scholarly journals An Iterative Procedure for Obtaining Maximum-Likelihood Estimates of the Parameters for a Mixture of Normal Distributions

1978 ◽  
Vol 35 (2) ◽  
pp. 362-378 ◽  
Author(s):  
B. Charles Peters, Jr. ◽  
Homer F. Walker
Keyword(s):  
Maximum Likelihood ◽  
Iterative Procedure ◽  
Maximum Likelihood Estimates ◽  
Normal Distributions ◽  
Mixture Of Normal Distributions

scholarly journals Maximum Likelihood Estimation of the Parameters of a System of Simultaneous Regression Equations

1988 ◽  
Vol 4 (1) ◽  
pp. 159-170 ◽  
Author(s):  
James Durbin
Keyword(s):  
Maximum Likelihood ◽  
Iterative Procedure ◽  
Least Squares Method ◽  
Likelihood Estimation ◽  
Maximum Likelihood Estimates ◽  
Regression Equations ◽  
Full Information ◽  
Full Information Maximum Likelihood ◽  
Three Stage Least Squares ◽  
Newton Raphson

Procedures for computing the full information maximum likelihood (FIML) estimates of the parameters of a system of simultaneous regression equations have been described by Koopmans, Rubin, and Leipnik, Chernoff and Divinsky, Brown, and Eisenpress. However, all of these methods are rather complicated since they are based on estimating equations that are expressed in an inconvenient form. In this paper, a transformation of the maximum likelihood (ML) equations is developed which not only leads to simpler computations but which also simplifies the study of the properties of the estimates. The equations are obtained in a form which is capable of solution by a modified Newton-Raphson iterative procedure. The form obtained also shows up very clearly the relation between the maximum likelihood estimates and those obtained by the three-stage least squares method of Zellner and Theil.

Evaluating hypotheses of instar-grouping in arthropods: a maximum likelihood approach

Paleobiology ◽  
2001 ◽  
Vol 27 (3) ◽  
pp. 466-484 ◽  
Author(s):  
Gene Hunt ◽  
Ralph E. Chapman
Keyword(s):  
Maximum Likelihood ◽  
Data Sets ◽  
Size Distributions ◽  
Normal Distributions ◽  
Maximum Likelihood Approach ◽  
Statistical Framework ◽  
Growth Increments ◽  
Likelihood Analysis ◽  
Mixture Of Normal Distributions ◽  
Likelihood Approach

The ontogeny of arthropod exoskeletons is punctuated by short periods of growth following each molt, separated by longer stages of unchanging morphology called instars. The recognition of instar clusters in size distributions has been important in understanding the growth and evolution of fossil arthropods. Generally, these clusters have been identified by inspection, but this approach has been criticized for its subjectivity. In this paper, we describe a statistical framework for evaluating hypotheses of clustering based on maximum likelihood analysis of mixture models. The approach assumes that individuals are normally distributed within instars; thus an arthropod size distribution can be considered a mixture of normal distributions. This methodology provides an objective framework to compare various plausible hypotheses of grouping, including the possibility that there is no significant grouping at all.We apply this method to evaluate clustering in two trilobite species, Ampyxina bellatula and Piochaspis sellata. Both of these data sets show statistically significant evidence of clustering, a phenomenon rarely documented for holaspid-stage trilobites. After consideration of alternative causes of clustering, we argue that the observed groupings are best explained as instar groups. In these two species, growth increments between molts were similar throughout the observed portion of ontogeny, although subtle yet significant variation can be seen within the ontogeny of Ampyxina bellatula.

scholarly journals Normal approximation for mixtures of normal distributions and the evolution of phenotypic traits

2021 ◽  
Vol 53 (1) ◽  
pp. 162-188
Author(s):  
Krzysztof Bartoszek ◽  
Torkel Erhardsson
Keyword(s):  
Normal Distribution ◽  
Phenotypic Trait ◽  
Phenotypic Traits ◽  
Normal Distributions ◽  
Average Value ◽  
Conditional Moments ◽  
Mixture Of Normal Distributions ◽  
Explicit Bounds ◽  
Mixtures Of Normal Distributions ◽  
Parameter Values

AbstractExplicit bounds are given for the Kolmogorov and Wasserstein distances between a mixture of normal distributions, by which we mean that the conditional distribution given some $\sigma$ -algebra is normal, and a normal distribution with properly chosen parameter values. The bounds depend only on the first two moments of the first two conditional moments given the $\sigma$ -algebra. The proof is based on Stein’s method. As an application, we consider the Yule–Ornstein–Uhlenbeck model, used in the field of phylogenetic comparative methods. We obtain bounds for both distances between the distribution of the average value of a phenotypic trait over n related species, and a normal distribution. The bounds imply and extend earlier limit theorems by Bartoszek and Sagitov.

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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