TAILIEUCHUNG - SAS/ETS 9.22 User's Guide 155

SAS/Ets User's Guide 155. 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 | 1532 F Chapter 22 The SEVERITY Procedure Experimental Given n observations of the severity value yi 1 i n the estimate of kth raw moment is denoted by mk and computed as 1 n mk n X yk i 1 The 100pth percentile is denoted by np 0 p 1 . By definition np satisfies F np- p F np where F np lim 0 F np h . PROC SEVERITY uses the following practical method of computing np. Let F y denote the empirical distribution function EDF estimate at a severity value y. This estimate is computed by PROC SEVERITY and supplied to the name_PARMINIT subroutine. Let y and y denote two consecutive values in the array of y values such that F y p and F yp p. Then the estimate rfp is computed as P - FP p yp 7 C yC - yp Fp Fp where FC F yC and Fp F y . Let e denote the smallest double-precision floating-point number such that 1 e 1. This machine precision constant can be obtained by using the CONSTANT function in Base SAS software. The details of how parameters are initialized for each predefined distribution model are as follows BURR The parameters are initialized by using the method of moments. The kth raw moment of the Burr distribution is E X k 6 1 kM a - kM -y k ây r a Three moment equations E Xk my k 1 2 3 need to be solved for initializing the three parameters of the distribution. In order to get an approximate closed form solution the second shape parameter y is initialized to a value of 2. If 2m3 3m1m2 0 then simplifying and solving the moment equations yields the following feasible set of initial values g j m2m3 i 3 m y 2 y 2m3 3m1m2 2m3 3m1m2 If 2m3 3m1m2 e then the parameters are initialized as follows 6 pm2 â 2 y 2 EXP The parameters are initialized by using the method of moments. The kth raw moment of the exponential distribution is E Xk 6kr k 1 k -1 Solving E X m1 yields the initial value of 6 m1. Predefined Distribution Models F 1533 GAMMA The parameter a is initialized by using its approximate maximum likelihood ML estimate. For a set of n iid observations yi 1 i n drawn from a

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