scholarly journals Non-linear failure rate: A comparison of the Bayesian and frequentist approaches to estimation

2018 ◽  
Vol 20 ◽  
pp. 03001 ◽  
Author(s):  
Tien Thanh Thach ◽  
Radim Briš
Keyword(s):  
Maximum Likelihood ◽  
Failure Rate ◽  
Likelihood Function ◽  
Maximum Likelihood Estimators ◽  
Entropy Method ◽  
Bayes Estimators ◽  
Estimation Of Parameters ◽  
Reliability Characteristics ◽  
Cross Entropy Method ◽  
Non Linear

In this article, a new generalization of linear failure rate called nonlinear failure rate is developed, analyzed, and applied to a real dataset. A comparison of Bayesian and frequentist approaches to the estimation of parameters and reliability characteristics of non-linear failure rate is investigated. The maximum likelihood estimators are obtained using the cross-entropy method to optimize the log-likelihood function. The Bayes estimators of parameters and reliability characteristics are obtained via Markov chain Monte Carlo method. A simulation study is performed in order to compare the proposed Bayes estimators with maximum likelihood estimators on the basis of their biases and mean squared errors. We demonstrate that the proposed model fits a well-known dataset better than other mixture models.

Inference on P(Y

2020 ◽  
Vol 72 (2) ◽  
pp. 89-110
Author(s):  
Manoj Chacko ◽  
Shiny Mathew
Keyword(s):  
Maximum Likelihood ◽  
Pareto Distribution ◽  
Generalized Pareto Distribution ◽  
Maximum Likelihood Estimators ◽  
Record Values ◽  
Asymptotic Distributions ◽  
Bayes Estimators ◽  
Generalized Pareto ◽  
Pareto Distributions ◽  
Generalized Pareto Distributions

In this article, the estimation of [Formula: see text] is considered when [Formula: see text] and [Formula: see text] are two independent generalized Pareto distributions. The maximum likelihood estimators and Bayes estimators of [Formula: see text] are obtained based on record values. The Asymptotic distributions are also obtained together with the corresponding confidence interval of [Formula: see text]. AMS 2000 subject classification: 90B25

Noninvertibility and Pseudo-Maximum Likelihood Estimation of Misspecified ARMA Models

1991 ◽  
Vol 7 (4) ◽  
pp. 435-449 ◽  
Author(s):  
B.M. Pötscher
Keyword(s):  
Maximum Likelihood ◽  
Likelihood Function ◽  
Moving Average ◽  
Maximum Likelihood Estimators ◽  
Likelihood Estimation ◽  
Arma Models ◽  
Limiting Behavior ◽  
Pseudo Likelihood ◽  
Moving Average Model ◽  
Pseudo Maximum Likelihood

Recently Tanaka and Satchell [11] investigated the limiting properties of local maximizers of the Gaussian pseudo-likelihood function of a misspecified moving average model of order one in case the spectral density of the data process has a zero at frequency zero. We show that pseudo-maximum likelihood estimators in the narrower sense, that is, global maximizers of the Gaussian pseudo-likelihood function, may exhibit behavior drastically different from that of the local maximizers. Some general results on the limiting behavior of pseudo-maximum likelihood estimators in potentially misspecified ARMA models are also presented.

Parametric estimation and inference

2019 ◽  
pp. 165-185
Author(s):  
M. D. Edge
Keyword(s):  
Maximum Likelihood ◽  
Likelihood Function ◽  
Maximum Likelihood Estimators ◽  
Wald Test ◽  
Parametric Model ◽  
Parametric Estimation ◽  
Ratio Test ◽  
Sampling Variance ◽  
Squared Error Loss Function ◽  
Squared Error

If it is reasonable to assume that the data are generated by a fully parametric model, then maximum-likelihood approaches to estimation and inference have many appealing properties. Maximum-likelihood estimators are obtained by identifying parameters that maximize the likelihood function, which can be done using calculus or using numerical approaches. Such estimators are consistent, and if the costs of errors in estimation are described by a squared-error loss function, then they are also efficient compared with their consistent competitors. The sampling variance of a maximum-likelihood estimate can be estimated in various ways. As always, one possibility is the bootstrap. In many models, the variance of the maximum-likelihood estimator can be derived directly once its form is known. A third approach is to rely on general properties of maximum-likelihood estimators and use the Fisher information. Similarly, there are many ways to test hypotheses about parameters estimated by maximum likelihood. This chapter discusses the Wald test, which relies on the fact that the sampling distribution of maximum-likelihood estimators is normal in large samples, and the likelihood-ratio test, which is a general approach for testing hypotheses relating nested pairs of models.

scholarly journals Bias in Zipf’s law estimators

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Charlie Pilgrim ◽  
Thomas T Hills
Keyword(s):  
Maximum Likelihood ◽  
Probability Distribution ◽  
Approximate Bayesian Computation ◽  
Likelihood Function ◽  
Maximum Likelihood Estimators ◽  
Power Laws ◽  
Frequency Data ◽  
Computationally Efficient ◽  
Complex Process ◽  
Approximate Bayesian

AbstractThe prevailing maximum likelihood estimators for inferring power law models from rank-frequency data are biased. The source of this bias is an inappropriate likelihood function. The correct likelihood function is derived and shown to be computationally intractable. A more computationally efficient method of approximate Bayesian computation (ABC) is explored. This method is shown to have less bias for data generated from idealised rank-frequency Zipfian distributions. However, the existing estimators and the ABC estimator described here assume that words are drawn from a simple probability distribution, while language is a much more complex process. We show that this false assumption leads to continued biases when applying any of these methods to natural language to estimate Zipf exponents. We recommend that researchers be aware of the bias when investigating power laws in rank-frequency data.

scholarly journals Estimation of Parameters of PTRC SRGM using Non-informative Priors

2022 ◽  
pp. 172-178
Author(s):  
Rajesh Singh ◽  
Pritee Singh ◽  
Kailash Kale
Keyword(s):  
Maximum Likelihood ◽  
Prior Knowledge ◽  
Fixed Number ◽  
Likelihood Method ◽  
Bayes Estimators ◽  
Estimation Of Parameters ◽  
Informative Priors ◽  
Monte Carlo Simulation Technique ◽  
Failure Intensity ◽  
Poisson Type

Reliability is an essentially important characteristic of software. The reliability of software has been assessed by considering Poisson Type occurrence of software failures and the failure intensity of one parameter say (η_1 ) Rayleigh class. Here, it is assumed that the software contains fixed number of inherent faults say (η_0 ). The scale parameter of Rayleigh density (η_1 ) and fixed number of inherent faults contained in software are the parameters of interest. The failure intensity and mean failure function of this Poisson Type Rayleigh Class (PTRC) Software Reliability Growth Model (SRGM) have been studied. The estimates of above parameters can be obtained by using maximum likelihood method. Bayesian technique has been used to about estimates of η_0 and η_1 if prior knowledge about these parameters is available. The prior knowledge about these parameters is considered in the form of non- informative priors for both the parameters. The proposed Bayes estimators are compared with their corresponding maximum likelihood estimators on the basis of risk efficiencies under squared error loss. The Monte Carlo simulation technique is used for calculating risk efficiencies. It is seen that both the proposed Bayes estimators can be preferred over corresponding MLEs for the proper choice of the values of execution time.

scholarly journals Bayesian estimation for Dagum distribution based on progressive type I interval censoring

PLoS ONE ◽  
2021 ◽  
Vol 16 (6) ◽  
pp. e0252556
Author(s):  
Refah Alotaibi ◽  
Hoda Rezk ◽  
Sanku Dey ◽  
Hassan Okasha
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimators ◽  
Interval Censoring ◽  
Type I ◽  
Bayes Estimators ◽  
Squared Error Loss ◽  
Squared Error ◽  
Dagum Distribution ◽  
Credible Intervals ◽  
Better Than

In this paper, we consider Dagum distribution which is capable of modeling various shapes of failure rates and aging criteria. Based on progressively type-I interval censoring data, we first obtain the maximum likelihood estimators and the approximate confidence intervals of the unknown parameters of the Dagum distribution. Next, we obtain the Bayes estimators of the parameters of Dagum distribution under the squared error loss (SEL) and balanced squared error loss (BSEL) functions using independent informative gamma and non informative uniform priors for both scale and two shape parameters. A Monte Carlo simulation study is performed to assess the performance of the proposed Bayes estimators with the maximum likelihood estimators. We also compute credible intervals and symmetric 100(1 − τ)% two-sided Bayes probability intervals under the respective approaches. Besides, based on observed samples, Bayes predictive estimates and intervals are obtained using one-and two-sample schemes. Simulation results reveal that the Bayes estimates based on SEL and BSEL performs better than maximum likelihood estimates in terms of bias and MSEs. Besides, credible intervals have smaller interval lengths than confidence interval. Further, predictive estimates based on SEL with informative prior performs better than non-informative prior for both one and two sample schemes. Further, the optimal censoring scheme has been suggested using a optimality criteria. Finally, we analyze a data set to illustrate the results derived.

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