A System-Independent Numerical Method to Obtain Maximum Likelihood Estimates of the Three Parameter Weibull Distribution

1980 ◽  
Vol 23 (4) ◽  
pp. 0955-0958 ◽  
Author(s):  
John C. Simon ◽  
Frank E. Woeste
Keyword(s):  
Maximum Likelihood ◽  
Weibull Distribution ◽  
Numerical Method ◽  
Maximum Likelihood Estimates

scholarly journals Estimation of the Inverse Weibull Distribution Parameters under Type-I Hybrid Censoring

2021 ◽  
Vol 50 (5) ◽  
pp. 38-51
Author(s):  
Mohammad Kazemi ◽  
Mina Azizpoor
Keyword(s):  
Maximum Likelihood ◽  
Weibull Distribution ◽  
Maximum Likelihood Estimates ◽  
Type I ◽  
Unknown Parameters ◽  
Bayes Estimates ◽  
Hybrid Censoring ◽  
Inverse Weibull Distribution ◽  
Distribution Parameters ◽  
Highest Posterior Density

The hybrid censoring is a mixture of type-I and type-II censoring schemes. This paper presents the statistical inferences of the inverse Weibull distribution parameters when the data are type-I hybrid censored. First, we consider the maximum likelihood estimates of the unknown parameters. It is observed that the maximum likelihood estimates can not be obtained in closed form. We further obtain the Bayes estimates and the corresponding highest posterior density credible intervals of the unknown parameters under the assumption of independent gamma priors using the importance sampling procedure. We also compute the approximate Bayes estimates using Lindley's approximation technique. The performance of the Bayes estimates have been compared with maximum likelihood estimates through the Monte Carlo Markov chain techniques. Finally, a real data set have been analysed for illustration purpose.

scholarly journals Bias-Corrected Maximum Likelihood Estimators of the Parameters of the Unit-Weibull Distribution

2021 ◽  
Vol 50 (3) ◽  
pp. 41-53
Author(s):  
Andre Menezes ◽  
Josmar Mazucheli ◽  
F. Alqallaf ◽  
M. E. Ghitany
Keyword(s):  
Maximum Likelihood ◽  
Weibull Distribution ◽  
Parametric Bootstrap ◽  
Real Data ◽  
Bias Reduction ◽  
Maximum Likelihood Estimates ◽  
Data Sets ◽  
Analytical Methodology ◽  
Finite Samples ◽  
Modeling Data

It is well known that the maximum likelihood estimates (MLEs) have appealing statistical properties. Under fairly mild conditions their asymptotic distribution is normal, and no other estimator has a smaller asymptotic variance.However, in finite samples the maximum likelihood estimates are often biased estimates and the bias disappears as the sample size grows.Mazucheli, Menezes, and Ghitany (2018b) introduced a two-parameter unit-Weibull distribution which is useful for modeling data on the unit interval, however its MLEs are biased in finite samples.In this paper, we adopt three approaches for bias reduction of the MLEs of the parameters of unit-Weibull distribution.The first approach is the analytical methodology suggested by Cox and Snell (1968), the second is based on parametric bootstrap resampling method, and the third is the preventive approach introduced by Firth (1993).The results from Monte Carlo simulations revealed that the biases of the estimates should not be ignored and the bias reduction approaches are equally efficient. However, the first approach is easier to implement.Finally, applications to two real data sets are presented for illustrative purposes.

scholarly journals AN APPLICATION OF THE POISSON-WEIBULL MODEL FOR CENSURED DATA

2021 ◽  
Vol 39 (4) ◽  
pp. 505-521
Author(s):  
Valdemiro Piedade VIGAS ◽  
Fábio PRATAVIERA ◽  
Giovana Oliveira SILVA
Keyword(s):  
Maximum Likelihood ◽  
Weibull Distribution ◽  
Survival Data ◽  
Real Data ◽  
Weibull Model ◽  
Maximum Likelihood Estimates ◽  
Data Set ◽  
Kaplan Meier ◽  
Proposed Model ◽  
The Mean

In this paper, we proposed the Poisson-Weibull distribution for the modeling of survival data. The motivation to study this model since, in addition to generalizing the Weibull distribution, which is widely used in several areas of knowledge among them the Survival and Reliability analysis, it presents great exibility in the forms of the hazard function. The Poisson-Weibull distribution was created in a composition of discrete and continuous distributions where there is no information about which factor was responsible for the component failure, only the minimum lifetime value among all risks is observed. The maximum likelihood approach was used to estimate the parameters of the model. Also was conducted a simulation study to examine the mean, the bias, and the root of the mean square error of the maximum likelihood estimates of the proposed model according to the censoring percentages and sample sizes. The model selection criteria were also applied, in addition to graphic techniques such as TTT-Plot and Kaplan-Meier. Application to the real data set was used to illustrate the usefulnessof the distribution.

scholarly journals Directional Selection and the Site-Frequency Spectrum

Genetics ◽  
2001 ◽  
Vol 159 (4) ◽  
pp. 1779-1788 ◽  
Author(s):  
Carlos D Bustamante ◽  
John Wakeley ◽  
Stanley Sawyer ◽  
Daniel L Hartl
Keyword(s):  
Maximum Likelihood ◽  
High Power ◽  
Negative Selection ◽  
Likelihood Estimation ◽  
Directional Selection ◽  
Likelihood Ratio Tests ◽  
Maximum Likelihood Estimates ◽  
Likelihood Methods ◽  
Ancestral States ◽  
Asymptotic Variances

Abstract In this article we explore statistical properties of the maximum-likelihood estimates (MLEs) of the selection and mutation parameters in a Poisson random field population genetics model of directional selection at DNA sites. We derive the asymptotic variances and covariance of the MLEs and explore the power of the likelihood ratio tests (LRT) of neutrality for varying levels of mutation and selection as well as the robustness of the LRT to deviations from the assumption of free recombination among sites. We also discuss the coverage of confidence intervals on the basis of two standard-likelihood methods. We find that the LRT has high power to detect deviations from neutrality and that the maximum-likelihood estimation performs very well when the ancestral states of all mutations in the sample are known. When the ancestral states are not known, the test has high power to detect deviations from neutrality for negative selection but not for positive selection. We also find that the LRT is not robust to deviations from the assumption of independence among sites.

scholarly journals An Integrated Framework for the Inference of Viral Population History From Reconstructed Genealogies

Genetics ◽  
2000 ◽  
Vol 155 (3) ◽  
pp. 1429-1437
Author(s):  
Oliver G Pybus ◽  
Andrew Rambaut ◽  
Paul H Harvey
Keyword(s):  
Maximum Likelihood ◽  
Sequence Data ◽  
Demographic History ◽  
Population History ◽  
Maximum Likelihood Estimates ◽  
Viral Population ◽  
True Parameter ◽  
Subtype B ◽  
Exponential Growth Model ◽  
Parameter Values

Abstract We describe a unified set of methods for the inference of demographic history using genealogies reconstructed from gene sequence data. We introduce the skyline plot, a graphical, nonparametric estimate of demographic history. We discuss both maximum-likelihood parameter estimation and demographic hypothesis testing. Simulations are carried out to investigate the statistical properties of maximum-likelihood estimates of demographic parameters. The simulations reveal that (i) the performance of exponential growth model estimates is determined by a simple function of the true parameter values and (ii) under some conditions, estimates from reconstructed trees perform as well as estimates from perfect trees. We apply our methods to HIV-1 sequence data and find strong evidence that subtypes A and B have different demographic histories. We also provide the first (albeit tentative) genetic evidence for a recent decrease in the growth rate of subtype B.

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