scholarly journals The Approximate Solution of Maximum Likelihood Function for Exponential Family of Distribution with Zero Class Missing

1993 ◽  
Vol 22 (1) ◽  
pp. 1-11
Author(s):  
Hideo TODA ◽  
Harumi ONO
Keyword(s):  
Approximate Solution ◽  
Maximum Likelihood ◽  
Exponential Family ◽  
Likelihood Function ◽  
Zero Class

scholarly journals On Bayesian procedure for maximum earthquake magnitude estimation

2012 ◽  
Vol 2 (1) ◽  
pp. 7 ◽  
Author(s):  
Andrzej Kijko
Keyword(s):  
Maximum Likelihood ◽  
Posterior Distribution ◽  
Ad Hoc ◽  
Likelihood Function ◽  
Fundamental Problem ◽  
Likelihood Estimation ◽  
Earthquake Magnitude ◽  
Bayesian Estimator ◽  
Bayesian Procedure ◽  
Posterior Estimate

This work is focused on the Bayesian procedure for the estimation of the regional maximum possible earthquake magnitude <em>m</em><sub>max</sub>. The paper briefly discusses the currently used Bayesian procedure for m<sub>max</sub>, as developed by Cornell, and a statistically justifiable alternative approach is suggested. The fundamental problem in the application of the current Bayesian formalism for <em>m</em><sub>max</sub> estimation is that one of the components of the posterior distribution is the sample likelihood function, for which the range of observations (earthquake magnitudes) depends on the unknown parameter <em>m</em><sub>max</sub>. This dependence violates the property of regularity of the maximum likelihood function. The resulting likelihood function, therefore, reaches its maximum at the maximum observed earthquake magnitude <em>m</em><sup>obs</sup><sub>max</sub> and not at the required maximum <em>possible</em> magnitude <em>m</em><sub>max</sub>. Since the sample likelihood function is a key component of the posterior distribution, the posterior estimate of <em>m^</em><sub>max</sub> is biased. The degree of the bias and its sign depend on the applied Bayesian estimator, the quantity of information provided by the prior distribution, and the sample likelihood function. It has been shown that if the maximum posterior estimate is used, the bias is negative and the resulting underestimation of <em>m</em><sub>max</sub> can be as big as 0.5 units of magnitude. This study explores only the maximum posterior estimate of <em>m</em><sub>max</sub>, which is conceptionally close to the classic maximum likelihood estimation. However, conclusions regarding the shortfall of the current Bayesian procedure are applicable to all Bayesian estimators, <em>e.g.</em> posterior mean and posterior median. A simple, <em>ad hoc</em> solution of this non-regular maximum likelihood problem is also presented.

Maximum likelihood estimation for symmetric α-stable Ornstein–Uhlenbeck processes

2020 ◽  
pp. 2150018
Author(s):  
Zhifen Chen ◽  
Xiaopeng Chen
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Transition Function ◽  
Closed Form Expression ◽  
Discrete Observations ◽  
Form Expression ◽  
Probability Density Function Pdf ◽  
Accuracy And Stability

In this paper, we consider the maximum likelihood estimation for the symmetric [Formula: see text]-stable Ornstein–Uhlenbeck (S[Formula: see text]S-OU) processes based on discrete observations. Since the closed-form expression of maximum likelihood function is hard to obtain in the Lévy case, we choose a mixture of Cauchy and Gaussian distribution to approximate the probability density function (PDF) of the S[Formula: see text]S distribution. By means of transition function and Laplace transform, we construct an explicit approximate sequence of likelihood function, which converges to the likelihood function of S[Formula: see text]S distribution. Based on the approximation of likelihood function we give an algorithm for computing maximum likelihood estimation. We also numerically simulate some experiments which demonstrate the accuracy and stability of the proposed estimator.

scholarly journals Maximum Likelihood Estimation for Three-Parameter Weibull Distribution Using Evolutionary Strategy

2019 ◽  
Vol 2019 ◽  
pp. 1-8 ◽  
Author(s):  
Fan Yang ◽  
Hu Ren ◽  
Zhili Hu
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Weibull Distribution ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Estimation Procedure ◽  
Evolutionary Strategy ◽  
Likelihood Method ◽  
Conventional Algorithm

The maximum likelihood estimation is a widely used approach to the parameter estimation. However, the conventional algorithm makes the estimation procedure of three-parameter Weibull distribution difficult. Therefore, this paper proposes an evolutionary strategy to explore the good solutions based on the maximum likelihood method. The maximizing process of likelihood function is converted to an optimization problem. The evolutionary algorithm is employed to obtain the optimal parameters for the likelihood function. Examples are presented to demonstrate the proposed method. The results show that the proposed method is suitable for the parameter estimation of the three-parameter Weibull distribution.

scholarly journals Exact sampling from conditional Boolean models with applications to maximum likelihood inference

2001 ◽  
Vol 33 (2) ◽  
pp. 339-353 ◽  
Author(s):  
M. N. M. van Lieshout ◽  
E. W. van Zwet
Keyword(s):  
Maximum Likelihood ◽  
Likelihood Function ◽  
Boolean Model ◽  
Boolean Models ◽  
Stochastic Version ◽  
The Past ◽  
Single Realization ◽  
The Em Algorithm ◽  
Monte Carlo Approximation ◽  
Do So

We are interested in estimating the intensity parameter of a Boolean model of discs (the bombing model) from a single realization. To do so, we derive the conditional distribution of the points (germs) of the underlying Poisson process. We demonstrate how to apply coupling from the past to generate samples from this distribution, and use the samples thus obtained to approximate the maximum likelihood estimator of the intensity. We discuss and compare two methods: one based on a Monte Carlo approximation of the likelihood function, the other a stochastic version of the EM algorithm.

Maximum likelihood estimation in nonlinear structured fisheries models using survey and catch-at-age data

2011 ◽  
Vol 68 (10) ◽  
pp. 1717-1731 ◽  
Author(s):  
Christian N. Brinch ◽  
Anne Maria Eikeset ◽  
Nils Chr. Stenseth
Keyword(s):  
Maximum Likelihood ◽  
Gadus Morhua ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Maximum Likelihood Estimates ◽  
Simulated Maximum Likelihood ◽  
True Parameter ◽  
Bayesian Techniques ◽  
Age Structured ◽  
Parameter Values

Age-structured population dynamics models play an important role in fisheries assessments. Such models have traditionally been estimated using crude likelihood approximations or more recently using Bayesian techniques. We contribute to this literature with three main messages. Firstly, we demonstrate how to estimate such models efficiently by simulated maximum likelihood using Laplace importance samplers for the likelihood function. Secondly, we demonstrate how simulated maximum likelihood estimates may be validated using different importance samplers known to approach the exact likelihood function in different regions of the parameter space. Thirdly, we show that our method works in practice by Monte Carlo simulations using parameter values as estimated from data on the Northeast Arctic cod ( Gadus morhua ) stock. The simulations suggest that we are able to recover the unknown true maximum likelihood estimates using moderate importance sample sizes and show that we are able to adequately recover the true parameter values.

Fatigue Limit Measurement of the Cast Steel for Chinese Railway Rolling Wagon Bogie Frames

2010 ◽  
Vol 118-120 ◽  
pp. 121-125 ◽  
Author(s):  
Lian You Yu ◽  
Yong Xiang Zhao
Keyword(s):  
Maximum Likelihood ◽  
Fatigue Limit ◽  
Likelihood Function ◽  
Fatigue Cracks ◽  
Cast Steel ◽  
Small Sample ◽  
Test Method ◽  
Maximum Likelihood Approach ◽  
Study Results ◽  
Likelihood Approach

Fatigue limit measurement is investigated experimentally on the grade B cast steel for Chinese railway freight car bogie frames. Small sample up-and-down test method was employed for the present study. Results reveal that fatigue cracks initiated mostly from the material cast shrinking cavities. Distinct river-like flowers and second cracks appeared on fracture surface in perpendicular to fatigue crack path. Lots of dimples are distributed in transient fracture district to indicate that present material is ductile. Maximum likelihood approach is applied for measuring the probabilistic fatigue limits, in which the limits are defined as the fatigue strengths following normal distribution at an expected fatigue life. Statistical parameters are then estimated by a likelihood function method. A comparison analysis to the existent conventional, Dixon-Mood and Zhang-Kececioglu approaches indicates that the maximum likelihood approach is the approach meeting the definition.

Asymptotic inference for maximum likelihood estimators under the special exponential family with double-truncation

2015 ◽  
Vol 58 (3) ◽  
pp. 877-909 ◽  
Author(s):  
Takeshi Emura ◽  
Ya-Hsuan Hu ◽  
Yoshihiko Konno
Keyword(s):  
Maximum Likelihood ◽  
Exponential Family ◽  
Maximum Likelihood Estimators ◽  
Asymptotic Inference ◽  
Double Truncation
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