Infinite Likelihood

2017 ◽  
Author(s):  
Russell Cheng
Keyword(s):  
Maximum Likelihood ◽  
Asymptotic Normality ◽  
Confidence Intervals ◽  
Parameter Estimates ◽  
Threshold Parameter ◽  
Bootstrap Confidence Intervals ◽  
Stable Law ◽  
Log Likelihood ◽  
Consistency And Asymptotic Normality

This chapter examines methods that overcome a difficulty with infinite likelihoods. In shifted threshold distributions where the PDF has the form f(y) ∼ k(b,c)(y−a)c−1, if y tends to the threshold parameter a, then the log-likelihood tends to infinity if c < 1 and a also tends to y(1) the smallest observation. The maximum likelihood (ML) method fails in this case, yielding parameter estimates that are not consistent. A method is described overcoming this problem, called the maximum product of spacings method. This yields parameter estimates with the same consistency and asymptotic normality properties as ML estimators when these exist, and which yield, when c < 1 where ML fails, consistent estimates with that for a hyper-efficient. Confidence intervals for a are difficult to obtain theoretically when c < 2. A method is given using percentiles of the stable law distribution and this is numerically compared with bootstrap confidence intervals.

Regression for randomly sampled spatial series: the trigonometric case

1986 ◽  
Vol 23 (A) ◽  
pp. 275-289 ◽  
Author(s):  
David R. Brillinger
Keyword(s):  
Maximum Likelihood ◽  
Asymptotic Normality ◽  
Point Process ◽  
Maximum Likelihood Estimates ◽  
Dimensional Parameter ◽  
Finite Dimensional ◽  
Spatial Series ◽  
Consistency And Asymptotic Normality ◽  
Image Position ◽  
Asymptotically Efficient

The model Y(t) = s(t | θ) + ε(t) is studied in the case that observations are made at scattered points τ j in a subset of Rp and θ is a finite-dimensional parameter. The particular cases of 0 = (α, β) and (α, β, ω) are considered in detail. Consistency and asymptotic normality results are developed assuming that the spatial series ε(·) and the point process {τ j} are independent, stationary and mixing. The estimates considered are equivalent to least squares asymptotically and are not generally asymptotically efficient.Contributions of the paper include: study of the Rp case, management of irregularly placed observations, allowance for abnormal domains of observation and the discovery that aliasing complications do not arise when the point process {τ j} is mixing. There is a brief discussion of the construction and properties of maximum likelihood estimates for the spatial-temporal case.

scholarly journals A New Maximum Likelihood Estimator Formulated in Pole-Residue Modal Model

2019 ◽  
Vol 9 (15) ◽  
pp. 3120
Author(s):  
Sandro Amador ◽  
Mahmoud El-Kafafy ◽  
Álvaro Cunha ◽  
Rune Brincker
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Maximum Likelihood Estimator ◽  
Confidence Intervals ◽  
Modal Parameters ◽  
Parameter Estimates ◽  
Modal Parameter ◽  
Likelihood Estimator ◽  
Pole Residue ◽  
Modal Model

Recently, a lot of efforts have been devoted to developing more precise Modal Parameter Estimation (MPE) techniques. This is explained by the necessity in civil, mechanical and aerospace engineering of obtaining accurate estimates for the modal parameters of the tested structures, as well as of determining reliable confidence intervals for these estimates. The Non-linear Least Squares (NLS) identification techniques based on Maximum Likelihood (ML) have been increasingly used in modal analysis to improve precision of estimates provided by the Least Squares (LS) based estimators when they are not accurate enough. Apart from providing more accurate estimates, the main advantage of the ML estimators, with regard to their LS counterparts, is that they allow for taking into account not only the measured Frequency Response Functions (FRFs) but also the noise information during the parametric identification process and, therefore, provide the modal parameters estimates together with their uncertainties bounds. In this paper, a new derivation of a Maximum Likelihood Estimator formulated in Pole-residue Modal Model (MLE-PMM) is presented. The proposed formulation is meant to be used in combination with the Least Squares Frequency Domain (LSCF) to improve the precision of the modal parameter estimates and compute their confidence intervals. Aiming at demonstrating the efficiency of the proposed approach, it is applied to two simulated examples in the final part of the paper.

Regression for randomly sampled spatial series: the trigonometric case

1986 ◽  
Vol 23 (A) ◽  
pp. 275-289
Author(s):  
David R. Brillinger
Keyword(s):  
Maximum Likelihood ◽  
Asymptotic Normality ◽  
Point Process ◽  
Maximum Likelihood Estimates ◽  
Dimensional Parameter ◽  
Finite Dimensional ◽  
Spatial Series ◽  
Consistency And Asymptotic Normality ◽  
Image Position ◽  
Asymptotically Efficient

The model Y(t) = s(t | θ) + ε(t) is studied in the case that observations are made at scattered points τ j in a subset of Rp and θ is a finite-dimensional parameter. The particular cases of 0 = (α, β) and (α, β, ω) are considered in detail. Consistency and asymptotic normality results are developed assuming that the spatial series ε(·) and the point process {τ j } are independent, stationary and mixing. The estimates considered are equivalent to least squares asymptotically and are not generally asymptotically efficient. Contributions of the paper include: study of the Rp case, management of irregularly placed observations, allowance for abnormal domains of observation and the discovery that aliasing complications do not arise when the point process {τ j } is mixing. There is a brief discussion of the construction and properties of maximum likelihood estimates for the spatial-temporal case.

Examples of Embedded Distributions

2017 ◽  
Author(s):  
Russell Cheng
Keyword(s):  
Probability Distributions ◽  
Information Matrix ◽  
Extreme Value ◽  
Inverse Gaussian ◽  
Score Statistic ◽  
Maclaurin Series ◽  
Threshold Parameter ◽  
Stable Law ◽  
Log Likelihood ◽  
Embedded Model

This chapter gives examples of probability distributions that, in their conventional parametrization, contain embedded models. Embeddedness is not intrinsic but depends on the parametrization. The simplest way to reveal and remove embeddedness is to reparametrize and make the log-likelihood, L, expandable as a Maclaurin series of one parameter, α‎: L = L0 + L1α‎ + L2α‎2 + … with L0 the log-likelihood of the embedded model hidden in the original parametrization. The quantity L1, rescaled using the information matrix, is the score statistic which can be used for formally comparing the original and embedded model fits. Embeddedness occurs in many distributions if there is a shifted threshold parameter. Examples given in the chapter are the Burr XII, gamma, generalized extreme value, inverse Gaussian, inverted gamma, logistic, loglogistic, lognormal, loggamma, Pareto, and Weibull distributions. Another interesting example occurs in early parametrizations of the stable law distribution.

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