dehaan               package:accuracy               R Documentation

_d_e_h_a_a_n _g_l_o_b_a_l _o_p_t_i_m_a_l_i_t_y _t_e_s_t

_D_e_s_c_r_i_p_t_i_o_n:

     Implements the de Haan test for identification of the global
     optimum of a likelihood surface.

_U_s_a_g_e:

        dehaan(llTest, llMax, pval=.05 )

_A_r_g_u_m_e_n_t_s:

  llTest: Vector of randomly generated likelihood values

   llMax: Value of the likelihood function for a candidate global
          optimum

    pval: p-value of the test

_D_e_t_a_i_l_s:

     _dehaan_ computes a (1-p) confidence interval for the global
     optimum of a likelihood surface from a vector of user-supplied
     randomly chosen likelihood values.   A user-supplied candidate
     maximum likelihood value is evaluated against confidence interval.

_V_a_l_u_e:

     _dehaan=TRUE_ if the candidate value is greater than the (1-p)
     confidence interval for the true optimum.

_N_o_t_e:

     The choice of the vector of user-supplied random likelihood values
     is somewhat of an art.  The parameter space should be large enough
     to capture the true optimum, but not so large as to include
     illegal or impractical parameter values.

     The generated confidence interval depends on the number of random
     draws from the parameter space.  The number should be large enough
     to allow the application of asympotic theory.  A recommended
     number of random evaluations of the likelihood function at a given
     set of parameter values is 500 or more, but depends on the size of
     the parameter space.  Users are encouraged to experiment with more
     and less draws and observe results.

_A_u_t_h_o_r(_s):

     Micah Altman Micah_Altman@harvard.edu <URL:
     http://www.hmdc.harvard.edu/micah_altman/> , Michael P. Mcdonald

_R_e_f_e_r_e_n_c_e_s:

     Altman, M., J. Gill and M. P. McDonald.  2003.  _Numerical Issues
     in Statistical Computing for the Social Scientist_.  John Wiley &
     Sons. <URL: http://www.hmdc.harvard.edu/numerical_issues/>

     de Haan, L. 1981.  ``Estimation of the Minimum of a Function Using
     Order Statistics.'' _Journal of the American Statistical
     Association_ *76*, 467-9.

     Veall, M. R. 1990.  ``Testing for a Global Maximum in an
     Econometric Context.'' _Econometrica_ *58* 1459-65.

_E_x_a_m_p_l_e_s:

       # The deHaan test is constructed as a maximum likelihood
       # test, with negative values for the likelihood.  The BOD problem
       # is a non-linear least squares minimization problem.  This test
       # is implemented using the negative of the sum of squares for consistency
       # with the deHaan framework of maximum likelihood.

       BOD <-
     structure(list(Time = c(1, 2, 3, 4, 5, 7), demand = c(8.3, 10.3, 
     19, 16, 15.6, 19.8)), .Names = c("Time", "demand"), row.names = c("1", 
     "2", "3", "4", "5", "6"), class = "data.frame", reference = "A1.4, p. 270")
       stval<-expand.grid(A = seq(10, 100, 10), lrc = seq(.5, .8, .1))
       llfun<-function(A,lrc,BOD)
             -sum((BOD$demand - A*(1-exp(-exp(lrc)*BOD$Time)))^2)
       lls<-NULL 
       for (i in 1:nrow(stval))  {
          lls = rbind(lls, llfun(stval[i,1], stval[i,2],BOD))
       }
       fm1 <- nls(demand ~ A*(1-exp(-exp(lrc)*Time)),  
                     data = BOD, start = c(A = 20, lrc = log(.35)))
       ss = -sum(resid(fm1)^2)
       dehaan(lls, ss)

