Precautionary Loss Function Research Articles

Bayesian estimation strategy for multi-component geometric life testing model under doubly type-1 censoring scheme

This study develops a Bayesian approach for estimating the unknown parameters of the 3-component mixture of geometric (3-CMG) model under a doubly type-I censoring scheme (DT1CS). The derivations of the Bayes estimators (BEs) and Bayes risks (BRs) are presented under square error loss function (SELF), precautionary loss function (PLF) and DeGroot loss function (DLF) using Beta prior under DT1CS. The strategy is evaluated through extensive simulation and real-life data analysis, showing the strength and efficiency of the newly proposed model. The study recommends that the SELF is the optimal choice for accurately estimating the unknown parameters of the 3-CMG model.

Pre-test Shrinkage Estimation for Reliability Function of Burr XII Distribution Using Progressive Type II Censored Sample under Precautionary Loss Function (PLF)

This article deal with the proposal of suggest and study of the properties of pre-test shrinkage estimators of Reliability Function for the Burr XII distribution using Progressive Type II censored sample. Since some difficulties to derive equations of risk function for proposed shrinkage estimators of reliability function under Precautionary Loss Function (PLF), we to study properties by using Monte-Carlo simulation. The numerical and Monte-Carlo simulations show that the performance of the proposed estimators is better than classical estimators in terms of relative risk.

Constrained Bayes in multiplicative area‐level models under the precautionary loss function

AbstractConsider the problem of benchmarking small‐area estimates under multiplicative models with positive parameters. The goal is to propose a loss function that guarantees positive constrained estimates of small‐area parameters in this situation. The weighted precautionary loss function is introduced to solve the problem. Compared with the weighted Kullback–Leibler (KL) loss function, our proposed loss function penalizes underestimation of the small‐area parameters of interest more for small values of parameters. This property is appealing when we estimate disease rates. It tends to give larger estimates of small‐area parameters compared with those obtained under the KL loss function. The hierarchical empirical Bayes and constrained hierarchical empirical Bayes estimates of small‐area parameters and their corresponding risk functions under the new proposed loss function are obtained. The performance of the proposed methods is investigated using simulation studies and a real dataset.

Bayesian estimation of 3-component mixture geometric distribution under different loss functions

This article focuses on discrete survival data analysis, employing a doubly Type-I censoring scheme (DT1SC) under a Bayesian framework for parameters estimation in a 3-component mixture geometric (3-CMG) distribution. The non-informative (uniform) prior is utilized under squared error, DeGroot, and precautionary loss functions. The time is considered as discrete in this paper, presenting a departure from continuous approaches in survival analysis. The methodology is tailored to address challenges that traditional survival analysis encounters when faced with limited or missing information. The proposed method effectively handles the complexities arising from incomplete or missing data, providing a robust mechanism for accurate parameter estimation. Through extensive simulations and application to real-world datasets, our approach demonstrates its effectiveness in managing uncertainties, particularly in scenarios involving sparse or missing data. This work represents a noteworthy contribution to methodological advancements in survival analysis, offering a valuable tool for researchers and practitioners navigating intricate dynamics within the 3-CMG under DT1CS with time being considered as discrete.

Estimation of a Shape Parameter of a Gompertz-lindley Distribution Using Bayesian and Maximum Likelihood Methods

The Gompertz-Lindley distribution is an extension of the Lindley distribution with three parameters. It was found to be more flexible for modeling real life events. The distribution contains two shape parameters and a scale parameter. Despite the necessity of parameter estimation theory in modeling, it has not been shown that a method of estimation method is better for any of these three parameters of the Gompertz-Lindley distribution. This paper identifies the best estimation method for the shape parameter of the Gompertz-Lindley distribution by deriving Bayesian estimators for the shape parameter of the distribution using two non-informative prior distributions (Uniform and Jeffery) and an informative prior (gamma) under squared error loss function (SELF), quadratic loss function (QLF) and precautionary loss function (PLF). These estimators were evaluated and the results compared with the maximum likelihood estimation method using Monte Carlo simulations with the mean square error (MSE) as a criterion for choosing the best estimator.

Estimation of Entropy for Generalized Rayleigh Distribution under Progressively Type-II Censored Samples

This paper investigates the problem of entropy estimation for the generalized Rayleigh distribution under progressively type-II censored samples. Based on progressively type-II censored samples, we first discuss the maximum likelihood estimation and interval estimation of Shannon entropy for the generalized Rayleigh distribution. Then, we explore the Bayesian estimation problem of entropy under three types of loss functions: K-loss function, weighted squared error loss function, and precautionary loss function. Due to the complexity of Bayesian estimation computation, we use the Lindley approximation and MCMC method for calculating Bayesian estimates. Finally, using a Monte Carlo statistical simulation, we compare the mean square errors to examine the superiority of maximum likelihood estimation and Bayesian estimation under different loss functions. An actual example is provided to verify the feasibility and practicality of various estimations.

Bayesian estimation of a geometric distribution using informative priors based on a Type-I censoring scheme

In this paper, the geometric distribution parameter is estimated under a type-I censoring scheme by means of the Bayesian estimation approach. The Beta and Kumaraswamy informative priors, as well as five loss functions are used for this purpose. Expressions of Bayes estimators and Bayes risks are derived under the Squared Error Loss Function (SELF), the Quadratic Loss Function (QLF), the Precautionary Loss Function (PLF), the Simple Asymmetric Precautionary Loss Function (SAPLF), and the DeGroot Loss Function (DLF) using the two aforementioned priors. The prior densities are obtained through prior predictive distributions. Simulation studies are carried out to make comparisons using Bayes risks. Finally, a real-life data example is used to verify the model’s efficiency.

Bayesian Estimation for Two Parameters of Exponential Distribution under Different Loss Functions

In this paper, two parameters for the Exponential distribution were estimated using theBayesian estimation method under three different loss functions: the Squared error loss function,the Precautionary loss function, and the Entropy loss function. The Exponential distribution priorand Gamma distribution have been assumed as the priors of the scale γ and location δ parametersrespectively. In Bayesian estimation, Maximum likelihood estimators have been used as the initialestimators, and the Tierney-Kadane approximation has been used effectively. Based on the MonteCarlosimulation method, those estimators were compared depending on the mean squared errors (MSEs).The results showed that the Bayesian estimation under the Entropy loss function,assuming Exponential distribution and Gamma distribution priors for the scale and locationparameters, respectively, is the best estimator for the scale parameter. The best estimation methodfor location is the Bayesian estimation under the Entropy loss function in case of a small value ofthe scale γ (say γ < 1). Bayesian estimation under the Precautionary loss function is the best incase of a relatively large value of the scale γ (say γ > 1).

Bayesian Estimation of Transmuted Lomax Mixture Model with an Application to Type-I Censored Windshield Data

Transmuted distributions have been centered of focus for researchers recently due to their flexibility and applicability in statistics. However, the only few contributions have considered estimation for mixture of transmuted lifetime models especially under Bayesian methods has been explored more recently. We have considered the Bayesian estimation of transmuted Lomax mixture model (TLMM) for type-I censored samples. The Bayes estimates (BEs) for informative and non-informative priors. The BEs and posterior risks (PRs) are evaluated using four different loss functions (LFs), two symmetric and two asymmetric, namely the squared error loss function (SELF), precautionary loss function (PLF), weighted balance loss function (WBLF), and general entropy loss function (GELF). Simulations are run using Lindley Approximation method to compare the BEs under various sample sizes and censoring rates. The estimates under informative prior and GELF were found superior to their counterparts. The applicability of the proposed estimates has been illustrated using the analysis of a real data regarding type-I censored failure times of windshields airplanes.

Bayesian Estimation of a Scale Parameter of the Gumbel-Lomax Distribution Using Informative and Non Informative Priors

Estimating the scale parameter of the Gumbel-Lomax Distribution using the Bayesian method of estimation and evaluating the estimators by assuming two non-informative prior distributions and one informative prior distribution is very important for the general application of the Gumbel-Lomax distribution. These estimators are obtained using the squared error loss function (SELF), Quadratic loss function (QLF) and precautionary loss function (PLF). The posterior distributions of the scale parameter of the Gumbel-Lomax distribution are derived and the Estimators are also obtained using the above mentioned priors and loss functions. Furthermore, a simulation using a package in R software is carried out to assess the performance of the estimators by making use of the Mean Squared Errors of the Estimators under the Bayesian approach and Maximum likelihood method. Our results show that Bayesian Method using PLF under all priors produces the best estimators of the scale parameter compared to estimators using the Maximum Likelihood method, SELF and QLF under all the priors irrespective of the values of the parameters and the different sample sizes. It is also discovered that the other parameters have no effect on the estimators of the scale parameter.

Some Bayes estimators for Exponentiated Rayleigh Distribution under Doubly Type II Censored Data

Bayesian analysis for the shape parameter of Exponentiated Rayleigh distribution or (Burr type x) has been considered in this paper, under two priors distributions [Inverted Levy and Quasi] using doubly type II censored data, with two loss functions [Weighted and Precautionary], we conducted a simulation study that evaluate the effectiveness of four estimators by six presumption experiments for varied values of distribution parameters and three sample sizes using mean squared error [MSE]. In general the best estimator of the shape parameter of the (ERD) using Doubly type II censored data is under Inverted Levy prior under Precautionary loss function in all sample sizes

Arctan-Based Family of Distributions: Properties, Survival Regression, Bayesian Analysis and Applications

In this paper, a new class of the continuous distributions is established via compounding the arctangent function with a generalized log-logistic class of distributions. Some structural properties of the suggested model such as distribution function, hazard function, quantile function, asymptotics and a useful expansion for the new class are given in a general setting. Two special cases of this new class are considered by employing Weibull and normal distributions as the parent distribution. Further, we derive a survival regression model based on a sub-model with Weibull parent distribution and then estimate the parameters of the proposed regression model making use of Bayesian and frequentist approaches. We consider seven loss functions, namely the squared error, modified squared error, weighted squared error, K-loss, linear exponential, general entropy, and precautionary loss functions for Bayesian discussion. Bayesian numerical results include a Bayes estimator, associated posterior risk, credible and highest posterior density intervals are provided. In order to explore the consistency property of the maximum likelihood estimators, a simulation study is presented via Monte Carlo procedure. The parameters of two sub-models are estimated with maximum likelihood and the usefulness of these sub-models and a proposed survival regression model is examined by means of three real datasets.

Stress-Strength Reliability and Randomly Censored Model of Two-Parameter Power Function Distribution

The power function distribution is a flexible waiting time model that may provide better fit for some failure data. This paper presents the Bayes estimates of two-parameter power function distribution under progressive censoring. Different progressive censoring schemes have been used for the analysis. The Bayes estimates are obtained, using conjugate priors, under five loss functions including square error, precautionary, weighted, LINEX, and DeGroot loss function. The Gibbs sampling algorithm and Tierney and Kadane’s Approximation are used for the Bayes estimates of model parameters, reliability function, and stress-strength reliability. The comparison of the Bayes estimates is considered through the root mean squared errors. One real-life dataset is analyzed to illustrate the applications of proposed estimates. The results from the simulation study and real data analysis suggest that the Bayes estimation was more efficient for the progressive censoring schemes with all the withdrawals at the time of first failure.

Bayesian Analysis for Two Parameter Lomax Distribution Under Different Loss Functions

Statistical analysis via Bayesian approach is a common practice applied to draw inference about unknown parameter(s) and reliability characteristics of the probability distribution. The article include Bayesian inference of unknown shape parameter of two-parameter Lomax distribution. An attempt has been made to obtain Bayes estimators by using extension of Jeffrey’s prior and Gamma prior under Entropy loss function and Precautionary loss function. Comparison has been made by using mean square error through simulation study with varying sample sizes.

Estimation Based on Adaptive Progressively Censored under Competing Risks Model with Engineering Applications

This paper has investigated the estimation problem for the competing risks model where the data are adaptive progressively type-II censored and follow the Rayleigh distribution. Maximum likelihood and Bayesian methods are used to estimate the unknown parameters. To generate the interval estimates of the model parameters, approximate confidence intervals and two bootstrap confidence intervals are also considered based on the classical setup. Bayes estimates are obtained using independent gamma priors based on various loss functions including squared error, LINEX, general entropy, weighted SE, and precautionary loss functions. Moreover, Bayes credible intervals and the highest posterior density intervals of the model parameters are obtained. A numerical investigation has been carried out to assess the performance of the classical and Bayes estimates as well as the associated confidence and credible intervals. Finally, one simulated and two real data sets, one for breaking strengths of wire connections and the other for times to failure of small electrical appliances, have been analyzed for illustrative purposes. The results showed that the Bayes estimates using the LINEX loss function provide more reasonable estimates than the classical and Bayes estimates using squared error, general entropy, and precautionary loss functions.

Bayesian Analysis of 3-Component Unit Lindley Mixture Model with Application to Extreme Observations

Bayesian inference of the 3-component unit Lindley right censored mixture is presented in this paper. The posterior distributions of the parameters are derived assuming informative (gamma) as well as noninformative (uniform and Jeffreys) priors. For the gamma (informative) prior, hyperparameters are elicited using prior predictive distribution. The Bayesian estimation has been carried out considering both symmetric and asymmetric loss functions (squared error, quadratic, weighted, and precautionary loss functions). Simulation studies for various sample sizes and different threshold values (test termination times) are considered to evaluate the performances of the Bayes estimators w.r.t their posterior risks under the said loss functions. Real life flood data from Naser Lake is also analyzed as a 3-component mixture for the sake of illustrative purpose. The simulation study and data analysis reveals that the estimates under informative prior perform better than the noninformative priors. Also, it is observed that the increase in sample size and the threshold value (test termination time) are inversely proportional to the posterior risks. Among the loss functions considered, the loss functions performance from the best to the least, w.r.t the posterior risk, is as follows: precautionary loss function < weighted loss function < quadratic loss function < squared error loss function. Posterior risks are directly proportional to the sample size and threshold value (test termination time).

Harmonic Mixture-G Family of Distributions: Survival Regression, Simulation by Likelihood, Bootstrap and Bayesian Discussion with MCMC Algorithm

To study the heterogeneous nature of lifetimes of certain mechanical or engineering processes, a mixture model of some suitable lifetime distributions may be more appropriate and appealing than simpler models. In this paper, a new mixture family of the lifetime distributions is introduced via harmonic weighted mean of an underlying distribution and the distribution of the proportional hazard model corresponding to the baseline model.The proposed class of distributions includes the general Marshall-Olkin family of distributions as a special case. Some important properties of the proposed model such as survival function, hazard function, order statistics and some results on stochastic ordering are obtained in a general setting. A special case of this new family is considered by employingWeibull distribution as the parent distribution. We derive several properties of the special distribution such as moments,hazard function survival regression and certain characterizations results. Moreover, we estimate the parameters of the model by using frequentist and Bayesian approaches. For Bayesian analysis, five loss functions, namely the squared error loss function (SELF), weighted squared error loss function (WSELF), modified squared error loss function (MSELF), precautionary loss function (PLF), and K-loss function (KLF) are considered. The beta prior as well as the gamma prior are used to obtain the Bayes estimators and posterior risk of the unknown parameters of the model. Furthermore, credible intervals (CIs) and highest posterior density (HPD) intervals are also obtained. A simulation study is presented via Monte Carlo to investigate the bias and mean square error of the maximum likelihood estimators. For illustrative purposes, two real-life applications of the proposed distribution to Kidney and cancer patients are provided.

Bayesian Risk Analysis for Length Biased Log Logistic Distribution Under Different Loss

The aim of this paper is parametric and reliability estimation for the two parameter length biased log-logistic distribution under squared error, generalized exponential, linear exponential and precautionary loss functions. Bayes estimates obtained under non informative priors through Lindleys approximation and through Markov Chain Monte Carlo are then compared with the classical parametric estimates. Bayesian risk analysis based on a simulated and a real data set are used to demonstrate application of the theoretic results.

Bayesian Estimation of Weighted Inverse Maxwell Distribution under Different Loss Functions

In this study, the shape parameter of the weighted Inverse Maxwell distribution is estimated by employing Bayesian techniques. To produce posterior distributions, the extended Jeffery's prior and the Erlang prior are utilised. The estimators are derived from the squared error loss function, the entropy loss function, the precautionary loss function, and the Linex loss function. Furthermore, an actual data set is studied to assess the effectiveness of various estimators under distinct loss functions.

Statistical Inference on 2-Component Mixture of Topp-Leone Distribution, Bayesian and non-Bayesian Estimation

To study the heterogeneous nature of lifetimes of certain mechanical or engineering processes, a mixture model of some suitable lifetime distributions may be more appropriate and appealing as compared to simple models. This paper considers mixture of Topp-Leone distributions under classical and Bayesian perspective based on complete sample. The new distribution which exhibits decreasing and upside down bathtub shaped density while the distribution has the ability to model lifetime data with decreasing, increasing and upside down bathtub shaped failure rates. We derive several properties of the new distribution such as moments, moment generating function, conditional moment, mean deviation, Bonferroni and Lorenz curves and the order statistics of the proposed distribution. Moreover, we estimate the parameters of the model by using frequentist and Bayesian approaches. For Bayesian analysis, five loss functions, namely the squared error loss function (SELF), weighted squared error loss function (WSELF), modified squared error loss function (MSELF), precautionary loss function (PLF), and K-loss function (KLF) and uniform as well as gamma priors are considered to obtain the Bayes estimators and posterior risk of the unknown parameters of the model. Furthermore, credible intervals (CIs) and highest posterior density (HPD) intervals are also obtained. Monte Carlo simulation study is done to access the behavior of these estimators. For the illustrative purposes, a real-life application of the proposed distribution to a tensile strength data set is provided