wbllike

Weibull negative loglikelihood

Syntax

nlogL = wbllike(params,x)

nlogL = wbllike(params,x,censoring)

nlogL = wbllike(params,x,censoring,freq)

[nlogL,aVar] = wbllike(___)

Description

nlogL = wbllike(params,x) returns the Weibull negative loglikelihood of the distribution parameters (params) given the sample data (x). params(1) and params(2) are the scale parameter a and shape parameter b, respectively.

nlogL = wbllike(params,x,censoring) specifies whether each value in x is right-censored or not. Use the logical vector censoring to specify the value 1 for observations that are right-censored and 0 for observations that are fully observed.

nlogL = wbllike(params,x,censoring,freq) specifies the frequency (or weights) of the observations. To specify freq without specifying censoring, you can pass [] for censoring.

[nlogL,aVar] = wbllike(___) also returns the inverse of the Fisher information matrix aVar, using any of the input argument combinations in the previous syntaxes. If the values in params are the maximum likelihood estimates (MLEs) of the parameters, the diagonal elements of aVar are their asymptotic variances. aVar is based on the observed Fisher information, not the expected information.

The Weibull negative loglikelihood for uncensored data is

$$\left(-\log L\right)=-\log {\displaystyle \prod _{i=1}f\left(a,b|x_i\right)=-{\displaystyle \sum _{i=1}^n\log f\left(a,b|x_i\right)}}$$

where f is the Weibull probability density function.

wbllike is a utility function for maximum likelihood estimation.

Examples

Negative Loglikelihood of MLEs

Find the maximum likelihood estimates (MLEs) of a random data set drawn from the Weibull distribution by using the mle function, and then find the negative loglikelihood of the MLEs by using the wbllike function.

Generate 1000 random numbers from the Weibull distribution with the scale parameter a=5 and shape parameter b=3.

rng(0,"twister") % For reproducibility
n = 1000; % Number of samples
a = 5;
b = 3;
x = wblrnd(a,b,[n,1]);

Find the MLEs for the distribution parameters.

pHat = mle(x,Distribution="Weibull")

pHat = 1×2

    5.0450    3.1937

Compute the negative loglikelihood of the MLEs and the inverse of the Fisher information matrix.

[nlogL,aVar] = wbllike(pHat,x)

nlogL = 
1.8556e+03

aVar = 2×2

    0.0028    0.0013
    0.0013    0.0063

Because pHat contains MLE values, the wbllike function returns their asymptotic variances in the diagonal elements of aVar.

Input Arguments

params — Weibull distribution parameters

vector of two positive scalar values

Weibull distribution parameters, specified as a vector of two positive scalar values. params(1) and params(2) are the scale and shape values, respectively.

Data Types: single | double

x — Sample data

numeric vector

Sample data, specified as a numeric vector.

Data Types: single | double

censoring — Indicator for censoring

array of 0s (default) | logical vector

Indicator for the censoring of each value in x, specified as a logical vector of the same size as x. Use 1 for observations that are right-censored and 0 for observations that are fully observed.

The default is an array of 0s, meaning that all observations are fully observed.

Data Types: logical

freq — Frequency (or weights) of observations

array of 1s (default) | nonnegative vector

Frequency (or weights) of the observations, specified as a nonnegative vector that is the same size as x. The freq input argument typically contains nonnegative integer counts for the corresponding elements in x, but can contain any nonnegative values.

To obtain the weighted negative loglikelihood for a data set with censoring, specify weights of observations, normalized to the number of observations in x.

The default is an array of 1s, meaning one observation per element of x.

Data Types: single | double

Output Arguments

nlogL — Negative loglikelihood

numeric scalar

Negative loglikelihood value of the distribution parameters (params) given the sample data (x), returned as a numeric scalar.

aVar — Inverse of Fisher information matrix

numeric matrix

Inverse of the Fisher information matrix, returned as a 2-by-2 numeric matrix. aVar is based on the observed Fisher information given the observed data (x), not the expected information.

If values in params are the MLEs of the parameters, aVar is an approximation to the asymptotic variance-covariance matrix (also known as the asymptotic covariance matrix). To find the MLEs, use mle.

Alternative Functionality

wbllike is a function specific to the Weibull distribution. Statistics and Machine Learning Toolbox™ also offers the generic functions mlecov, fitdist, negloglik, and proflik and the Distribution Fitter app, which support various probability distributions.

• mlecov returns the asymptotic covariance matrix of the MLEs of the parameters for a distribution specified by a custom probability density function. For example, mlecov(params,x,'pdf',@wblpdf) returns the asymptotic covariance matrix of the MLEs for the Weibull distribution.

• Create a WeibullDistribution probability distribution object by fitting the distribution to data using the fitdist function or the Distribution Fitter app. The object property ParameterCovariance stores the covariance matrix of the parameter estimates. To obtain the negative loglikelihood of the parameter estimates and the profile of the likelihood function, pass the object to negloglik and proflik, respectively.

References

[1] Patel, J. K., C. H. Kapadia, and D. B. Owen. Handbook of Statistical Distributions. New York: Marcel Dekker, 1976.

Version History

Introduced before R2006a