Nonlinear Optimization Examples: Example 11.4: MLEs for Two-Parameter Weibull Distribution

Nonlinear Optimization Examples

Example 11.4: MLEs for Two-Parameter Weibull Distribution

This example considers a data set given in Lawless (1982). The data are the number of days it took rats painted with a carcinogen to develop carcinoma. The last two observations are censored. Maximum likelihood estimates (MLEs) and confidence intervals for the parameters of the Weibull distribution are computed. In the following code, the data set is given in the vector CARCIN, and the variables P and M give the total number of observations and the number of uncensored observations. The set represents the indices of the observations.

  
    proc iml; 
       carcin = { 143  164  188  188  190  192  206 
                  209  213  216  220  227  230  234 
                  246  265  304  216  244 }; 
       p = ncol(carcin); m = p - 2;

The three-parameter Weibull distribution uses three parameters: a scale parameter, a shape parameter, and a location parameter. This example computes MLEs and corresponding 95% confidence intervals for the scale parameter, $\sigma$ , and the shape parameter, , for a constant value of the location parameter, $\theta =0$ . The program can be generalized to estimate all three parameters. Note that Lawless (1982) denotes $\sigma$ , , and $\theta$ by $\alpha$ , $\beta$ , and $\mu$ , respectively.

The observed likelihood function of the three-parameter Weibull distribution is

$l(\theta,\sigma,c) = \frac{c^m}{\sigma^m} \prod_{i \in d} ( \frac{t_i - \theta}{\sigma} )^{c-1} \prod_{i=1}^p \exp \{ -( \frac{t_i - \theta}{\sigma} )^c \}$

The log likelihood, $\ell(\theta,\sigma,c) = \log l(\theta,\sigma,c)$ , is

$\ell(\theta,\sigma,c) = m \log c - mc \log \sigma + (c-1) \sum_{i \in d} \log(t_i - \theta) - \sum_{i=1}^p (\frac{t_i - \theta}{\sigma})^c$

The log-likelihood function, $\ell(\theta,\sigma,c)$ , for $\theta =0$ is the objective function to be maximized to obtain the MLEs $(\hat{\sigma},\hat{c})$ . The following statements define the function:

  
    start f_weib2(x) global(carcin,thet); 
       /* x[1]=sigma and x[2]=c */ 
       p = ncol(carcin); m = p - 2; 
       sum1 = 0.; sum2 = 0.; 
       do i = 1 to p; 
          temp = carcin[i] - thet; 
          if i <= m then sum1 = sum1 + log(temp); 
          sum2 = sum2 + (temp / x[1])##x[2]; 
       end; 
       f = m*log(x[2]) - m*x[2]*log(x[1]) + (x[2]-1)*sum1 - sum2; 
       return(f); 
    finish f_weib2;

The derivatives of $\ell$ with respect to the parameters $\theta$ , $\sigma$ , and are given in Lawless (1982). The following code specifies a gradient module, which computes $\partial \ell / \partial \sigma$ and $\partial \ell / \partial c$ :

  
    start g_weib2(x) global(carcin,thet); 
       /* x[1]=sigma and x[2]=c */ 
       p = ncol(carcin); m = p - 2; 
       g = j(1,2,0.); 
       sum1 = 0.; sum2 = 0.; sum3 = 0.; 
       do i = 1 to p; 
          temp = carcin[i] - thet; 
          if i <= m then sum1 = sum1 + log(temp); 
          sum2 = sum2 + (temp / x[1])##x[2]; 
          sum3 = sum3 + ((temp / x[1])##x[2]) * (log(temp / x[1])); 
       end; 
       g[1] = -m * x[2] / x[1] + sum2 * x[2] / x[1]; 
       g[2] = m / x[2] - m * log(x[1]) + sum1 - sum3; 
       return(g); 
    finish g_weib2;

The MLEs are computed by maximizing the objective function with the trust-region algorithm in the NLPTR subroutine. The following code specifies starting values for the two parameters, $c = \sigma =0.5$ , and to avoid infeasible values during the optimization process, it imposes lower bounds of $c, \sigma \gt=10^{-6}$ . The optimal parameter values are saved in the variable XOPT, and the optimal objective function value is saved in the variable FOPT.

  
    n = 2; thet = 0.; 
    x0 = j(1,n,.5); 
    optn = {1 2}; 
    con = { 1.e-6 1.e-6 , 
             .     .    }; 
    call nlptr(rc,xres,"f_weib2",x0,optn,con,,,,"g_weib2"); 
    /*--- Save result in xopt, fopt ---*/ 
    xopt = xres`; fopt = f_weib2(xopt);

The results shown in Output 11.4.1 are the same as those given in Lawless (1982).

Output 11.4.1: Parameter Estimates for Carcinogen Data

Optimization Results
Parameter Estimates
N	Parameter	Estimate	Gradient Objective Function
1	X1	234.318611	1.3363283E-9
2	X2	6.083147	-7.850915E-9

Value of Objective Function = -88.23273515

The following code computes confidence intervals based on the asymptotic normal distribution. These are compared with the profile-likelihood-based confidence intervals computed in the next example. The diagonal of the inverse Hessian (as calculated by the NLPFDD subroutine) is used to calculate the standard error.

  
    call nlpfdd(f,g,hes2,"f_weib2",xopt,,"g_weib2"); 
    hin2 = inv(hes2); 
    /* quantile of normal distribution */ 
    prob = .05; 
    noqua = probit(1. - prob/2); 
    stderr = sqrt(abs(vecdiag(hin2))); 
    xlb = xopt - noqua * stderr; 
    xub = xopt + noqua * stderr; 
    print "Normal Distribution Confidence Interval"; 
    print xlb xopt xub;

Output 11.4.2: Confidence Interval Based on Normal Distribution

XLB	XOP2	XUB
215.41298	234.31861	253.22425
3.9894574	6.0831471	8.1768368

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