scholarly journals BAYESIAN ESTIMATION OF PARETO SURVIVAL MODEL WITH INFORMATIVE PRIOR ON CENSORED DATA

2019 ◽  
Vol 3 (2) ◽  
pp. 64
Author(s):  
Setyo Wira Rizki ◽  
Shantika Martha
Keyword(s):  
Bayesian Estimation ◽  
Censored Data ◽  
Loss Function ◽  
Posterior Distribution ◽  
Likelihood Function ◽  
Estimation Method ◽  
Survival Model ◽  
Squared Error Loss Function ◽  
Hazard Functions ◽  
The Best Approximation

This research conducts a case of the cancer patients in censored data using Bayesian methodology. There are three types of loss function in Bayesian estimation method such as squared error loss function (self), linear exponential loss function (lelf) and general entropy loss function (gelf). Pareto survival model is selected as presentation data. To construct a posterior distribution, framing a likelihood function of Pareto and a prior, requires the prior distribution. An exponential distribution is chosen as a prior that describes parameter character of the Pareto. The posterior distribution is used to discover estimators in three loss functions of Bayesian methods. There are estimators held down by Bayesian self , Bayesian lelf  and Bayesian gelf  which substance 3.79, 3.78 and 3.90 correspondingly. After getting those estimators, the hazard functions  ,  and  and survival functions   ,  and  can be determined. The result shows that all of survival values under Bayesian approaches are lower than the real survival value. It means the result is more trusted because as a prior, the parameter is defined more precisely than before. The hazard function confirmations a same shape in all approaches. The rates of hazard are decreasing along with survival values which show the same behavior. The curves are strictly dropping after first data. This occurrence because due to a heavy-tailed character of Pareto.  The result indicates that MSE of parameter estimation under the Bayesian self, lelf and gelf are 1.3x10-2, 1.2x10-2 and 0 respectively. The mse of survival estimation under the Bayesian self, lelf and gelf are 10-4, 1.1x10-4 and 3x10-5 individually. It concludes that the Bayesian gelf  is the best approximation.

scholarly journals Estimating the Parameters of the Two-Parameter Rayleigh Distribution Based on Adaptive Type II Progressive Hybrid Censored Data with Competing Risks

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1783
Author(s):  
Shuhan Liu ◽  
Wenhao Gui
Keyword(s):  
Maximum Likelihood ◽  
Competing Risks ◽  
Censored Data ◽  
Loss Function ◽  
Likelihood Function ◽  
Rayleigh Distribution ◽  
Type Ii ◽  
Squared Error Loss Function ◽  
Two Parameter ◽  
Bayesian Estimators

This paper attempts to estimate the parameters for the two-parameter Rayleigh distribution based on adaptive Type II progressive hybrid censored data with competing risks. Firstly, the maximum likelihood function and the maximum likelihood estimators are derived before the existence and uniqueness of the latter are proven. Further, Bayesian estimators are considered under symmetric and asymmetric loss functions, that is the squared error loss function, the LINEXloss function, and the general entropy loss function. As the Bayesian estimators cannot be obtained explicitly, the Lindley method is applied to compute the approximate Bayesian estimates. Finally, a simulation study is conducted, and a real dataset is analyzed for illustrative purposes.

scholarly journals Bayesian Estimation of the Parameter of the p-Dimensional Size-Biased Rayleigh Distribution

2017 ◽  
Vol 56 (1) ◽  
pp. 88-91
Author(s):  
Arun Kumar Rao ◽  
Himanshu Pandey ◽  
Kusum Lata Singh
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Bayesian Estimation ◽  
Density Function ◽  
Rayleigh Distribution ◽  
Survival Model ◽  
Loss Functions ◽  
Bayes Estimators ◽  
Hazard Functions ◽  
Squared Error

In this paper, we have derived the probability density function of the size-biased p-dimensional Rayleigh distribution and studied its properties. Its suitability as a survival model has been discussed by obtaining its survival and hazard functions. We also discussed Bayesian estimation of the parameter of the size-biased p-dimensional Rayleigh distribution. Bayes estimators have been obtained by taking quasi-prior. The loss functions used are squared error and precautionary.

Bayesian Reliability Estmation with a Truncated Normal Prior

1988 ◽  
Vol 37 (3-4) ◽  
pp. 227-231 ◽  
Author(s):  
Samir K. Bhattacharya ◽  
Ravindar K. Tyagi
Keyword(s):  
Bayesian Estimation ◽  
Parameter Space ◽  
Loss Function ◽  
Exponential Model ◽  
Life Testing ◽  
Squared Error Loss Function ◽  
Squared Error Loss ◽  
Squared Error ◽  
Life Tests ◽  
Truncated Normal

Beyesian reliebility estimation for the exponential model. based on life tests that are terminated after a preassigned number of failures, is carried out under the assumption of the squared error loss function and a truncated normal priod density on the parameter space. The Bayesian estimation of reliability for the case of ‘attribute life testing’ is also discussed.

scholarly journals Bayesian Estimation and Prediction of Discrete Gompertz Distribution

2021 ◽  
pp. 1-21
Author(s):  
M. A. Hegazy ◽  
R. E. Abd El-Kader ◽  
A. A. El-Helbawy ◽  
G. R. Al-Dayian
Keyword(s):  
Bayesian Estimation ◽  
Loss Function ◽  
Hazard Rate ◽  
Squared Error Loss Function ◽  
Squared Error Loss ◽  
Gompertz Distribution ◽  
Estimation And Prediction ◽  
Error Loss ◽  
Squared Error ◽  
Rate Functions

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.

scholarly journals Bayes Estimators of Exponentiated Inverse Rayleigh Distribution using Lindleys Approximation

2021 ◽  
pp. 60-71
Author(s):  
Bashiru Omeiza Sule ◽  
Taiwo Mobolaji Adegoke ◽  
Kafayat Tolani Uthman
Keyword(s):  
Loss Function ◽  
Posterior Distribution ◽  
Real Life ◽  
Rayleigh Distribution ◽  
Loss Functions ◽  
Bayes Estimators ◽  
Shape Parameters ◽  
True Parameter ◽  
Squared Error Loss Function ◽  
Scale Parameters

In this paper, Bayes estimators of the unknown shape and scale parameters of the Exponentiated Inverse Rayleigh Distribution (EIRD) have been derived using both the frequentist and bayesian methods. The Bayes theorem was adopted to obtain the posterior distribution of the shape and scale parameters of an Exponentiated Inverse Rayleigh Distribution (EIRD) using both conjugate and non-conjugate prior distribution under different loss functions (such as Entropy Loss Function, Linex Loss Function and Scale Invariant Squared Error Loss Function). The posterior distribution derived for both shape and scale parameters are intractable and a Lindley approximation was adopted to obtain the parameters of interest. The loss function were employed to obtain the estimates for both scale and shape parameters with an assumption that the both scale and shape parameters are unknown and independent. Also the Bayes estimate for the simulated datasets and real life datasets were obtained. The Bayes estimates obtained under dierent loss functions are close to the true parameter value of the shape and scale parameters. The estimators are then compared in terms of their Mean Square Error (MSE) using R programming language. We deduce that the MSE reduces as the sample size (n) increases.

scholarly journals Evidential Estimation of an Uncertain Mixed Exponential Distribution under Progressive Censoring

Entropy ◽  
2020 ◽  
Vol 22 (10) ◽  
pp. 1106
Author(s):  
Kuang Zhou ◽  
Yimin Shi
Keyword(s):  
Exponential Distribution ◽  
Mixture Models ◽  
Censored Data ◽  
Likelihood Function ◽  
Finite Mixture Models ◽  
Expectation Maximization Algorithm ◽  
Estimation Method ◽  
Optimization Method ◽  
Uncertain Information ◽  
Mixed Exponential Distribution

In this paper, the evidential estimation method for the parameters of the mixed exponential distribution is considered when a sample is obtained from Type-II progressively censored data. Different from the traditional statistical inference methods for censored data from mixture models, here we consider a very general form where there is some uncertain information about the sub-class labels of units. The partially specified label information, as well as the censored data are represented in a united frame by mass functions within the theory of belief functions. Following that, the evidential likelihood function is derived based on the completely observed failures and the uncertain information included in the data. Then, the optimization method using the evidential expectation maximization algorithm (E2M) is introduced. A general form of the maximal likelihood estimates (MLEs) in the sense of the evidential likelihood, named maximal evidential likelihood estimates (MELEs), can be obtained. Finally, some Monte Carlo simulations are conducted. The results show that the proposed estimation method can incorporate more information than traditional EM algorithms, and this confirms the interest in using uncertain labels for the censored data from finite mixture models.

scholarly journals Bayesian estimation of the shape parameter of generalized Rayleigh distribution under non-informative prior

2015 ◽  
Vol 4 (1) ◽  
pp. 1
Author(s):  
Yakubu Aliyu ◽  
Abubakar Yahaya
Keyword(s):  
Bayesian Estimation ◽  
Loss Function ◽  
Shape Parameter ◽  
Rayleigh Distribution ◽  
Skewed Distribution ◽  
Time Data ◽  
Squared Error Loss Function ◽  
Squared Error ◽  
Informative Prior ◽  
Generalized Rayleigh Distribution

<p>A decade ago, two-parameter Burr Type X distribution was introduced by Surles and Padgett [14] which was described as Generalized Rayleigh Distribution (GRD). This skewed distribution can be used quiet effectively in modelling life time data. In this work, Bayesian estimation of the shape parameter of GRD was considered under the assumption of non-informative prior. The estimates were obtained under the squared error, Entropy and Precautionary loss functions. Extensive Monte Carlo simulations were carried out to compare the performances of the Bayes estimates with that of MLEs. It was observed that the estimate under the Entropy loss function is more stable than the estimates under squared error loss function, Precautionary loss function and MLEs.</p>

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

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 636 ◽  
Author(s):  
Ali Algarni ◽  
Abdullah M. Almarashi ◽  
Hassan Okasha ◽  
Hon Keung Tony Ng
Keyword(s):  
Bayesian Estimation ◽  
Loss Function ◽  
Scale Parameter ◽  
Type I ◽  
Squared Error Loss Function ◽  
Squared Error Loss ◽  
Type I Censoring ◽  
Squared Error ◽  
Censoring Scheme ◽  
Bayesian Estimators

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.

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