve MCMC sampler convergence rate and decrease sampler run-time by reducing parameter spaceý. ýThe idea here is to assumeý, ýquite reasonablyý, ýthat the spatial information available from the entire set of observed locations can be summarized in terms of a smallerý, ýbut representativeý, ýsets of locationsý, ýor ‘knots’ý. ýThat isý, ýwe still use all of the data but we represent the spatial structure through a dimension reductioný. ýSoý, ýagainý, ýin implementing the reductioný, ýwe need to design the knotsý. ýConsideration of this issue forms the balance of the articleý. ýTo evaluate the performance of this class of modelsý, ýwe conduct a simulation study as well as analysis of a real data set regarding the quality of underground mineral water of a large area in Golestan provinceý, ýIraný.

ýLatent class analysis (LCA) is a method of evaluating non sampling errorsý, ýespecially measurement error in categorical dataý. ýBiemer (2011) introduced four latent class modeling approachesý: ýprobability model parameterizationý, ýlog linear modelý, ýmodified path modelý, ýand graphical model using path diagramsý. ýThese models are interchangeableý. ýLatent class probability models express likelihood of cross-classification tables in term of conditional and marginal probabilities for each cellý. ýIn this approach model parameters are estimated using EM algorithmý. ýTo test latent class model chi-square statistic is used as a measure of goodness-of-fitý. ýIn this paper we use LCA and data from a small-scale survey to estimate misclassification error (as a measurement error) of students who had at least a failing grade as well as misclassification error of students with average grades below 14ý.