Squared Error Loss Function Research Articles

Estimation and Prediction for the Power-Exponential Hazard Rate Distribution Based on Record Data

Synoptic The problems of classical and Bayesian estimation of the parameters of the power-exponential hazard rate distribution (P-EHRD) based on record values and the prediction of future record values are considered. The parameters of P-EHRD are estimated by the maximum likelihood and the least squares methods, and the Bayes estimates are obtained by the Metropolis-Hastings method under the squared error loss and LINEX loss functions. Also, an asymptotic confidence interval, two bootstrap confidence intervals and the highest posterior density (HPD) credible interval for the unknown parameters are constructed. The problem of predicting the future record values from the P-EHRD based on the past record values is considered using the maximum likelihood and Bayesian approaches. To investigate and compare the performance of the different proposed methods, a Monte Carlo simulation study is conducted. Finally, an example is presented to illustrate the estimation and prediction procedures.

On E-Bayesian estimations for the cumulative hazard rate and mean residual life under generalized inverted exponential distribution and type-II censoring

Estimation of reliability and hazard rate is necessary in many applications. To this aim, different methods of estimation have been employed. Each method suffers from its own problems such as complexity of calculations, high risk and so on. Toward this end, this study employed a new method, E-Bayesian, for estimating the parametric functions of the Generalized Inverted Exponential distribution, which is one of the most noticeable distributions in lifetime studies. Relations are derived under a squared error loss function, type-II censoring and a conjugate prior. E-Bayesian estimations are obtained based on different priors of the hyperparameters to investigate the influence of different prior distributions on these estimations. The asymptotic behaviors of E-Bayesian estimations and relations among them have been investigated. Finally, a comparison among the maximum likelihood, Bayes and E-Bayesian estimations in different sample sizes are made, using a real data and the Monte Carlo simulation. Simulations show that the new presented method is more efficient than previous methods and is also easy to operate. Also, some comparisons among the results of Generalized Inverted Exponential distribution, Exponential distribution and Generalized Exponential distribution are provided.

Inferences for the DUS-Exponential Distribution Based on Upper Record Values

This article considers the problem of estimation of the parameter of DUS-exponential distribution based on upper records through classical as well as Bayesian procedures. Maximum likelihood estimator is calculated under the classical scheme and Bayes estimator is obtained under the squared error loss function by using MCMC technique. The performances of both the estimators are compared on the basis of their estimated mean squared errors. The asymptotic and HPD intervals for the unknown parameter is also calculated. In addition to that, the entropy of ith upper record and joint entropy based on m upper records are obtained for considered distribution. A real data set is considered to illustrate the suitability of the proposed methodology.

Three-Stage Estimation of the Mean and Variance of the Normal Distribution with Application to an Inverse Coefficient of Variation with Computer Simulation

This paper considers sequentially two main problems. First, we estimate both the mean and the variance of the normal distribution under a unified one decision framework using Hall’s three-stage procedure. We consider a minimum risk point estimation problem for the variance considering a squared-error loss function with linear sampling cost. Then we construct a confidence interval for the mean with a preassigned width and coverage probability. Second, as an application, we develop Fortran codes that tackle both the point estimation and confidence interval problems for the inverse coefficient of variation using a Monte Carlo simulation. The simulation results show negative regret in the estimation of the inverse coefficient of variation, which indicates that the three-stage procedure provides better estimation than the optimal.

Classical and Bayesian Estimation of Stress-Strength Reliability from Generalized Inverted Exponential Distribution based on Upper Records

This paper deals with the problem of classical and Bayesian estimation of stress-strength reliability (R=P(X<Y)) based on upper record values from generalized inverted exponential distribution (GIED). Hassan {et al.} (2018) discussed the maximum likelihood estimator (MLE) and Bayes estimator of $R$ by considering that the scale parameter to be known for defined distribution while we consider the case when all the parameters of GIED are unknown. In the classical approach, we have discussed MLE and uniformly minimum variance estimator (UMVUE). In Bayesian approach, we have considered the Bays estimator of R by considering the squared error loss function. Further, based on upper records, we have considered the Asymptotic confidence interval based on MLE, Bayesian credible interval and bootstrap confidence interval for $R$. Finally, Monte Carlo simulations and real data applications are being carried out for comparing the performances of the estimators of R.

Bayesian estimation of the survival characteristics for Hjorth distribution under progressive type-II censoring

In this paper, the Bayes estimation procedure for the parameters and survival characteristics (survival and hazard functions) of the Hjorth distribution has been proposed with progressively type-II censored data. The Bayes estimators are derived with gamma prior and evaluated under squared error loss function. It is known that the censored observations create the complexity in Bayes estimation procedures. Therefore, two approximation techniques, namely Tierney–Kadane approximation method and Markov Chain Monte Carlo method have been used to compute the approximate Bayes estimators. The proposed estimates are compared with the usual maximum likelihood estimators through Monte Carlo simulations. Lastly, a medical data set has been considered to show the applicability of the proposed model as well study in real life scenario.

On the Bayesian Analysis of Two-Component Mixture of Transmuted Weibull Distribution

Transmuted distributions are skewed distributions and recently attracted a great attention of researchers due totheir applications in reliability and statistics. In this article, our main focus is on the Bayesian estimation of two-component mixture of Transmuted Weibull Distribution (TWD) under type-I right censored sampling scheme. In order to estimate the unknown parameters, non-informative and informative priors under Squared Error Loss Function (SELF), Precautionary Loss Function (PLF) and Quadratic Loss Function (QLF) are assumed when computing the posterior estimations. In addition the Bayesian credible intervals (BCI) were also constructed. Markov Chain Monte Carlo (MCMC) technique is adopted to generate samples from the posterior distributions and in turn computing different posterior summaries including Bayes estimates(BEs), posterior risks(PRs) and Bayesian credible intervals (BCI). As an illustration comparision of these Bayes estimators are made through simulated under different loss functions in terms of their respective posterior risks assuming different sample sizes and censoring rates. Two real-life examples; the first being the survival times of hepatitis B & C patientswhile the second being the hole diameter of 12 mm and the sheet thickness is 3.15 mm are also discussed to illustrate the potential application of the proposed methodology.

Bayesian estimation of the reliability characteristic of Shanker distribution

In this study, we discussed the Bayesian property of unknown parameter and reliability characteristic of the Shanker distribution. The maximum likelihood estimate is calculated. The approximate confidence interval of the unknown parameter is constructed based on the asymptotic normality of maximum likelihood estimator. Two bootstrap confidence intervals for the unknown parameter are also computed. Bayesian estimates of parameter and reliability characteristic against squared error loss function are obtained. Lindley’s approximation and Metropolis-Hastings algorithm are applied to obtain the Bayes estimates. In consequence, we also construct the highest posterior density intervals. A numerical comparison is also made to compare different methods through a Monte Carlo simulation study. Finally, two real data sets are also analyzed using the proposed methods.

Inference of accelerated dependent competing risks model for Marshall–Olkin bivariate Weibull distribution with nonconstant parameters

In this paper, we discuss the statistical inference of constant-stress accelerated dependent competing risks model under Type-II hybrid censoring schemes. The dependency structure is modeled by a Marshall–Olkin bivariate Weibull distribution. Both the shape and the scale parameters in the model are assumed to be dependent on the stress levels through a log-linear relationship. The maximum likelihood estimates (MLEs) of the model parameters are derived. Confidence intervals (CIs) of the model parameters are constructed based on the asymptotic normality of MLEs and the bias-corrected percentile bootstrap method. Bayes estimates with the squared error loss function and the highest posterior density (HPD) credible intervals (CIs) are obtained by using an importance sampling method. In addition, the estimates for the accelerated coefficients and the reliability under normal use stress level at mission time are derived. A Monte Carlo simulation study is used to evaluate the performance of the proposed statistical inference methods. A real data example is used to illustrate the methodologies proposed in this paper and to compare the proposed model with the copula model.

مقارنة مقدرات بيز لدالة المعولية لتوزيع باريتو من النوع الاول باستعمال دوال معلوماتية مضاعفة مختلفة

The comparison of double informative priors which are assumed for the reliability function of Pareto type I distribution. To estimate the reliability function of Pareto type I distribution by using Bayes estimation, will be used two different kind of information in the Bayes estimation; two different priors have been selected for the parameter of Pareto type I distribution . Assuming distribution of three double prior’s chi- gamma squared distribution, gamma - erlang distribution, and erlang- exponential distribution as double priors. The results of the derivaties of these estimators under the squared error loss function with two different double priors. Using the simulation technique, to compare the performance for each estimator, several cases from pareto type I distribution for data generating, and for different samples sizes (small, medium, and large). 
 It has been obtained from the simulation results the double prior distribution of gamma-erlang distribution with give a good estimation for reliability function when the true value for for all .Also the double prior distribution chi- gamma square distribution with give good estimation for reliability function when the true value for all t. And the same thing for with the values of the parameters and for all t except t=1.3. It has obtained a good estimation for reliability function (), when the double prior distribution is chi-gamma square distribution with at the true value for for all t.

Bayesian estimation and prediction for Burr‐Rayleigh mixture model using censored data

SummaryIn this study, Burr‐XII and Rayleigh distributions are combined to form a new mixture model that is considered to model heterogeneous data. Our objective is to estimate parameters of the proposed mixture model using Bayesian technique under type‐I censoring. Bayesian parameter estimation for the said mixture model is conducted by using informative priors, ie, gamma and squared root inverted gamma (SRIG) as well as noninformative prior, ie, Jeffrey's prior. Squared error loss function (SELF) and quadratic loss function (QLF) are employed to obtain and Bayes estimators. Properties of the proposed Bayes estimators are highlighted through a simulation study. When prior distributions and loss functions utilized in the study are compared in terms of posterior risks, informative prior found to be more suitable and decision turns out to be in favor of QLF. Prediction limits for the single sample case and two sample case are obtained to provide an insight into future sample data. Application of the proposed model is also elaborated using a real‐life example.

Statistical Inference of the Rayleigh Distribution Based on Progressively Type II Censored Competing Risks Data

A competing risks model under progressively type II censored data following the Rayleigh distribution is considered. We establish the maximum likelihood estimation for unknown parameters and compute the observed information matrix and the expected Fisher information matrix to construct the asymptotic confidence intervals. Moreover, we obtain the Bayes estimation based on symmetric and non-symmetric loss functions, that is, the squared error loss function and the general entropy loss function, and the highest posterior density intervals are also derived. In addition, a simulation study is presented to assess the performances of different methods discussed in this paper. A real-life data set analysis is provided for illustration purposes.

Estimation in Inverse Weibull Distribution Based on Randomly Censored Data

This article deals with the estimation of the parameters and reliability characteristics in inverse Weibull (IW) distribution based on the random censoring model. The censoring distribution is also taken as an IW distribution. Maximum likelihood estimators of the parameters, survival and failure rate functions are derived. Asymptotic confidence intervals of the parameters based on the Fisher information matrix are constructed. Bayes estimators of the parameters, survival and failure rate functions under squared error loss function using non-informative and gamma informative priors are developed. Furthermore, Bayes estimates are obtained using Tierney-Kadane's approximation method and Markov chain Monte Carlo (MCMC) techniques. Also, highest posterior density (HPD) credible intervals of the parameters based on MCMC techniques are constructed. A simulation study is conducted to compare the performance of various estimates. Finally, a randomly censored real data set supports the estimation procedures developed in this article.

Bayesian Reliability Analysis of Exponential Distribution Model Based on Zero-failure Data and Scale Squared Error Loss Function

Based on the zero failure data, this paper studies the Bayes estimation of the failure rate and reliability of the exponential distribution model under the scale squared error loss function. When the prior distribution of the parameters is the Gamma distribution, the Bayes estimators and E-Bayes estimators of the failure rate and reliability of the exponential model can be obtained. Finally, a practical example is utilized to illustrate the effectiveness of the estimation.

Approximate Bayes Estimation for Log-Dagum Distribution

In this article, the approximate Bayes estimation problem for the log-Dagum distribution with three parameters is considered. Firstly, the maximum likelihood estimators and asymptotic confidence intervals based on these estimators for unknown parameters of log-Dagum distribution are constructed. In addition, approximate Bayes estimators under squared error loss function for unknown parameters of this distribution are obtained using Tierney and Kadane approximation. A Monte-Carlo simulation study is performed to compare performances of maximum likelihood and approximate Bayes estimators in terms of mean square errrors and biases. Finally, real data analysis for this distribution is performed.

Parameter estimation of the inverted exponentiated Rayleigh distribution based on progressively first-failure censored samples

We study statistical inference of the inverted exponentiated Rayleigh distribution under progressively first-failure censoring samples in our paper. Specifically, we deal with Maximum likelihood and Bayes estimators of parameters. The observed Fisher matrix is conducive to obtain asymptotic confidence interval. Parametric bootstrap methods are applied to provide the confidence intervals. Bayes estimators in terms of squared error loss function are derived with Metropolis–Hastings technique, which are helpful to construct highest posterior density credible intervals. We compare the behavior of various estimators by conducting Monte Carlo simulations. A set of actual data is analyzed to introduce the proposed methods.

Bayesian inference on reliability in a multicomponent stress-strength bathtub-shaped model based on record values

In the literature, there are a well-developed estimation techniques for the reliability assessment in multicomponent stress-strength models when the information about all the experimental units are available. However, in real applications, only observations that exceed (or fall below) the current value may be recorded. In this paper, assuming that the components of the system follow bathtub-shaped distribution, we investigate Bayesian estimation of the reliability of a multicomponent stress-strength system when the available data are reported in terms of record values. Considering squared error, linex and entropy loss functions, various Bayes estimates of the reliability are derived. Because there are not closed forms for the Bayes estimates, we will use Lindley’s method to calculate the approximate Bayes estimates. Further, for comparison purposes, the maximum likelihood estimate of the reliability parameter is obtained. Finally, simulation studies are conducted in order to evaluate the performances of the proposed procedures and analysis of real data sets is provided.

Ruin Probability Functions and Severity of Ruin as a Statistical Decision Problem

It is known that the classical ruin function under exponential claim-size distribution depends on two parameters, which are referred to as the mean claim size and the relative security loading. These parameters are assumed to be unknown and random, thus, a loss function that measures the loss sustained by a decision-maker who takes as valid a ruin function which is not correct can be considered. By using squared-error loss function and appropriate distribution function for these parameters, the issue of estimating the ruin function derives in a mixture procedure. Firstly, a bivariate distribution for mixing jointly the two parameters is considered, and second, different univariate distributions for mixing both parameters separately are examined. Consequently, a catalogue of ruin probability functions and severity of ruin, which are more flexible than the original one, are obtained. The methodology is also extended to the Pareto claim size distribution. Several numerical examples illustrate the performance of these functions.

Numerical study of robust Bayesian analysis of generalized inverted family of distributions based on progressive type II right censoring

In this article, we have developed the robust Bayesian inference for the generalized inverted family of distributions (GIFD) under an -contamination class of prior distributions for the shape parameter α, with different possibilities of known and unknown scale parameter, based on progressive type II censoring. We have derived the ML-II Bayes estimators of the parameters, reliability function and hazard function under the general entropy loss function (GELF) and linear exponential loss function (LLF). Results under squared error loss functions (SELF) are derived as a special case of GELF. We have also presented simulation study and analysis of a real data set.

Bayesian Approach of Estimating Shape Parameter of Log Logistic Distribution

In this research, Log logistic distribution under Bayesian paradigm is studied. Posterior distribution has been derived using Uniform and Jeffrey’s’ priors. The three loss functions taken up are squared error loss function (SELF), weighted balanced loss function (WBLF) and precautionary loss function (PLF). The performance of an estimator is assessed on the basis of its relative posterior risk. An extensive simulation was carried out to illustrate the applicability of this research and to compare the performance of different estimators. The study indicated that for estimation of the shape parameter using Bayesian estimation technique, the precautionary loss function can effectively be employed. Keywords : Bayesian estimation, Log logistic distribution, Uniform prior, Jeffrey’s prior, loss functions.