Maximum Likelihood Method

(redirected from Maximum likelihood)
Also found in: Dictionary, Medical, Acronyms, Wikipedia.

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 ?
Maximum likelihood (ML) tree constructed from of the mtCOI-II sequences of 10 Bemisia tabaci populations.
Maximum likelihood estimation and model selection in contingency tables with missing data, J.
Maximum likelihood estimation of familial relationship indicated these two individuals have a probability 11.
When measured by TMSE, the EB estimator was clearly superior to both the maximum likelihood (ML) and global unstratified (GU) estimators (Table 1).
Maximum likelihood estimation method was used instead of two step generalized least squares to estimate the regression parameters in equations (1) and (2).
Although Mercury was able to greatly reduce processing time while preserving clarity by optimizing the maximum likelihood expectation maximization algorithm, more processing power was still needed.
Footnote AFDVD(TM) = All Format DVD SoC = System-on-Chip PRML = Partial Response Maximum likelihood ECC = Error correction code CD = Compact Disc DVD = Digital Versatile Disc CIRC = Cross Interleaved Reed-Solomon Code 1.
A maximum likelihood factor analysis with a varimax rotation yielded four factors: self-confidence; value of mathematics; enjoyment of mathematics; and motivation.
Phylogenetic trees were constructed using parsimony, maximum likelihood and distance methods.
Maximum likelihood and maximum parsimony analyses of the 12s sequences suggested that the abyssal species of Coryphaenoides indeed form a sister taxon to the slope-dwelling species of Coryphaenoides, and might represent a separate radiation.
In order to achieve this purpose, Monte Carlo simulations were used to characterize the statistical properties of three evaluation methods: maximum likelihood method, moments method, and linear regression.
On the other hand, the Maximum Likelihood Hierarchical Model(MLH)is a probabilistic version of its predecessor HIARC, which is also a multiattribute disaggregate lexicographic model (Gensch & Svestka, 1979).

Full browser ?