Likelihood Function

Let denote the random response variable, and let denote its value recorded in an observation in the input data set. Let denote the censoring indicator: indicates that the observation is uncensored (sometimes referred to as an event observation) and indicates that the observation is right-censored. When , the recorded value of is assumed to be the censoring limit, denoted by . Let denote the left-truncation threshold. Let and denote the PDF and CDF respectively, evaluated at for a set of parameter values . Then, the set of input observations can be categorized into the following four subsets within each BY group:

• : the set of uncensored observations that are not left-truncated. The likelihood of an observation is

• : the set of uncensored observations that are left-truncated. The likelihood of an observation is

• : the set of right-censored observations that are not left-truncated. The likelihood of an observation is

• : the set of right-censored observations that are left-truncated. The likelihood of an observation is

Note that . Also, the sets and are empty when left-truncation is not specified, and the sets and are empty when right-censoring is not specified.

Given this, the likelihood of the data is as follows:

The maximum likelihood procedure used by PROC SEVERITY finds an optimal set of parameter values that maximizes subject to the boundary constraints on parameter values. Note that for a distribution dist, such boundary constraints can be specified by using the dist_LOWERBOUNDS and dist_UPPERBOUNDS subroutines. Some aspects of the optimization process can be controlled by using the NLOPTIONS statement.

Probability of Observability and Likelihood

If probability of observability is specified for the left-truncation, then PROC SEVERITY uses a modified likelihood function for each truncated observation. If the probability of observability is , then for each left-truncated observation with truncation threshold , there exist observations with a response variable value less than or equal to . Each such observation has a probability of . Thus, following the notation of the section Likelihood Function, the likelihood of the data is as follows:

Note that the likelihood of the observations that are not left-truncated (observations in sets and ) is not affected.

Estimating Covariance and Standard Errors

PROC SEVERITY computes an estimate of the covariance matrix of the parameters by using the asymptotic theory of the maximum likelihood estimators (MLE). If denotes the number of observations used for estimating a parameter vector , then the theory states that as , the distribution of , the estimate of , converges to a normal distribution with mean and covariance such that , where is the information matrix for the likelihood of the data, . The covariance estimate is obtained by using the inverse of the information matrix.

In particular, if denotes the Hessian matrix of the log likelihood, then the covariance estimate is computed as

where is a denominator determined by the VARDEF= option. If VARDEF=N, then , which yields the asymptotic covariance estimate. If VARDEF=DF, then , where is number of parameters (the model’s degrees of freedom). The VARDEF=DF option is the default, because it attempts to correct the potential bias introduced by the finite sample.

The standard error of the parameter is computed as the square root of the th diagonal element of the estimated covariance matrix; that is, .

Note that covariance and standard error estimates might not be available if the Hessian matrix is found to be singular at the end of the optimization process. This can especially happen if the optimization process stops without converging.