Maximum likelihood estimation
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In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of a statistical model, given observations, by maximizing a likelihood function. The resulting arguments at the maximum are called maximum likelihood estimates, which is also abbreviated as MLE.
The method of maximum likelihood is used with a wide range of statistical analyses. As an example, suppose that we are interested in the heights of adult female penguins, but are unable to measure the height of every penguin in a population (due to cost or time constraints). Assuming that the heights are normally distributed with some unknown mean and variance, the mean and variance can be estimated with MLE while only knowing the heights of some sample of the overall population. MLE would accomplish that by taking the mean and variance as parameters and finding particular parametric values that make the observed results the most probable given the normal model.
From the point of view of Bayesian inference, MLE is a special case of maximum a posteriori estimation (MAP) that assumes a uniform prior distribution of the parameters. In frequentist inference, MLE is one of several methods to get estimates of parameters without using prior distributions. Priors are avoided by not making probability statements about the parameters, but only about their estimates, whose properties are fully defined by the observations and the statistical model.
Contents
Principles
The method of maximum likelihood is based on the likelihood function, . We are given a statistical model, i.e. a family of distributions , where denotes the (possibly multidimensional) parameter for the model. The method of maximum likelihood finds the values of the model parameter, , that maximize the likelihood function, . Intuitively, this selects the parameter values that make the data most probable.
The method defines a maximum likelihood estimate:
if a maximum exists.
In practice, it is often convenient to work with the natural logarithm of the likelihood function, called the loglikelihood:
or the average loglikelihood:
The hat over indicates that it is akin to an estimator. Indeed, estimates the expected loglikelihood of a single observation in the model.
An MLE is the same regardless of whether we maximize the likelihood or the loglikelihood, because log is a strictly increasing function.
For many models, a maximum likelihood estimator can be found as an explicit function of the observed data . For many other models, however, no closedform solution to the maximization problem is known or available, and an MLE can only be found via numerical global optimization. For some problems, there may be multiple values that maximize the likelihood. For other problems, no maximum likelihood estimate exists: either the loglikelihood function increases without ever reaching a supremum value, or the supremum does exist but is outside the bounds of , the set of acceptable parameter values.
Properties
A maximum likelihood estimator is an extremum estimator obtained by maximizing, as a function of θ, the objective function (cf. loss function) . If the data are independent and identically distributed, then we have
this being the sample analogue of the expected loglikelihood , where this expectation is taken with respect to the true density.
Maximumlikelihood estimators have no optimum properties for finite samples, in the sense that (when evaluated on finite samples) other estimators may have greater concentration around the true parametervalue.^{[1]} However, like other estimation methods, maximum likelihood estimation possesses a number of attractive limiting properties: As the sample size increases to infinity, sequences of maximum likelihood estimators have these properties:
 Consistency: the sequence of MLEs converges in probability to the value being estimated.
 Efficiency, i.e. it achieves the Cramér–Rao lower bound when the sample size tends to infinity. This means that no consistent estimator has lower asymptotic mean squared error than the MLE (or other estimators attaining this bound).
 Secondorder efficiency after correction for bias.
Consistency
Under the conditions outlined below, the maximum likelihood estimator is consistent. The consistency means that if the data were generated by and we have a sufficiently large number of observations n, then it is possible to find the value of θ_{0} with arbitrary precision. In mathematical terms this means that as n goes to infinity the estimator converges in probability to its true value:
Under slightly stronger conditions, the estimator converges almost surely (or strongly):
Note that, in practical applications, data is never generated by . Rather, is a model, often in idealized form, of the process that generated the data. It is a common aphorism in statistics that all models are wrong. Thus, true consistency does not occur in practical applications. Nevertheless, consistency is often considered to be a desirable property for an estimator to have.
To establish consistency, the following conditions are sufficient.^{[2]}

Identification of the model:
In other words, different parameter values θ correspond to different distributions within the model. If this condition did not hold, there would be some value θ_{1} such that θ_{0} and θ_{1} generate an identical distribution of the observable data. Then we would not be able to distinguish between these two parameters even with an infinite amount of data—these parameters would have been observationally equivalent.
The identification condition is absolutely necessary for the ML estimator to be consistent. When this condition holds, the limiting likelihood function ℓ(θ·) has unique global maximum at θ_{0}.

Compactness: the parameter space Θ of the model is compact.
The identification condition establishes that the loglikelihood has a unique global maximum. Compactness implies that the likelihood cannot approach the maximum value arbitrarily close at some other point (as demonstrated for example in the picture on the right).
Compactness is only a sufficient condition and not a necessary condition. Compactness can be replaced by some other conditions, such as:
 both concavity of the loglikelihood function and compactness of some (nonempty) upper level sets of the loglikelihood function, or
 existence of a compact neighborhood N of θ_{0} such that outside of N the loglikelihood function is less than the maximum by at least some ε > 0.

Continuity: the function ln f(xθ) is continuous in θ for almost all values of x:

Dominance: there exists D(x) integrable with respect to the distribution f(xθ_{0}) such that
The dominance condition can be employed in the case of i.i.d. observations. In the noni.i.d. case, the uniform convergence in probability can be checked by showing that the sequence is stochastically equicontinuous. If one wants to demonstrate that the ML estimator converges to θ_{0} almost surely, then a stronger condition of uniform convergence almost surely has to be imposed:
Additionally, if (as assumed above) the data were generated by , then under certain conditions, it can also be shown that the maximum likelihood estimator converges in distribution to a normal distribution. Specifically,^{[3]}
where I is the Fisher information matrix.
Functional invariance
The maximum likelihood estimator selects the parameter value which gives the observed data the largest possible probability (or probability density, in the continuous case). If the parameter consists of a number of components, then we define their separate maximum likelihood estimators, as the corresponding component of the MLE of the complete parameter. Consistent with this, if is the MLE for , and if is any transformation of , then the MLE for is by definition
It maximizes the socalled profile likelihood:
The MLE is also invariant with respect to certain transformations of the data. If where is one to one and does not depend on the parameters to be estimated, then the density functions satisfy
and hence the likelihood functions for and differ only by a factor that does not depend on the model parameters.
For example, the MLE parameters of the lognormal distribution are the same as those of the normal distribution fitted to the logarithm of the data.
Higherorder properties
As noted above, the maximum likelihood estimator is √n consistent and asymptotically efficient, meaning that it reaches the Cramér–Rao bound:
where is the Fisher information matrix:
In particular, it means that the bias of the maximum likelihood estimator is equal to zero up to the order ^{1}⁄_{√n }. However, when we consider the higherorder terms in the expansion of the distribution of this estimator, it turns out that θ_{mle} has bias of order ^{1}⁄_{n}. This bias is equal to (componentwise)^{[4]}
where denotes the (j,k)th component of the inverse Fisher information matrix , and
Using these formulae it is possible to estimate the secondorder bias of the maximum likelihood estimator, and correct for that bias by subtracting it:
This estimator is unbiased up to the terms of order ^{1}⁄_{n}, and is called the biascorrected maximum likelihood estimator.
This biascorrected estimator is secondorder efficient (at least within the curved exponential family), meaning that it has minimal mean squared error among all secondorder biascorrected estimators, up to the terms of the order ^{1}⁄_{n ² }. It is possible to continue this process, that is to derive the thirdorder biascorrection term, and so on. However, as was shown by Kano (1996), the maximum likelihood estimator is not thirdorder efficient.
Relation to Bayesian inference
A maximum likelihood estimator coincides with the most probable Bayesian estimator given a uniform prior distribution on the parameters. Indeed, the maximum a posteriori estimate is the parameter θ that maximizes the probability of θ given the data, given by Bayes' theorem:
where is the prior distribution for the parameter θ and where is the probability of the data averaged over all parameters. Since the denominator is independent of θ, the Bayesian estimator is obtained by maximizing with respect to θ. If we further assume that the prior is a uniform distribution, the Bayesian estimator is obtained by maximizing the likelihood function . Thus the Bayesian estimator coincides with the maximum likelihood estimator for a uniform prior distribution .
Examples
It has been suggested that Maximum likelihood estimation with flow data be merged into this article. (Discuss) Proposed since October 2017.

Discrete uniform distribution
Consider a case where n tickets numbered from 1 to n are placed in a box and one is selected at random (see uniform distribution); thus, the sample size is 1. If n is unknown, then the maximum likelihood estimator of n is the number m on the drawn ticket. (The likelihood is 0 for n < m, ^{1}⁄_{n} for n ≥ m, and this is greatest when n = m. Note that the maximum likelihood estimate of n occurs at the lower extreme of possible values {m, m + 1, ...}, rather than somewhere in the "middle" of the range of possible values, which would result in less bias.) The expected value of the number m on the drawn ticket, and therefore the expected value of , is (n + 1)/2. As a result, with a sample size of 1, the maximum likelihood estimator for n will systematically underestimate n by (n − 1)/2.
Discrete distribution, finite parameter space
Suppose one wishes to determine just how biased an unfair coin is. Call the probability of tossing a ‘head’ p. The goal then becomes to determine p.
Suppose the coin is tossed 80 times: i.e. the sample might be something like x_{1} = H, x_{2} = T, ..., x_{80} = T, and the count of the number of heads "H" is observed.
The probability of tossing tails is 1 − p (so here p is θ above). Suppose the outcome is 49 heads and 31 tails, and suppose the coin was taken from a box containing three coins: one which gives heads with probability p = ^{1}⁄_{3}, one which gives heads with probability p = ^{1}⁄_{2} and another which gives heads with probability p = ^{2}⁄_{3}. The coins have lost their labels, so which one it was is unknown. Using maximum likelihood estimation the coin that has the largest likelihood can be found, given the data that were observed. By using the probability mass function of the binomial distribution with sample size equal to 80, number successes equal to 49 but for different values of p (the "probability of success"), the likelihood function (defined below) takes one of three values:
The likelihood is maximized when p = ^{2}⁄_{3}, and so this is the maximum likelihood estimate for p.
Discrete distribution, continuous parameter space
Now suppose that there was only one coin but its p could have been any value 0 ≤ p ≤ 1. The likelihood function to be maximised is
and the maximisation is over all possible values 0 ≤ p ≤ 1.
One way to maximize this function is by differentiating with respect to p and setting to zero:
which has solutions p = 0, p = 1, and p = ^{49}⁄_{80}. The solution which maximizes the likelihood is clearly p = ^{49}⁄_{80} (since p = 0 and p = 1 result in a likelihood of zero). Thus the maximum likelihood estimator for p is ^{49}⁄_{80}.
This result is easily generalized by substituting a letter such as s in the place of 49 to represent the observed number of 'successes' of our Bernoulli trials, and a letter such as n in the place of 80 to represent the number of Bernoulli trials. Exactly the same calculation yields ^{s}⁄_{n} which is the maximum likelihood estimator for any sequence of n Bernoulli trials resulting in s 'successes'.
Continuous distribution, continuous parameter space
For the normal distribution which has probability density function
the corresponding probability density function for a sample of n independent identically distributed normal random variables (the likelihood) is
or more conveniently,
where is the sample mean.
This family of distributions has two parameters: θ = (μ, σ); so we maximize the likelihood, , over both parameters simultaneously, or if possible, individually.
Since the logarithm function itself is a continuous strictly increasing function over the range of the likelihood, the values which maximize the likelihood will also maximize its logarithm (the loglikelihood itself is not necessarily strictly increasing). The loglikelihood can be written as follows:
(Note: the loglikelihood is closely related to information entropy and Fisher information.)
We now compute the derivatives of this loglikelihood as follows.
This is solved by
This is indeed the maximum of the function, since it is the only turning point in μ and the second derivative is strictly less than zero. Its expected value is equal to the parameter μ of the given distribution,
which means that the maximum likelihood estimator is unbiased.
Similarly we differentiate the loglikelihood with respect to σ and equate to zero:
which is solved by
Inserting the estimate we obtain
To calculate its expected value, it is convenient to rewrite the expression in terms of zeromean random variables (statistical error) . Expressing the estimate in these variables yields
Simplifying the expression above, utilizing the facts that and , allows us to obtain
This means that the estimator is biased. However, is consistent.
Formally we say that the maximum likelihood estimator for is
In this case the MLEs could be obtained individually. In general this may not be the case, and the MLEs would have to be obtained simultaneously.
The normal loglikelihood at its maximum takes a particularly simple form:
This maximum loglikelihood can be shown to be the same for more general least squares, even for nonlinear least squares. This is often used in determining likelihoodbased approximate confidence intervals and confidence regions, which are generally more accurate than those using the asymptotic normality discussed above.
Nonindependent variables
It may be the case that variables are correlated, that is, not independent. Two random variables X and Y are independent only if their joint probability density function is the product of the individual probability density functions, i.e.
Suppose one constructs an ordern Gaussian vector out of random variables , where each variable has means given by . Furthermore, let the covariance matrix be denoted by .
The joint probability density function of these n random variables is then given by:
In the two variable case, the joint probability density function is given by:
In this and other cases where a joint density function exists, the likelihood function is defined as above, in the section Principles, using this density.
Iterative procedures
Consider problems where both states and parameters such as require to be estimated. Iterative procedures such as Expectationmaximization algorithms may be used to solve joint stateparameter estimation problems.
For example, suppose that n samples of state estimates together with a sample mean have been calculated by either a minimumvariance Kalman filter or a minimumvariance smoother using a previous variance estimate . Then the next variance iterate may be obtained from the maximum likelihood estimate calculation
The convergence of MLEs within filtering and smoothing EM algorithms has been studied in the literature.^{[5]}^{[6]}
History
Early users of maximum likelihood were Carl Friedrich Gauss, PierreSimon Laplace, Thorvald N. Thiele, and Francis Ysidro Edgeworth.^{[7]}
However its widespread use arose between 1912 and 1922 when Ronald Fisher recommended, widely popularized, and carefully analyzed maximumlikelihood estimation (with fruitless attempts at proofs).^{[8]}
Maximumlikelihood estimation finally transcended heuristic justification in a proof published by Samuel S. Wilks in 1938, now called Wilks' theorem.^{[9]} The theorem shows that the error in the logarithm of likelihood values for estimates from multiple independent samples is asymptotically χ ² distributed, which enables convenient determination of a confidence region around any one estimate of the parameters. The only difficult part of Wilks’ proof depends on the expected value of the Fisher information matrix, which ironically is provided by a theorem proven by Fisher.^{[10]} Wilks continued to improve on the generality of the theorem throughout his life, with his most general proof published in 1962.^{[11]}
Some of the theory behind maximum likelihood estimation was developed for Bayesian statistics.^{[8]}
Reviews of the development of maximum likelihood estimation have been provided by a number of authors.^{[12]}
See also

Other estimation methods
 Generalized method of moments are methods related to the likelihood equation in maximum likelihood estimation
 Mestimator, an approach used in robust statistics
 Maximum a posteriori (MAP) estimator, for a contrast in the way to calculate estimators when prior knowledge is postulated
 Maximum spacing estimation, a related method that is more robust in many situations
 Maximum entropy estimation
 Method of moments (statistics), another popular method for finding parameters of distributions
 Method of support, a variation of the maximum likelihood technique
 Minimum distance estimation
 Partial likelihood methods for panel data
 Quasimaximum likelihood estimator, an MLE estimator that is misspecified, but still consistent
 Restricted maximum likelihood, a variation using a likelihood function calculated from a transformed set of data

Related concepts:
 Akaike information criterion, a criterion to compare statistical models, based on MLE
 Extremum estimator, a more general class of estimators to which MLE belongs
 Fisher information, information matrix, its relationship to covariance matrix of ML estimates
 Mean squared error, a measure of how 'good' an estimator of a distributional parameter is (be it the maximum likelihood estimator or some other estimator)
 RANSAC, a method to estimate parameters of a mathematical model given data that contains outliers
 Rao–Blackwell theorem, which yields a process for finding the best possible unbiased estimator (in the sense of having minimal mean squared error); the MLE is often a good starting place for the process
 Wilks’ Theorem provides a means of estimating the size and shape of the region of roughly equallyprobable estimates for the population's parameter values, using the information from a single sample, using a chisquared distribution
Notes
 ^ Pfanzagl (1994, p. 206)
 ^ Newey & McFadden (1994, Theorem 2.5.)
 ^ Newey & McFadden (1994, Theorem 3.3.)
 ^ Cox & Snell (1968, formula (20))
 ^ Einicke, G. A.; Falco, G.; Malos, J. T. (May 2010). "EM Algorithm State Matrix Estimation for Navigation". IEEE Signal Processing Letters. 17 (5): 437–440. doi:10.1109/LSP.2010.2043151.
 ^ Einicke, G. A.; Falco, G.; Dunn, M. T.; Reid, D. C. (May 2012). "Iterative SmootherBased Variance Estimation". IEEE Signal Processing Letters. 19 (5): 275–278. doi:10.1109/LSP.2012.2190278.
 ^ Edgeworth & September 1908 and Edgeworth & December 1908
 ^ ^{a} ^{b} Pfanzagl, Johann, with the assistance of R. Hamböker (1994). Parametric Statistical Theory. Walter de Gruyter. pp. 207–208. ISBN 3110138638.CS1 maint: Multiple names: authors list (link)
 ^ Wilks (1938)
 ^ Owen, Art B. (2001). Empirical Likelihood. London: Chapman & Hall/Boca Raton, FL: CRC Press. ISBN 9781584880714.
 ^ Wilks, Samuel S. (1962), Mathematical Statistics, New York: John Wiley & Sons. ISBN 9780471946502.
 ^ Savage (1976), Pratt (1976), Stigler (1978, 1986, 1999), Hald (1998, 1999), and Aldrich (1997)
References: nonhistorical
 Cox, David R.; Snell, E. Joyce (1968). "A general definition of residuals". Journal of the Royal Statistical Society, Series B. 30: 248–275. JSTOR 2984505.
 Kano, Yutaka (1996). "Thirdorder efficiency implies fourthorder efficiency". Journal of the Japan Statistical Society. 26: 101–117. doi:10.14490/jjss1995.26.101. ^{[permanent dead link]}
 Newey, Whitney K.; McFadden, Daniel (1994). "Chapter 35: Large sample estimation and hypothesis testing". In Engle, Robert; McFadden, Dan. Handbook of Econometrics, Vol.4. Elsevier Science. pp. 2111–2245. ISBN 0444887660.
References: historical
 Aldrich, John (1997). "R. A. Fisher and the making of maximum likelihood 1912–1922". Statistical Science. 12 (3): 162–176. doi:10.1214/ss/1030037906. MR 1617519.
 Edgeworth, Francis Y. (Sep 1908). "On the probable errors of frequencyconstants". Journal of the Royal Statistical Society. 71 (3): 499–512. doi:10.2307/2339293. JSTOR 2339293.
 Edgeworth, Francis Y. (Dec 1908). "On the probable errors of frequencyconstants". Journal of the Royal Statistical Society. 71 (4): 651–678. doi:10.2307/2339378. JSTOR 2339378.
 Hald, Anders (1998). A history of mathematical statistics from 1750 to 1930. New York, NY: Wiley. ISBN 0471179124.
 Hald, Anders (1999). "On the history of maximum likelihood in relation to inverse probability and least squares". Statistical Science. 14 (2): 214–222. doi:10.1214/ss/1009212248. JSTOR 2676741.
 Pratt, John W. (1976). "F. Y. Edgeworth and R. A. Fisher on the efficiency of maximum likelihood estimation". The Annals of Statistics. 4 (3): 501–514. doi:10.1214/aos/1176343457. JSTOR 2958222.
 Savage, Leonard J. (1976). "On rereading R. A. Fisher". The Annals of Statistics. 4 (3): 441–500. doi:10.1214/aos/1176343456. JSTOR 2958221.
 Stigler, Stephen M. (1978). "Francis Ysidro Edgeworth, statistician". Journal of the Royal Statistical Society, Series A. 141 (3): 287–322. doi:10.2307/2344804. JSTOR 2344804.
 Stigler, Stephen M. (1986). The history of statistics: the measurement of uncertainty before 1900. Harvard University Press. ISBN 0674403401.
 Stigler, Stephen M. (1999). Statistics on the table: the history of statistical concepts and methods. Harvard University Press. ISBN 0674836014.
 Wilks, S. S. (1938), "The LargeSample Distribution of the Likelihood Ratio for Testing Composite Hypotheses", Annals of Mathematical Statistics, 9: 60–62, doi:10.1214/aoms/1177732360.
Further reading
 Andersen, Erling B. (1970); "Asymptotic Properties of Conditional Maximum Likelihood Estimators", Journal of the Royal Statistical Society B 32, 283–301
 Andersen, Erling B. (1980); Discrete Statistical Models with Social Science Applications, North Holland, 1980
 Basu, Debabrata (1988); Statistical Information and Likelihood : A Collection of Critical Essays by Dr. D. Basu; in Ghosh, Jayanta K., editor; Lecture Notes in Statistics, Volume 45, SpringerVerlag, 1988
 Einicke, G.A. (2012). Smoothing, Filtering and Prediction: Estimating the Past, Present and Future. Rijeka, Croatia: Intech. ISBN 9789533077529.
 Ferguson, Thomas S. (1982). "An inconsistent maximum likelihood estimate". Journal of the American Statistical Association. 77 (380): 831–834. doi:10.1080/01621459.1982.10477894. JSTOR 2287314.
 Ferguson, Thomas S. (1996). A Course in Large Sample Theory. Chapman & Hall. ISBN 0412043718.
 Le Cam, Lucien (1990). "Maximum likelihood — an introduction". ISI Review. 58 (2): 153–171. doi:10.2307/1403464.
 Le Cam, Lucien; Yang, Grace Lo (2000). Asymptotics in Statistics: some basic concepts (Second ed.). Springer. ISBN 0387950362.
 Lehmann, Erich L.; Casella, George (1998). Theory of Point Estimation, 2nd ed. Springer. ISBN 0387985026.
 Millar, R. B. (2011). Maximum Likelihood Estimation and Inference. Wiley.
 Ruppert, David (2010). Statistics and Data Analysis for Financial Engineering. Springer. p. 98. ISBN 9781441977861.
 van der Vaart, Aad W. (1998). Asymptotic Statistics. ISBN 0521784506.
External links
 Hazewinkel, Michiel, ed. (2001) [1994], "Maximumlikelihood method", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Maximum Likelihood Estimation Primer (an excellent tutorial)
 Implementing MLE for your own likelihood function using R
 Myung, I. J. (2003). "Tutorial on maximum likelihood estimation". Journal of Mathematical Psychology. CiteSeerX 10.1.1.74.671.