Estimation of a Parameter of Bivariate Pareto Distribution by Ranked Set Sampling

Ranked set sampling is applicable whenever ranking of a set of sampling units can be done easily by a judgement method or based on the measurement of an auxiliary variable on the units selected. In this work, we derive different estimators of a parameter associated with the distribution of the study variate Y, based on a ranked-set sample obtained by using an auxiliary variable X correlated with Y for ranking the sample units, when (X, Y) follows a bivariate Pareto distribution. Efficiency comparisons among these estimators are also made. Real-life data have been used to illustrate the application of the results obtained.     

A control chart for multivariate Poisson distribution using repetitive sampling

Control charts using repetitive group sampling have attracted a great deal of attention during the last few years. In the present article, we attempt to develop a control chart for the multivariate Poisson distribution using the repetitive group sampling scheme. In the proposed control chart, the monitoring statistic from the multivariate Poisson distribution has been used for the quick detection of the deteriorated process to avoid losses. The control coefficients have been estimated using the specified in-control average run lengths. The procedure of the proposed control chart has been explained by using the real-world example and a simulated data set. It has been observed that the proposed control chart is an efficient development for the quick detection of the nonrandom change in the manufacturing process.     

Bayesian and maximum likelihood estimations of the inverted exponentiated half logistic distribution under progressive Type II censoring

In this paper, the estimation of parameters, reliability and hazard functions of a inverted exponentiated half logistic distribution (IEHLD) from progressive Type II censored data has been considered. The Bayes estimates for progressive Type II censored IEHLD under asymmetric and symmetric loss functions such as squared error, general entropy and linex loss function are provided. The Bayes estimates for progressive Type II censored IEHLD parameters, reliability and hazard functions are also obtained under the balanced loss functions. However, the Bayes estimates cannot be obtained explicitly, Lindley approximation method and importance sampling procedure are considered to obtain the Bayes estimates. Furthermore, the asymptotic normality of the maximum likelihood estimates is used to obtain the approximate confidence intervals. The highest posterior density credible intervals of the parameters based on importance sampling procedure are computed. Simulations are performed to see the performance of the proposed estimates. For illustrative purposes, two data sets have been analyzed.     

Bayesian inference for multivariate gamma distributions

The paper considers the multivariate gamma distribution for which the method of moments has been considered as the only method of estimation due to the complexity of the likelihood function. With a non-conjugate prior, practical Bayesian analysis can be conducted using Gibbs sampling with data augmentation. The new methods are illustrated using artificial data for a trivariate gamma distribution as well as an application to technical inefficiency estimation.     

Bayesian estimation of quantile distributions

Use of Bayesian modelling and analysis has become commonplace in many disciplines (finance, genetics and image analysis, for example). Many complex data sets are collected which do not readily admit standard distributions, and often comprise skew and kurtotic data. Such data is well-modelled by the very flexibly-shaped distributions of the quantile distribution family, whose members are defined by the inverse of their cumulative distribution functions and rarely have analytical likelihood functions defined. Without explicit likelihood functions, Bayesian methodologies such as Gibbs sampling cannot be applied to parameter estimation for this valuable class of distributions without resorting to numerical inversion. Approximate Bayesian computation provides an alternative approach requiring only a sampling scheme for the distribution of interest, enabling easier use of quantile distributions under the Bayesian framework. Parameter estimates for simulated and experimental data are presented.     

Bayes estimation for the Block and Basu bivariate and multivariate Weibull distributions

Block and Basu bivariate exponential distribution is one of the most popular absolute continuous bivariate distributions. Recently, Kundu and Gupta [A class of absolute continuous bivariate distributions. Statist Methodol. 2010;7:464–477] introduced Block and Basu bivariate Weibull (BBBW) distribution, which is a generalization of the Block and Basu bivariate exponential distribution, and provided the maximum likelihood estimators using EM algorithm. In this paper, we consider the Bayesian inference of the unknown parameters of the BBBW distribution. The Bayes estimators are obtained with respect to the squared error loss function, and the prior distributions allow for prior dependence among the unknown parameters. Prior independence also can be obtained as a special case. It is observed that the Bayes estimators of the unknown parameters cannot be obtained in explicit forms. We propose to use the importance sampling technique to compute the Bayes estimates and also to construct the associated highest posterior density credible intervals. The analysis of two data sets has been performed for illustrative purposes. The performances of the proposed estimators are quite satisfactory. Finally, we generalize the results for the multivariate case.     

Estimation of parameters and the mean life of a mixed failure time distribution

This paper deals with estimation of parameters and the mean life of a mixed failure time distribution that has a discrete probability mass at zero and an exponential distribution with mean O for positive values. A new sampling scheme similar to Jayade and Prasad (1990) is proposed for estimation of parameters. We derive expressions for biases and mean square errors (MSEs) of the maximum likelihood estimators (MLEs). We also obtain the uniformly minimum variance unbiased estimators (UMVUEs) of the parameters. We compare the estimator of O and mean life fj based on the proposed sampling scheme with the estimators obtained by using the sampling scheme of Jayade and Prasad (1990).     

The Kumaraswamy skew-t distribution and its related properties

Skew normal distribution is an alternative distribution to the normal distribution to accommodate asymmetry. Since then extensive studies have been done on applying Azzalini’s skewness mechanism to other well-known distributions, such as skew-t distribution, which is more flexible and can better accommodate long tailed data than the skew normal one. The Kumaraswamy generalized distribution (Kw ? F) is another new class of distribution which is capable of fitting skewed data that can not be fitted well by existing distributions. Such a distribution has been widely studied and various versions of generalization of this distribution family have been introduced. In this article, we introduce a new generalization of the skew-t distribution based on the Kumaraswamy generalized distribution. The new class of distribution, which we call the Kumaraswamy skew-t (KwST) has the ability of fitting skewed, long, and heavy-tailed data and is more flexible than the skew-t distribution as it contains the skew-t distribution as a special case. Related properties of this distribution family such as mathematical properties, moments, and order statistics are discussed. The proposed distribution is applied to a real dataset to illustrate the estimation procedure.     

Waiting time paradox and size biased sampling

Feller's (1971) 'Waiting time for the bus paradox1 is explained by the fact that the long Interarrival times have a better chance of covering an arbitrary point in time than do short interarrival times, This gives rise to size biased sampling and the length of the Interarrival time that contains the arbitrary point has a moment distribution, This paper deals with the characterization of the exponential distribution based on this 'Waiting time paradox'. Similar characterizations of the binomial, the negative binomial, the Poisson and the geometric distributions are obtained.     

负二项分布的两种近似分布及其比较

负二项分布是一个重要的离散型随机变量的分布,可以用泊松分布和正态分布作为其近似分布,通过对两种近似分布进行比较分析,结果表明：在参数q很小时,泊松近似的精度好于正态近似,而且在参数q很小时,即便r不是很大,用泊松分布也能获得负二项分布较好的近似;当参数q较大时,泊松近似效果不好,相比之下,正态近似的结果不错。     

The Skew‐normal Distribution and Related Multivariate Families*

Abstract. This paper provides an introductory overview of a portion of distribution theory which is currently under intense development. The starting point of this topic has been the so‐called skew‐normal distribution, but the connected area is becoming increasingly broad, and its branches include now many extensions, such as the skew‐elliptical families, and some forms of semi‐parametric formulations, extending the relevance of the field much beyond the original theme of ‘skewness’. The final part of the paper illustrates connections with various areas of application, including selective sampling, models for compositional data, robust methods, some problems in econometrics, non‐linear time series, especially in connection with financial data, and more.     

Inference and optimal censoring schemes for progressively censored Birnbaum–Saunders distribution

The aim of this paper is twofold. First we discuss the maximum likelihood estimators of the unknown parameters of a two-parameter Birnbaum–Saunders distribution when the data are progressively Type-II censored. The maximum likelihood estimators are obtained using the EM algorithm by exploiting the property that the Birnbaum–Saunders distribution can be expressed as an equal mixture of an inverse Gaussian distribution and its reciprocal. From the proposed EM algorithm, the observed information matrix can be obtained quite easily, which can be used to construct the asymptotic confidence intervals. We perform the analysis of two real and one simulated data sets for illustrative purposes, and the performances are quite satisfactory. We further propose the use of different criteria to compare two different sampling schemes, and then find the optimal sampling scheme for a given criterion. It is observed that finding the optimal censoring scheme is a discrete optimization problem, and it is quite a computer intensive process. We examine one sub-optimal censoring scheme by restricting the choice of censoring schemes to one-step censoring schemes as suggested by Balakrishnan (2007), which can be obtained quite easily. We compare the performances of the sub-optimal censoring schemes with the optimal ones, and observe that the loss of information is quite insignificant.     

Confidence regions for two proportions from independent negative binomial distributions

The negative binomial distribution offers an alternative view to the binomial distribution for modeling count data. This alternative view is particularly useful when the probability of success is very small, because, unlike the fixed sampling scheme of the binomial distribution, the inverse sampling approach allows one to collect enough data in order to adequately estimate the proportion of success. However, despite work that has been done on the joint estimation of two binomial proportions from independent samples, there is little, if any, similar work for negative binomial proportions. In this paper, we construct and investigate three confidence regions for two negative binomial proportions based on three statistics: the Wald (W), score (S) and likelihood ratio (LR) statistics. For large-to-moderate sample sizes, this paper finds that all three regions have good coverage properties, with comparable average areas for large sample sizes but with the S method producing the smaller regions for moderate sample sizes. In the small sample case, the LR method has good coverage properties, but often at the expense of comparatively larger areas. Finally, we apply these three regions to some real data for the joint estimation of liver damage rates in patients taking one of two drugs.     

Asymptotic distribution of (h, φ)-entropies

In this paper, (h,φ)-entropies are presented as a generalization of φ-entropies, Havrda-Charvat entropies and the Renyi entropy among others. For this functional, asymptotic distribution for simple random sampling and stratified .sampling with proportional affixing is obtained.     

A note on the posterior inference for the Yule–Simon distribution

The Yule–Simon distribution has been out of the radar of the Bayesian community, so far. In this note, we propose an explicit Gibbs sampling scheme when a Gamma prior is chosen for the shape parameter. The performance of the algorithm is illustrated with simulation studies, including count data regression, and a real data application to text analysis. We compare our proposal to the frequentist counterparts showing better performance of our algorithm when a small sample size is considered.     

Bayesian Inference in Generalized Error and Generalized Student-t Regression Models

This study takes up inference in linear models with generalized error and generalized t distributions. For the generalized error distribution, two computational algorithms are proposed. The first is based on indirect Bayesian inference using an approximating finite scale mixture of normal distributions. The second is based on Gibbs sampling. The Gibbs sampler involves only drawing random numbers from standard distributions. This is important because previously the impression has been that an exact analysis of the generalized error regression model using Gibbs sampling is not possible. Next, we describe computational Bayesian inference for linear models with generalized t disturbances based on Gibbs sampling, and exploiting the fact that the model is a mixture of generalized error distributions with inverse generalized gamma distributions for the scale parameter. The linear model with this specification has also been thought not to be amenable to exact Bayesian analysis. All computational methods are applied to actual data involving the exchange rates of the British pound, the French franc, and the German mark relative to the U.S. dollar.     

Bayesian inference and prediction of order statistics for a Type-II censored Weibull distribution

This paper describes the Bayesian inference and prediction of the two-parameter Weibull distribution when the data are Type-II censored data. The aim of this paper is twofold. First we consider the Bayesian inference of the unknown parameters under different loss functions. The Bayes estimates cannot be obtained in closed form. We use Gibbs sampling procedure to draw Markov Chain Monte Carlo (MCMC) samples and it has been used to compute the Bayes estimates and also to construct symmetric credible intervals. Further we consider the Bayes prediction of the future order statistics based on the observed sample. We consider the posterior predictive density of the future observations and also construct a predictive interval with a given coverage probability. Monte Carlo simulations are performed to compare different methods and one data analysis is performed for illustration purposes.     

Bayesian acceptance sampling plans following economic criteria: An application to paper pulp manufacturing

The main purposes of this paper are to derive Bayesian acceptance sampling plans regarding the number of defects per unit of product, and to illustrate how to apply the methodology to the paper pulp industry. The sampling plans are obtained following an economic criterion: minimize the expected total cost of quality. It has been assumed that the number of defects per unit of product follows a Poisson distribution with process average 5 , whose prior information is described either for a gamma or for a non- informative distribution. The expected total cost of quality is composed of three independent components: inspection, acceptance and rejection. Both quadratic and step-loss functions have been used to quantify the cost incurred for the acceptance of a lot containing units with defects. Combining the prior information on 5 with the loss functions, four different sampling plans are obtained. When the quadratic-loss function is used, an analytical relation between the optimum settings of the sample size and the acceptance number is derived. The robustness analysis indicates that the sampling plans obtained are robust with respect to the prior distribution of the process average as well as to the misspecification of its mean and variance.     

Comparison of two sampling schemes for generating record-breaking data from the proportional hazard rate models

ABSTRACT

In this article, we consider a sampling scheme in record-breaking data set-up, as record ranked set sampling. We compare the proposed sampling with the well-known sampling scheme in record values known as inverse sampling scheme when the underlying distribution follows the proportional hazard rate model. Various point estimators are obtained in each sampling schemes and compared with respect to mean squared error and Pitman measure of closeness criteria. It is observed in most of the situations that the new sampling scheme provides more efficient estimators than their counterparts. Finally, one data set has been analyzed for illustrative purposes.     

On bivariate discrete Weibull distribution

Recently, Lee and Cha proposed two general classes of discrete bivariate distributions. They have discussed some general properties and some specific cases of their proposed distributions. In this paper we have considered one model, namely bivariate discrete Weibull distribution, which has not been considered in the literature yet. The proposed bivariate discrete Weibull distribution is a discrete analogue of the Marshall–Olkin bivariate Weibull distribution. We study various properties of the proposed distribution and discuss its interesting physical interpretations. The proposed model has four parameters, and because of that it is a very flexible distribution. The maximum likelihood estimators of the parameters cannot be obtained in closed forms, and we have proposed a very efficient nested EM algorithm which works quite well for discrete data. We have also proposed augmented Gibbs sampling procedure to compute Bayes estimates of the unknown parameters based on a very flexible set of priors. Two data sets have been analyzed to show how the proposed model and the method work in practice. We will see that the performances are quite satisfactory. Finally, we conclude the paper.