TAILIEUCHUNG - SAS/ETS 9.22 User's Guide 156

SAS/Ets User's Guide 156. Provides detailed reference material for using SAS/ETS software and guides you through the analysis and forecasting of features such as univariate and multivariate time series, cross-sectional time series, seasonal adjustments, multiequational nonlinear models, discrete choice models, limited dependent variable models, portfolio analysis, and generation of financial reports, with introductory and advanced examples for each procedure. You can also find complete information about two easy-to-use point-and-click applications: the Time Series Forecasting System, for automatic and interactive time series modeling and forecasting, and the Investment Analysis System, for time-value of money analysis of a variety of investments | 1542 F Chapter 22 The SEVERITY Procedure Experimental Given this the likelihood of the data L is as follows l n f .yi n i 2E j 2Ei 1 - F tj m - F ck n k2C m2Ci 1 - F cm 1 - F tm The maximum likelihood procedure used by PROC SEVERITY finds an optimal set of parameter values 0 that maximizes log L subject to the boundary constraints on parameter values. Note that for a distribution dist such boundary constraints can be specified by using the dist_LOWHRBOUNDS 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 p 2 then for each left-truncated observation with truncation threshold t there exist 1 p p observations with a response variable value less than or equal to t. Each such observation has a probability of Pr Y t F@ t . Thus following the notation of the section Likelihood Function on page 1541 the likelihood of the data is as follows n1 p 1 p f yi n f y F tj HI 1 - F cfc n 1 - F m - - L i 2E j eEl keC meCl Note that the likelihood of the observations that are not left-truncated observations in sets E and C 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 N denotes the number of observations used for estimating a parameter vector 6 then the theory states that as N 1 the distribution of 6 the estimate of 6 converges to anormal distribution with mean 6 and covariance C such that I 6 C 1 where I 6 E V2 log L 6 is the information matrix for the likelihood of the data L 6 . The covariance estimate is obtained by using the inverse of the information matrix. In particular if G V2 log L 6

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