Statistics Toolbox™

mle - Maximum likelihood estimates

Syntax

phat = mle(data)
[phat,pci] = mle(data)
[...] = mle(data,'distribution',dist)
[...] = mle(data,...,name1,val1,name2,val2,...)
[...] = mle(data,'pdf',pdf,'cdf',cdf,'start',start,...)
[...] = mle(data,'logpdf',logpdf,'logsf',logsf,'start',start,...)
[...] = mle(data,'nloglf',nloglf,'start',start,...)

Description

phat = mle(data) returns maximum likelihood estimates (MLEs) for the parameters of a normal distribution, computed using the sample data in the vector data.

[phat,pci] = mle(data) returns MLEs and 95% confidence intervals for the parameters.

[...] = mle(data,'distribution',dist) computes parameter estimates for the distribution specified by dist. Acceptable strings for dist are:

• 'beta'

• 'bernoulli'

• 'binomial'

• 'discrete uniform'

• 'exponential'

• 'extreme value'

• 'gamma'

• 'generalized extreme value'

• 'generalized pareto'

• 'geometric'

• 'lognormal'

• 'negative binomial'

• 'normal'

• 'poisson'

• 'rayleigh'

• 'uniform'

• 'weibull'

[...] = mle(data,...,name1,val1,name2,val2,...) specifies optional argument name/value pairs chosen from the following list.

mle can also fit custom distributions that you define using distribution functions, in one of three ways.

[...] = mle(data,'pdf',pdf,'cdf',cdf,'start',start,...) returns MLEs for the parameters of the distribution defined by the probability density and cumulative distribution functions pdf and cdf. pdf and cdf are function handles created using the @ sign. They accept as inputs a vector data and one or more individual distribution parameters, and return vectors of probability density values and cumulative probability values, respectively. If the 'censoring' name/value pair is not present, you can omit the 'cdf' name/value pair. mle computes the estimates by numerically maximizing the distribution's log-likelihood, and start is a vector containing initial values for the parameters.

[...] = mle(data,'logpdf',logpdf,'logsf',logsf,'start',start,...) returns MLEs for the parameters of the distribution defined by the log probability density and log survival functions logpdf and logsf. logpdf and logsf are function handles created using the @ sign. They accept as inputs a vector data and one or more individual distribution parameters, and return vectors of logged probability density values and logged survival function values, respectively. This form is sometimes more robust to the choice of starting point than using pdf and cdf functions. If the 'censoring' name/value pair is not present, you can omit the 'logsf' name/value pair. start is a vector containing initial values for the distribution's parameters.

[...] = mle(data,'nloglf',nloglf,'start',start,...) returns MLEs for the parameters of the distribution whose negative log-likelihood is given by nloglf. nloglf is a function handle, specified using the @ sign, that accepts the four input arguments:

• params - a vector of distribution parameter values

• data - a vector of data

• cens - a Boolean vector of censoring values

• freq - a vector of integer data frequencies

nloglf must accept all four arguments even if you do not supply the 'censoring' or 'frequency' name/value pairs (see above). However, nloglf can safely ignore its cens and freq arguments in that case. nloglf returns a scalar negative log-likelihood value and, optionally, a negative log-likelihood gradient vector (see the 'GradObj' statset parameter below). start is a vector containing initial values for the distribution's parameters.

pdf, cdf, logpdf, logsf, or nloglf can also be cell arrays whose first element is a function handle as defined above, and whose remaining elements are additional arguments to the function. mle places these arguments at the end of the argument list in the function call.

The following optional argument name/value pairs are valid only when 'pdf' and 'cdf', 'logpdf' and 'logcdf', or 'nloglf' are given:

• 'lowerbound' — A vector the same size as start containing lower bounds for the distribution parameters. The default is -Inf.

• 'upperbound' — A vector the same size as start containing upper bounds for the distribution parameters. The default is Inf.

• 'optimfun' — A string, either 'fminsearch' or 'fmincon', naming the optimization function to be used in maximizing the likelihood. The default is 'fminsearch'. You can only specify 'fmincon' if Optimization Toolbox software is available.

When fitting a custom distribution, use the 'options' parameter to control details of the maximum likelihood optimization. See statset('mlecustom') for parameter names and default values. mle interprets the following statset parameters for custom distribution fitting as follows:

Example

The following returns an MLE and a 95% confidence interval for the success probability of a binomial distribution with 20 trials:

data = binornd(20,0.75,100,1); % Simulated data, p = 0.75

[phat,pci] = mle(data,'distribution','binomial',...
                 'alpha',.05,'ntrials',20)
phat =
    0.7370
pci =
    0.7171
    0.7562

See Also

betafit, binofit, evfit, expfit, gamfit, gevfit, gpfit, lognfit, nbinfit, normfit, mlecov, poissfit, raylfit, statset, unifit, wblfit

mhsample

mlecov