Alternative Dirichlet Priors for Estimating Entropy Via a Power Sum Functional

Entropy is a functional of probability and is a measurement of information contained in a system; however, the practical problem of estimating entropy in applied settings remains a challenging and relevant problem. The Dirichlet prior is a popular choice in the Bayesian framework for estimation of entropy when considering a multinomial likelihood. In this work, previously unconsidered Dirichlet type priors are introduced and studied. These priors include a class of Dirichlet generators as well as a noncentral Dirichlet construction, and in both cases includes the usual Dirichlet as a special case. These considerations allow for flexible behaviour and can account for negative and positive correlation. Resultant estimators for a particular functional, the power sum, under these priors and assuming squared error loss, are derived and represented in terms of the product moments of the posterior. This representation facilitates closed-form estimators for the Tsallis entropy, and thus expedite computations of this generalised Shannon form. Select cases of these proposed priors are considered to investigate the impact and effect on the estimation of Tsallis entropy subject to different parameter scenarios.

Bayesian estimation for Dagum distribution based on progressive type I interval censoring

In this paper, we consider Dagum distribution which is capable of modeling various shapes of failure rates and aging criteria. Based on progressively type-I interval censoring data, we first obtain the maximum likelihood estimators and the approximate confidence intervals of the unknown parameters of the Dagum distribution. Next, we obtain the Bayes estimators of the parameters of Dagum distribution under the squared error loss (SEL) and balanced squared error loss (BSEL) functions using independent informative gamma and non informative uniform priors for both scale and two shape parameters. A Monte Carlo simulation study is performed to assess the performance of the proposed Bayes estimators with the maximum likelihood estimators. We also compute credible intervals and symmetric 100(1 − τ)% two-sided Bayes probability intervals under the respective approaches. Besides, based on observed samples, Bayes predictive estimates and intervals are obtained using one-and two-sample schemes. Simulation results reveal that the Bayes estimates based on SEL and BSEL performs better than maximum likelihood estimates in terms of bias and MSEs. Besides, credible intervals have smaller interval lengths than confidence interval. Further, predictive estimates based on SEL with informative prior performs better than non-informative prior for both one and two sample schemes. Further, the optimal censoring scheme has been suggested using a optimality criteria. Finally, we analyze a data set to illustrate the results derived.

Bayesian Estimation and Prediction of Discrete Gompertz Distribution

In this paper, Bayesian inference is used to estimate the parameters, survival, hazard and alternative hazard rate functions of discrete Gompertz distribution. The Bayes estimators are derived under squared error loss function as a symmetric loss function and linear exponential loss function as an asymmetric loss function. Credible intervals for the parameters, survival, hazard and alternative hazard rate functions are obtained. Bayesian prediction (point and interval) for future observations of discrete Gompertz distribution based on two-sample prediction are investigated. A numerical illustration is carried out to investigate the precision of the theoretical results of the Bayesian estimation and prediction on the basis of simulated and real data. Regarding the results of simulation seems to perform better when the sample size increases and the level of censoring decreases. Also, in most cases the results under the linear exponential loss function is better than the corresponding results under squared error loss function. Two real lifetime data sets are used to insure the simulated results.

Estimation of Reliability in a Multicomponent Stress–Strength Model Based on Power Transformed Half-Logistic Distribution

This paper deals with the study of the reliability of multicomponent stress–strength model assuming that the strength components are independently and identically distributed as power transformed half-logistic (PHL) distribution. The strength components which are subject to a common stress are assumed to be independent with either Weibull distribution or PHL distribution. The maximum likelihood estimates of the multicomponent stress–strength reliability and its asymptotic confidence interval under the above said conditions are obtained. The Bayes estimates of the multicomponent stress–strength reliability are also obtained under squared error loss function and using gamma priors for the parameters. To evaluate the performance of the procedure, a simulation study is considered. For illustration purpose of the proposed model, two real life examples are given.

E-Bayesian Estimation of Chen Distribution Based on Type-I Censoring Scheme

In this paper, E-Bayesian estimation of the scale parameter, reliability and hazard rate functions of Chen distribution are considered when a sample is obtained from a type-I censoring scheme. The E-Bayesian estimators are obtained based on the balanced squared error loss function and using the gamma distribution as a conjugate prior for the unknown scale parameter. Also, the E-Bayesian estimators are derived using three different distributions for the hyper-parameters. Some properties of E-Bayesian estimators based on balanced squared error loss function are discussed. A simulation study is performed to compare the efficiencies of different estimators in terms of minimum mean squared errors. Finally, a real data set is analyzed to illustrate the applicability of the proposed estimators.

Parameter Estimation of the Temporal Point Process Model through the Bayesian Approach (Case Study : Malaria Disease Data from Wahidin Hospital in Makassar City)

Penelitian ini bertujuan mengestimasi parameter melalui pendekatan Bayesian dari model temporal point process. Paramater intensitas bersyarat model tersebut dipandang sebagai suatu renewal process yang selanjutnya digunakan melalui pendekatan Squared Error Loss Function (SELF). Parameter intensitas bersyarat model temporal point process diestimasi menggunakan metode maximum likelihood estimation melalui persamaan likelihood point process. Selain itu, penelitian ini mengkaji metode estimasi maksimum likelihood dan metode Bayes untuk menganalis fungsi resiko dari hasil penaksir parameter intensitas bersyarat. Pada aplikasi estimasi parameter ini, studi kasus yang digunakan adalah menganalisa data orang yang terkena penyakit malaria yang datanya berasal dari Rumah Sakit Wahidin Kota Makassar. Studi kasus tersebut menghasilkan nilai yang merupakan nilai resiko penaksir MLE yang lebih tinggi dibandingkan dengan menggunakan Metode Bayes sedangkan nilai merupakan hasil nilai resiko dari penaksir MLE yang lebih kecil dibandingkan dengan menggunakan Metode Bayes. This study parameter estimation of conditional intensity with temporal point process model by Bayesian approach. The conditional intensity with temporal point process model derived as a renewal process where inter event time is defined as its random variable. Squared Error Loss Function (SELF) approach is used to estimate the parameter of conditional intensity with temporal point process model which is happened as a renewal process using Bayesian. The other outlines of this paper is to determine the Risk Function as the result of estimation of conditional intensity by Bayesian and by Maximum likelihood Estimation (MLE). The application taking an analysis of Malaria at a place, which is properly conclude that the estimation using MLE method is more risky than the Bayesian it self.

Bayesian Estimation of Parameters of Weibull Distribution Using Linex Error Loss Function

This paper develops a Bayesian analysis of the scale parameter in the Weibull distribution with a scale parameter  θ  and shape parameter  β (known). For the prior distribution of the parameter involved, inverted Gamma distribution has been examined. Bayes estimates of the scale parameter, θ  , relative to LINEX loss function are obtained. Comparisons in terms of risk functions of those under LINEX loss and squared error loss functions with their respective alternate estimators, viz: Uniformly Minimum Variance Unbiased Estimator (U.M.V.U.E) and Bayes estimators relative to squared error loss function are made. It is found that Bayes estimators relative to squared error loss function dominate the alternative estimators in terms of risk function.

Minimax estimation of a bivariate cumulative distribution function

The problem of estimating a bivariate cumulative distribution function F under the weighted squared error loss and the weighted Cramer–von Mises loss is considered. No restrictions are imposed on the unknown function F. Estimators, which are minimax among procedures being affine transformation of the bivariate empirical distribution function, are found. Then it is proved that these procedures are minimax among all decision rules. The result for the weighted squared error loss is generalized to the case where F is assumed to be a continuous cumulative distribution function. Extensions to higher dimensions are briefly discussed.