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

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 898 ◽  
Author(s):  
Hongyi Liao ◽  
Wenhao Gui
Keyword(s):  
Competing Risks ◽  
Loss Function ◽  
Information Matrix ◽  
Real Life ◽  
Rayleigh Distribution ◽  
Bayes Estimation ◽  
Type Ii ◽  
Data Set ◽  
Squared Error Loss Function ◽  
Highest Posterior Density

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.

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 Bayesian Methods and Maximum Likelihood Estimations of Exponential Censored Time Distribution with Cure Fraction

2021 ◽  
pp. 106-112
Author(s):  
Dr. Al Omari Mohammed Ahmed
Keyword(s):  
Maximum Likelihood ◽  
Exponential Distribution ◽  
Censored Data ◽  
Loss Function ◽  
Bayesian Approach ◽  
Likelihood Estimation ◽  
Right Censored Data ◽  
Squared Error Loss Function ◽  
Linex Loss ◽  
Cure Fraction

This paper is focused on estimating the parameter of Exponential distribution under right-censored data with cure fraction. The maximum likelihood estimation and Bayesian approach were used. The Bayesian method is implemented using gamma, Jeffreys, and extension of Jeffreys priors with two loss functions, which are; squared error loss function and Linear Exponential Loss Function (LINEX). The methods of the Bayesian approach are compared to maximum likelihood counterparts and the comparisons are made with respect to the Mean Square Error (MSE) to determine the best for estimating the parameter of Exponential distribution under right-censored data with cure fraction. The results show that the Bayesian with gamma prior under LINEX loss function is a better estimation of the parameter of Exponential distribution with cure fraction based on right-censored data.

scholarly journals Bayesian Estimation of Gumbel Type-II Distribution under Type-II Censoring with Medical Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Kamran Abbas ◽  
Zamir Hussain ◽  
Noreen Rashid ◽  
Amjad Ali ◽  
Muhammad Taj ◽  
...  
Keyword(s):  
Clinical Trials ◽  
Loss Function ◽  
Prior Information ◽  
Randomized Clinical Trials ◽  
Bayesian Framework ◽  
Recurrence Time ◽  
Type Ii ◽  
Squared Error Loss Function ◽  
Squared Error ◽  
Bayesian Estimators

The time to event or survival time usually follows certain skewed probability distributions. These distributions encounter vital role using the Bayesian framework to analyze and project the maximum life expectancy in order to inform decision-making. The Bayesian method provides a flexible framework for monitoring the randomized clinical trials to update what is already known using prior information about specific phenomena under uncertainty. Additionally, medical practitioners can use the Bayesian estimators to measure the probability of time until tumor recurrence, time until cardiovascular death, and time until AIDS for HIV patients by considering the prior information. However, in clinical trials and medical studies, censoring is present when an exact event occurrence time is not known. The present study aims to estimate the parameters of Gumbel type-II distribution based on the type-II censored data using the Bayesian framework. The Bayesian estimators cannot be obtained in explicit forms, and therefore we use Lindley’s approximation based on noninformative prior and various loss functions such as squared error loss function, general entropy loss function, and LINEX (linear exponential) loss function. The maximum likelihood and Bayesian estimators are compared in terms of mean squared error by using the simulation study. Furthermore, two data sets about remission times (in months) of bladder cancer patients and survival times in weeks of 61 patients with inoperable adenocarcinoma of the lung are analyzed for illustration purposes.

Estimation of P(X

2017 ◽  
Vol 34 (7) ◽  
pp. 1111-1122 ◽  
Author(s):  
Soumya Roy ◽  
Biswabrata Pradhan ◽  
E.V. Gijo
Keyword(s):  
Maximum Likelihood ◽  
Censored Data ◽  
Quality Characteristic ◽  
Logistic Distribution ◽  
Type Ii ◽  
Finite Sample ◽  
Data Set ◽  
Content Type ◽  
Type Ii Censored Data ◽  
Interval Estimators

Purpose The purpose of this paper is to compare various methods of estimation of P(X<Y) based on Type-II censored data, where X and Y represent a quality characteristic of interest for two groups. Design/methodology/approach This paper assumes that both X and Y are independently distributed generalized half logistic random variables. The maximum likelihood estimator and the uniformly minimum variance unbiased estimator of R are obtained based on Type-II censored data. An exact 95 percent maximum likelihood estimate-based confidence interval for R is also provided. Next, various Bayesian point and interval estimators are obtained using both the subjective and non-informative priors. A real life data set is analyzed for illustration. Findings The performance of various point and interval estimators is judged through a detailed simulation study. The finite sample properties of the estimators are found to be satisfactory. It is observed that the posterior mean marginally outperform other estimators with respect to the mean squared error even under the non-informative prior. Originality/value The proposed methodology can be used for comparing two groups with respect to a suitable quality characteristic of interest. It can also be applied for estimation of the stress-strength reliability, which is of particular interest to the reliability engineers.

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