Maximum Likelihood Method


Also found in: Dictionary, Acronyms.

maximum likelihood method

[′mak·sə·məm ′līk·lē‚hu̇d ‚meth·əd]
(statistics)
A technique in statistics where the likelihood distribution is so maximized as to produce an estimate to the random variables involved.

Maximum Likelihood Method

 

a method of finding statistical estimates of the unknown parameters of a distribution. According to the maximum likelihood method, we select as the estimates of the parameters those values for which the data resulting from observations are “most likely.” It is assumed that the results of observations X1, …, Xn are mutually independent random variables with identical probability distributions, all depending on the same unknown parameter θ ε ө, where ө is the set of admissible values of θ. To assign an exact meaning to the concept of “most likely,” we proceed by introducing a function

L(x1, …, xn; θ) = p(x1; θ) … p(xn; θ)

where p(t;θ) for a continuous distribution is interpreted as the probability density of the random variable X, and in the discrete case as the probability that the random variable X takes the value t. The function L(X1, …, Xn;θ) of the random variables X1,…, Xn is called the likelihood function, and the maximum likelihood estimate of the parameter θ is that value img0194 (X1, …, Xn) (which is itself a random variable) of θ for which the likelihood function attains the largest possible value. Since the maximum point for log L is the same as that for L, it is usually sufficient to solve the so-called likelihood equation

in order to find the maximum likelihood estimates.

The maximum likelihood method does not always lead to acceptable results but in some sense is the best method for a broad set of cases of practical importance. For example, we may assert that if there exists an efficient unbiased estimate θ* for the parameter θ in a sample of size n, then the likelihood equation will have the unique solution θ = θ*. In dealing with the asymptotic behavior of maximum likelihood estimates for large n, it is well known that the maximum likelihood method leads under certain general conditions to a consistent estimate that is asymptotically normal and asymptotically efficient. The definitions given above can be generalized to the case of several unknown parameters and to the case of samples from multivariate distributions.

The maximum likelihood method in its modern form was proposed by the British statistician R. Fisher in 1912, although particular forms of the method were used by K. Gauss; even earlier, in the 18th century, J. Lambert and D. Bernoulli came close to the idea of the method.

REFERENCES

Cramer, H. Matematicheskie metody slalisliki. Moscow, 1948. (Translated from English.)
Rao, C. R. Lineinye statisticheskie metody i ikh primeneniia. Moscow, 1968. (Translated from English.)
Hudson, D. Statistika dlia fizikov. Moscow, 1970. (Translated from English.)

A. V. PROKHOROV

References in periodicals archive ?
Using a semi-parametric estimator, the authors show that the price elasticity of giving to all organizations is substantially lower with this method than with maximum likelihood methods, which are inappropriate in the data.
Since the maximum likelihood method leads to the least dispersion, i.
1-7 suggest that a minimum number of 30 specimens is required to have an acceptable degree of accuracy in obtaining the Weibull modulus by the maximum likelihood method and moment method.
On the basis of the nucleotide sequences, we reconstructed a phylogenetic tree by using maximum likelihood methods with the Tamura 3-parameter model as the evolutionary model; rates among sites were heterogeneous, and gamma distribution was used for the relative rate (7).
Estimates of the baseline hazard function and regression parameters may be obtained by maximum likelihood methods.
Income data are often given in grouped (frequency) format, so that standard maximum likelihood methods of estimation for exponential family distributions, as described in Lye and Martin (1994b), cannot be used.
7) These maximum likelihood methods make the parametric models more efficient than the nonparametric models, but the estimation of the parametric model is also more computationally burdensome.
and Morelli, a research engineer at NASA, describe the elements of system theory, including modeling and parameter estimation, a mathematical model and the ways interconnecting and separate systems are assessed with it, estimation theory, regression methods, maximum likelihood methods, frequency domain methods, experiment design, data compatibility, data analysis, and applications of MATLAB software.
The glmM phylogenetic trees based on the neighbor-joining (data not shown) and maximum likelihood methods (Figure 4) grouped most of the sequences from the Hong Kong strains in the same branch.
Using maximum likelihood methods (7), we estimated the average annual force of infection of ACL to be 0.
When maximum likelihood methods (6) are used, the average annual force of ACL infection ([lambda],) was estimated to be 0.

Full browser ?