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Topic: Maximum likelihood estimation

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	8.4.1.2. Maximum likelihood estimation
		Maximum likelihood estimation begins with writing a mathematical expression known as the Likelihood Function of the sample data.
		Loosely speaking, the likelihood of a set of data is the probability of obtaining that particular set of data, given the chosen probability distribution model.
		Maximum likelihood estimation is a totally analytic maximization procedure.
	www.itl.nist.gov /div898/handbook/apr/section4/apr412.htm (577 words)

	Maximum likelihood -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-04)
		Maximum Likelihood Estimation (or MLE) is a popular (additional info and facts about statistical) statistical method used to make inferences about parameters of the underlying (additional info and facts about probability distribution) probability distribution of a given (additional info and facts about data set) data set.
		This is in contrast to seeking other estimators, such as an (additional info and facts about unbiased estimator) unbiased estimator of, which may not necessarily yield the most likely value of but which will yield a value that (on average) will neither tend to over-estimate nor under-estimate the true value of.
		Maximum likelihood estimators have minimum variance (that is (additional info and facts about mean squared error) mean squared error when the MLE is unbiased) for infinite sample size.
	www.absoluteastronomy.com /encyclopedia/m/ma/maximum_likelihood.htm (1138 words)

	Statistics 5102 (Geyer, Spring 2007) Examples: Maximum Likelihood Estimation
		In order to do maximum likelihood estimation (MLE) using the computer we need to write the likelihood function or log likelihood function (usually the latter) as a function in the computer language we are using.
		In our particular problem, maximum likelihood for the shape parameter of the gamma distribution, a good estimate of the shape parameter α is the sample mean, since the theoretical mean of the gamma distribution is α / β where β is the
		Maximum likelihood is the only well-known method that is not computer intensive.
	www.stat.umn.edu /geyer/5102/examp/rlike.html (1972 words)

	OhioLINK ETD: Wang, Qiang
		The maximum likelihood estimator (MLE) of evolutionary parameters is shown to be asymptotically efficient with the number of sites approaching infinity, when the tree is assumed to be known.
		The large-sample property of the joint MLE for tree topology, branch lengths vector and evolutionary parameters is established.
		In order to estimate the variability of the MLE for evolutionary parameters, a standard bootstrap algorithm and a bootstrap method motivated by the conditional variance principle (CVB) are proposed for practical implementation.
	rave.ohiolink.edu /etdc/view?acc_num=osu1083177084 (248 words)

	Stats: Maximum likelihood estimation
		Maximum likelihood is an approach that looks at a large class of distributions and then chooses the "best" distribution.
		The log of the likelihood function often simplifies many of the calculations, and if you find the maximum of the log likelihood that also has to be the maximum of the likelihood itself.
		I won't show all the equations, but the maximum likelihood estimate of mu ends up equalling the sample mean and the maximum likelihood estimate of sigma ends up equalling, not the sample standard deviation exactly, but something very close where you replace n-1 with n in the formula.
	www.cmh.edu /stats/ask/mle.asp (611 words)

	Stats: Maximum likelihood estimation (May 6, 2003)
		Maximum likelihood is an approach that looks at a large class of distributions and then chooses the "best" distribution.
		The log of the likelihood function often simplifies many of the calculations, and if you find the maximum of the log likelihood that also has to be the maximum of the likelihood itself.
		I won't show all the equations, but the maximum likelihood estimate of mu ends up equalling the sample mean and the maximum likelihood estimate of sigma ends up equalling, not the sample standard deviation exactly, but something very close where you replace n-1 with n in the formula.
	www.childrens-mercy.org /stats/ask/mle.asp (601 words)

	MLE (Maximum Likelihood) Parameter Estimation for Complete Data
		From a statistical point of view, the method of maximum likelihood estimation is, with some exceptions, considered to be the most robust of the parameter estimation techniques discussed here.
		It is known, for example, that MLE estimates of the shape parameter for the Weibull distribution are badly biased for small sample sizes, and the effect can be increased depending on the amount of censoring.
		One of these is estimating the location parameter for the three-parameter Weibull distribution when the shape parameter has a value close to 1.
	www.weibull.com /LifeDataWeb/mle_for_complete_data.htm (646 words)

	Maximum Likelihood Estimation
		For the likelihood to be properly thought of as a density, a Bayesian approach is required.
		The MLE rule is an implementation of the likelihood principle.
		The iteration procedes until a local maximum of the likelihood is attained, although in the case of the first two methods, such convergence is not guaranteed.
	cnx.org /content/m11446/latest (1914 words)

	Maximum likelihood
		In statistics, the method of maximum likelihood, pioneered by geneticist/statistician Sir Ronald A. Fisher, is a method of point estimation, that uses as an estimate of an unobservable population parameter the member of the parameter space that maximizes the likelihood function.
		denote the unobservable population parameter to be estimated.
		If n is unknown, then the maximum-likelihood estimator of n is X, even though the expectation of X is only n/2; we can only be certain that n is at least X and is probably more.
	www.brainyencyclopedia.com /encyclopedia/m/ma/maximum_likelihood_1.html (405 words)

	BGIM : Maximum Likelihood Estimation Primer
		The aim of maximum likelihood estimation is to find the parameter value(s) that makes the observed data most likely.
		This is because the likelihood of the parameters given the data is defined to be equal to the probability of the data given the parameters
		The likelihood framework conceptually takes all of this in its stride, however, and this is what makes it the work-horse of many modern statistical methods.
	statgen.iop.kcl.ac.uk /bgim/mle/sslike_3.html (658 words)

	Maximum Likelihood
		Likelihood: The likelihood function is used to obtain maximum likelihood estimates (MLEs) of parameters and is the probabiity of the observed sequences as a function
		The maximum likelihood under the null hypothesis is logL0 = -7628.025.
		estimate of phylogeny is consistent with the monophyly of mammals and amniotes (though the tree is unrooted).
	ucjeps.berkeley.edu /bryolab/ib200/maximumlikelihood.html (2941 words)

	Variance Components and Mixed Model ANOVA/ANCOVA
		The basic goal of variance component estimation is to estimate the population covariation between random factors and the dependent variable.
		Maximum likelihood methods for estimating variance components are based on quadratic forms, and typically, but not always, require iteration to find a solution.
		For ANOVA methods for estimating variance components, a solution is found for the system of equations relating the estimated population variances and covariances among the random factors to the estimated population covariances between the random factors and the dependent variable.
	www.statsoft.com /textbook/stvarcom.html (2537 words)

	Maximum Likelihood
		The method of maximum likelihood was first proposed by the English statistician and population geneticist R. Fisher.
		The maximum likelihood method finds the estimate of a parameter that maximizes the probability of observing the data given a specific model for the data.
		Here, the likelihood calculated under the null hypothesis is in the numerator and the likelihood calculated under the alternative hypothesis is in the denominator.
	www.sciencemag.org /feature/data/phylo/coin/coin.html (1384 words)

	Maximum Likelihood Estimation (MLE)
		Once data have been collected and the likelihood function of a model given the data is determined, one is in a position to make statistical inferences about the population, that is, the probability distribution that underlies the data.
		The principle of maximum likelihood estimation (MLE), originally developed by R. Fisher in the 1920s, states that the desired probability distribution be the one that makes the observed data most likely, which is obtained by seeking the value of the parameter vector that maximizes the likelihood function L(θ).
		When finding a maximum likelihood estimator, it is often easier to find the value of parameter that minimizes the natural logarithm of the likelihood function rather than the value of the parameter that minimizes the likelihood function itself.
	cnx.org /content/m13501/latest (892 words)

	1.3.6.5.2. Maximum Likelihood
		Maximum likelihood estimation begins with the mathematical expression known as a likelihood function of the sample data.
		Loosely speaking, the likelihood of a set of data is the probability of obtaining that particular set of data given the chosen probability model.
		Except for a few cases where the maximum likelihood formulas are in fact simple, it is generally best to rely on high quality statistical software to obtain maximum likelihood estimates.
	www.itl.nist.gov /div898/handbook/eda/section3/eda3652.htm (583 words)

	Some notes on Maximum Likelihood Estimation
		Likelihood function is identical to pdf except viewed as a function of the parameters of the probability distribution, instead of the data.
		MLE is the one in the pool of possible values for the population (or process) parameter; it is the one for which the likelihood function obtains its maximum value.
		If data values are independent from each one, we can multiply likelihood functions of each data value and use product of those functions to find the likelihood for the entire sample of data values.
	www2.tltc.ttu.edu /Westfall/images/5347/f03h6sln.htm (324 words)

	[No title]
		An common alternative to the least squares loss function is to maximize the likelihood or log-likelihood function (or to minimize the negative log-likelihood function; the term maximum likelihood was first used by Fisher, 1922a).
		The method of maximum likelihood (the term first used by Fisher, 1922a) is a general method of estimating parameters of a population by values that maximize the likelihood (L) of a sample.
		The maximum unconfounding criterion specifies that design generators should be chosen such that the maximum number of interactions of less than or equal to the crucial order, given the resolution, are unconfounded with all other interactions of the crucial order.
	www.statsoft.com /textbook/glosm.html (5420 words)

	MLE (Maximum Likelihood) Parameter Estimation
		The idea behind maximum likelihood parameter estimation is to determine the parameters that maximize the probability (likelihood) of the sample data.
		From a statistical point of view, the method of maximum likelihood is considered to be more robust (with some exceptions) and yields estimators with good statistical properties.
		Although the methodology for maximum likelihood estimation is simple, the implementation is mathematically intense.
	www.weibull.com /AccelTestWeb/mle_maximum_likelihood_parameter_estimation.htm (665 words)

	Apophenia: Apophenia: Maximum likelihood estimation
		Most of the action with regards to maximum likelihood estimation is in the function apop_maximum_likelihood and the model objects.
		The likelihood function is taken from the model, the metric is the Manhattan metric, the copy/destroy functions are just the usual vector-handling fns., et cetera.
		An apop_model that is the output from a prior MLE estimation.
	apophenia.sourceforge.net /doc/group__mle.html (1156 words)

	Maximum likelihood estimation
		It has been proved in [14] that the maximum likelihood estimator (MLE) has the asymptotic properties (for large data records) of being unbiased and achieving the Cramér-Rao lower bound .
		The first problem that arises in almost all practical estimation situations is that the measured data are disturbed by random errors which are caused by imperfect measurement equipment and, in the MRI practice, are introduced by the patient.
		The maximum likelihood method is based on the principle that I should be estimated by its most likely value given the observations S, i.e.
	dutnsic.tn.tudelft.nl:8080 /main/node17.html (1093 words)

	Maximum-Likelihood Estimation of Relatedness -- Milligan 163 (3): 1153 -- Genetics
		In contrast, the bias of the likelihood estimator is relatively
		Dependence of the bias of the likelihood estimator on both the number of loci sampled and the number of alleles segregating at each.
		estimates of relatedness to subsequent estimation of quantitative
	www.genetics.org /cgi/content/full/163/3/1153 (6091 words)

	Maximum Likelihood Estimation
		The point where the derivative of the likelihood function is zero and the second derivative is negative is the maximum.
		The results for the log-likelihood function hold for the likelihood function (the maximum is the same), so logging it does no damage.
		The log of the likelihood will always be negative or zero, so multiplying it by a negative number will create a positive number.
	www.columbia.edu /~ag2319/teaching/G4075_Outline/node13.html (811 words)

	MLE++ : Find the Maximum Likelihood Estimator with a C++ Class Library
		MLE++ is a C++ class library for the estimation of econometric models.
		Since full, commented source code is included, the coded models may be used as guides to related models that the user may see in the literature, or which he or she may develop.
		These languages are interpreted, and in maximum likelihood estimation this means placing an interpreter in the inner loop of a maximization routine.
	www.magma.ca /~cahill/Mlepp.htm (1686 words)

	Maximum Likelihood Procedures
		Consistency means that as the sample size increases, the maximum likelihood estimate tends in probability to the true parameter value.
		Similarly, the t-statistics produced by SST maximum likelihood procedures are "asymptotic t-statistics" in the sense that they have an approximate t-distribution for large samples.
		The maximum likelihood estimator is found by setting these partial derivatives equal to zero and finding a solution to the equations.
	emlab.berkeley.edu /sst/max.like.html (5326 words)

	Maximum-likelihood Estimation and Dual Kalman Filtering
		A better motivated approach is to consider the problem of finding the maximum-likelihood estimates of the speech and the model parameters given the noisy data.
		However, to avoid using a single model to describe the entire nonstationary speech signal (or requiring the complexity of model-switching methods), the speech is windowed into approximately stationary segments, with a different model used for each segment.
		With a state-space representation of the speech model, the EKF method discussed in Section 14.4.2 gives the maximum-likelihood estimate of the speech assuming the model is known.
	cslu.cse.ogi.edu /nsel/wan_manuscript/node17.html (815 words)

	Maximum Likelihood Estimation (Site not responding. Last check: 2007-10-04)
		Estimate parameters by the method of maximum likelihood.
		Be careful to note that the argument is -log L (not -2 log L).
		It is for the user to ensure that the likelihood is correct, and that asymptotic likelihood inference is valid.
	www.omegahat.org /RDoc/examples/mle.xml (96 words)

	SSRN-Maximum-Likelihood Estimation of Discretely Sampled Diffusions: A Closed-Form Approach by Yacine Ait-Sahalia
		When a continuous-time diffusion is observed only at discrete dates, not necessarily close together, the likelihood function of the observations is in most cases not explicitly computable.
		Researchers have relied on simulations of sample paths in between the observation points, or numerical solutions of partial differential equations, to obtain estimates of the function to be maximized.
		We then show that maximizing the sequence instead of the true function results in an estimator which converges to the true maximum-likelihood estimator and shares its asymptotic properties of consistency, asymptotic normality and efficiency.
	papers.ssrn.com /sol3/papers.cfm?abstract_id=94135 (321 words)

	Maximum Likelihood Estimation (Site not responding. Last check: 2007-10-04)
		The values of the parameters that maximize L are the maximum likelihood estimators.
		The advantages of MLE are that the framework is completely general hypothesis testing is simplified using likelihood ratio tests and in principle the standard deviations of the parameter estimates can be calculated.
		The likelihood function attempts to accommodate the observational error in the data.
	www.soest.hawaii.edu /PFRP/seminar/tsld013.htm (149 words)

	Maximization and Maximum Likelihood Estimation
		The material in this section is somewhat more advanced as it assumes you are familiar with the basic concepts of maximum likelihood estimation.
		Closed form maximum likelihood estimates are not available for the shape parameter of this distribution, but we can use
		Approximate standard errors of the maximum likelihood estimates are given by the square roots of the diagonal entries of the inverse of the negative Hessian matrix at the maximum:
	www.nku.edu /~longa/public_html/classes/techreport/node57.html (773 words)

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