Journal of Statistical Modelling: Theory and Applications
Volume:1 Issue: 2, Summer and Autumn 2020

The main measure of the uncertainty contained in random variable X is the Shannon entropy H(X) = −E(log(f(X)). The cumulative entropy is an information measure which is alternative to the Shannon entropy and is connected with reliability theory. The cumulative residual entropy (CRE) introduced by Rao et al. (2004) is a generalized measure of uncertainty which is applied in reliability. Asadi and Zohrevand (2007) defined a dynamic version of the CRE by ε(X,t). In this paper, weighted residual entropy and weighted cumulative residual entropy are discussed. The properties of weighted entropy, cumulative residual entropy, weighted residual entropy, weighted cumulative residual entropy, weighted past entropy, weighted cumulative past entropy, dynamic cumulative residual entropy, dynamic cumulative past entropy, are also given.

In this paper, some recurrence relations are presented for the single and product moments of progressively Type-II right censored order statistics from a Pareto distribution. These relations are obtained for a progressively censored sample from Pareto distribution with fixed and random removals, where in the random case, the number of units removed at each failure time follows a binomial distribution. In addition,Thomas-Wilson’s Mixture Formula for Moments are obtained with with fixed and random removals. Finally, a numerical study is carried out to compare real and simulation results based on biases and MSEs of the expected termination time.

This article deals with the problem of characterizing the parent distribution on the basis of the cumulative residual entropy of sequential order statistics under a conditional proportional hazard rates model. It is shown that the equality of the cumulative residual entropy in the first sequential order statistics determines uniquely the parent distribution. Subsequently, we characterize the Weibull distribution on the basis of the ratio of the cumulative residual entropy of first sequential order statistics to the corresponding mean. Also, we consider characterizations based on the dynamic cumulative residual entropy and derive some bounds for the cumulative residual entropy of residual lifetime of the sequential order statistics.

One of the practical and important issue in statistics is the fitness of regression models. Optimal design is a way to obtain suitable fitness of this type of models. In addition, we need to use some criteria for attaining optimal design in regression models. The D-optimality criterion is one of the most famous criteria which is used here. An appropriate method to obtain the optimal designs is the Bayesian method that need to the prior distribution for the parameters of the model (coefficient regression). In this paper, by using Bayesian methods, D-optimal designs are obtained for quadratic beta regression model. Also, uniform and normal distributions are considered as the prior distributions and obtained results are analyzed.

This paper aims to conduct a model-based analysis of the spatial patterns of three tree species in a Hyrcanian forest and investigate their associations. There are many known and unknown mechanisms that influence the spatial forest structure and species associations. These complex and mainly unobservable mechanisms can be modeled by hidden Gaussian random fields and log-Gaussian Cox process models are appropriate for linking them to the spatial patterns of tree species. We consider a multivariate log-Gaussian Cox process model that can take into account the overall mixed effects of all influential factors on spatial distributions of species and quantify species associations in terms of some parameters. This construction provides a suitable framework for modeling and analyzing spatial patterns of several species. We also discuss modeling tree diameters, parameter estimation and goodness of fit methods and apply them to the data. Results from fitting the model to the data show that there is a significant negative association between two light-demanding species. Finally, a Gamma intensity-dependent model is considered to model spatial correlation in tree diameters of one of the species.

This paper introduces the problem of interval estimation for stress strength reliability parameter P(X<Y)‎, ‎where random variables X and Y stand for stress and strength‎, ‎respectively‎. ‎In most of the research papers‎, ‎the authors assumed that X and Y come from the same family of distribution‎. ‎By taking into account some situations arise‎, ‎in this paper we assume that X and Y follow exponential and inverted exponential distributions‎, ‎respectively‎. ‎Our goal is to construct a confidence interval for reliability parameter in this model by using some (approximately) exact and strong methods such as bootstrap‎, ‎generalized and highest posterior distribution approaches‎. ‎Also‎, ‎we compare these methods by means of the expected length and coverage probability criteria‎. ‎Finally‎, ‎a real data set is given and we apply the above methods of estimation on it to inference about the parameter of interest.

Reducing income inequality is one of the major steps toward economic development. When the level of inequality in the distribution of income and wealth is high in the society, many economic, social and even political problems might happen. So, many studies in the economic literature tried to find the determinants of income inequality and propose some policies to decline it. In this paper, we will address the analysis of income inequality panel data across different countries through 2011 to 2015. One of the commonly used methodologies to analyze panel data is the linear mixed effects model. Since the linearity assumption might be violated, recently, the idea of mixed effect models are combined with the flexibility of tree-based estimation methods which allows for potential higher order interactions as well. In this paper, we apply the resulting estimation method, called the RE-EM tree, to the income inequality panel data. The results show that the RE-EM tree is less sensitive to parametric assumptions and provides improved predictive power compared to simple regression trees without random effects. This is due to the fact that each country applies its own specific poverty reduction measures handled via country-specific random coefficients of RE-EM tree.

Maximum likelihood estimators are usually biased. The first order bias term of the maximum likelihood estimators can be large for a small or medium sample size, and this bias may have a significant effect on distribution performance. Different methods may be used to reduce this bias. These methods have inspired many scholars to study this field over the past years, but the use of Bartlett’s method requires the expected value of third power derivatives of the likelihood function. Consequently, because this quantity (the expected value of third power derivatives of the likelihood function) is not necessarily calculable in some distributions, in this paper we propose a new method based on algebraic approximation of the maximum likelihood estimator bias which needless the expected value of third power derivatives of the likelihood function. In addition, as an application of this method, we will consider a bias correction for estimating parameters of Maxwell distribution.

A new generalized version of the Weibull distribution, which is called the perturbed Weibull distribution, is introduced in this paper. The present distribution provide enough flexibility for analyzing different types of data with increasing, decreasing, constant, bathtub shaped, unimodal, increasing-decreasing-increasing and decreasing-increasing-decreasing hazard functions in comparing with former extensions of the Weibull distribution. We study its properties including servival and hazard functions, moments, moment generating and characteristic functions, quantiles and Renyi entropy. Estimation of parameters using the methods of moment and maximum likelihood is obtained. We show the consistency of the moments and maximum likelihood estimators using some simulation study. Finally, the flexibility of the new distribution is illustrated in an application to two real data sets.

The article addresses different estimators of the probability density function and the cumulative distribution function for the two-parameter exponential distribution for type-II censored sample. Following estimation methods are considered: maximum likelihood estimator, uniformly minimum variance unbiased estimator and plug-in uniformly minimum variance unbiased estimator. Analysis of real data sets are performed to compare the performances of the proposed methods of estimation. The maximum likelihood estimators of the PDF and the CDF are performing better in mean squared error sense. In case of unknown location and known scale parameter, plug-in uniformly minimum variance unbiased estimator is performing better.

In this paper, a new discrete distribution is studied based on geometric odds ratio. This new distribution has three parameters and can be a unimodal or a bimodal discrete distribution. Some important distributional properties are studied. For example, moments, the behaviour of the hazard rate function, stochastic orders, mixing processes, infinite divisibility, Rényi and Shannon entropies and the distributions of order statistics are investigated. We will see that the hazard rate function of the new discrete distribution can be monotonically increasing and decreasing and bathtub-shaped. The parameters of the distribution are estimated by the maximum likelihood method, and a real data set is analyzed in order to show the effectiveness of the model.

‎A new family of continuous distributions namely‎ ‎new power generalized Weibull-G family of distributions is proposed‎. ‎Some special sub-models of the new family are provided‎. ‎Some statistical properties of the new family of distributions are obtained including the quantile function‎, ‎ordinary and incomplete moments‎, ‎probability weighted moments‎, ‎distribution of the order statistics and Renyi entropy‎. ‎The maximum likelihood method is used for estimating model parameters‎. ‎A simulation study is employed to check the consistency of the maximum likelihood estimates‎. ‎The flexibility of a sub-model of the generated family is illustrated by means of two applications to real data sets.