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.

Research on the Distributed Characteristics of the Delay for Network Coding Based Real-Time Multicast in MANETs

2014 ◽  
Vol 530-531 ◽  
pp. 768-772
Author(s):  
Guo Ping Tan ◽  
Lin Feng Tan ◽  
Lei Cao ◽  
Mei Yan Ju
Keyword(s):  
Maximum Likelihood ◽  
Probability Distribution ◽  
Network Coding ◽  
Real Time ◽  
Lognormal Distribution ◽  
Ad Hoc ◽  
Estimation Method ◽  
Likelihood Estimation ◽  
Actual Distribution ◽  
Delay Distribution

For the study of the applications of partial network coding based real-time multicast protocol (PNCRM) in Mobile Ad hoc networks, the researches should be developed in the probability distribution of delay. In this paper, NS2 is used to obtain the delay of data packets through simulations. Because the delay does not obey the strict normal distribution, the maximum likelihood estimate method based on the lognormal distribution is used to process the data. Using MATLAB to obtain the actual distribution of the natural logarithm of delay, then drawing the delay distribution with the maximum likelihood estimation method based on the lognormal distribution, the conclusion that the distributions obtained by the above mentioned methods are basically consistent can be obtained. So the delay distribution of PNCRM meets the lognormal distribution and the characteristic of delay probability distribution can be estimated.

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.

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.

AN ANALYSIS OF ACCELERATED PERFORMANCE DEGRADATION TESTS ASSUMING THE ARRHENIUS STRESS-RELATIONSHIP

2008 ◽  
Vol 25 (06) ◽  
pp. 847-864 ◽  
Author(s):  
TAE HYOUNG KANG ◽  
SANG WOOK CHUNG ◽  
WON YOUNG YUN
Keyword(s):  
Maximum Likelihood ◽  
Normal Distribution ◽  
Exposure Time ◽  
Failure Time ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Location Parameter ◽  
Performance Degradation ◽  
Maximum Likelihood Estimates ◽  
Degradation Tests

An analytical model is developed for accelerated performance degradation tests. The performance degradations of products at a specified exposure time are assumed to follow a normal distribution. It is assumed that the relationship between the location parameter of normal distribution and the exposure time is a linear function of the exposure time that the slope coefficient of the linear relationship has an Arrhenius dependence on temperature, and that the scale parameter of the normal distribution is constant and independent of temperature or exposure time. The method of maximum likelihood estimation is used to estimate the parameters involved. The likelihood function for the accelerated performance degradation data is derived. The approximated variance-covariance matrix is also derived for calculating approximated confidence intervals of maximum likelihood estimates. Finally we use two real examples for estimating the failure-time distribution, technically defined as the time when performance degrades below a specified level.

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.

scholarly journals ON MOMENT CONDITIONS FOR QUASI-MAXIMUM LIKELIHOOD ESTIMATION OF MULTIVARIATE ARCH MODELS

2012 ◽  
Vol 29 (3) ◽  
pp. 545-566 ◽  
Author(s):  
Marco Avarucci ◽  
Eric Beutner ◽  
Paolo Zaffaroni
Keyword(s):  
Maximum Likelihood ◽  
Asymptotic Normality ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Multivariate Garch ◽  
Moment Condition ◽  
Arch Models ◽  
Moment Conditions ◽  
True Parameter ◽  
Quasi Maximum Likelihood

This paper questions whether it is possible to derive consistency and asymptotic normality of the Gaussian quasi-maximum likelihood estimator (QMLE) for possibly the simplest multivariate GARCH model, namely, the multivariate ARCH(1) model of the Baba, Engle, Kraft, and Kroner form, under weak moment conditions similar to the univariate case. In contrast to the univariate specification, we show that the expectation of the log-likelihood function is unbounded, away from the true parameter value, if (and only if) the observable has unbounded second moment. Despite this nonstandard feature, consistency of the Gaussian QMLE is still warranted. The same moment condition proves to be necessary and sufficient for the stationarity of the score when evaluated at the true parameter value. This explains why high moment conditions, typically bounded sixth moment and above, have been used hitherto in the literature to establish the asymptotic normality of the QMLE in the multivariate framework.

scholarly journals The maximum likelihood degree of Fermat hypersurfaces

2015 ◽  
Vol 6 (2) ◽  
Author(s):  
Daniele Agostini ◽  
Davide Alberelli ◽  
Francesco Grande ◽  
Paolo Lella
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Projective Space ◽  
Critical Points ◽  
Projective Variety ◽  
Likelihood Function ◽  
Likelihood Estimation ◽  
Topological Invariant ◽  
Fermat Curve ◽  
Computational Comparison

We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section. 

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