expexp                 package:VGAM                 R Documentation

_E_x_p_o_n_e_n_t_i_a_t_e_d _E_x_p_o_n_e_n_t_i_a_l _D_i_s_t_r_i_b_u_t_i_o_n

_D_e_s_c_r_i_p_t_i_o_n:

     Estimates the two parameters of the exponentiated exponential
     distribution by maximum likelihood estimation.

_U_s_a_g_e:

     expexp(lshape = "loge", lscale = "loge",
            eshape=list(), escale=list(),
            ishape = 1.1, iscale = NULL, 
            tolerance = 1.0e-6, zero = NULL)

_A_r_g_u_m_e_n_t_s:

lshape, lscale: Parameter link functions for the shape and scale
          parameters. See 'Links' for more choices. The defaults ensure
          both parameters are positive.

eshape, escale: List. Extra argument for each of the links. See 'earg'
          in 'Links' for general information.

  ishape: Initial value for the shape  parameter. If convergence fails
          try setting a different value for this argument.

  iscale: Initial value for the scale parameter. By default, an initial
          value is chosen internally using 'ishape'.

tolerance: Numeric. Small positive value for testing whether values are
          close enough to 1 and 2.

    zero: An integer-valued vector specifying which linear/additive
          predictors are modelled as intercepts only. The default is
          none of them. If used, choose one value from the set {1,2}.

_D_e_t_a_i_l_s:

     The exponentiated exponential distribution is an alternative to
     the Weibull and the gamma distributions. The formula for the
     density is

 f(y;shape,scale) = shape scale (1-exp(-scale y))^(shape-1) exp(-scale y)

     where y>0, shape>0 and scale>0. The mean of Y is
     (psi(shape+1)-psi(1))/scale (returned as the fitted values) where
     psi is the digamma function. The variance of Y is
     (psi'(1)-psi'(shape+1))/ scale^2 where psi' is the trigamma
     function.

     This distribution has been called the two-parameter generalized
     exponential distribution by Gupta and Kundu (2006). A special case
     of the exponentiated exponential distribution: shape=1 is the
     exponential distribution.

_V_a_l_u_e:

     An object of class '"vglmff"' (see 'vglmff-class'). The object is
     used by modelling functions such as 'vglm' and 'vgam'.

_W_a_r_n_i_n_g:

     Practical experience shows that reasonably good initial values
     really helps. In particular, try setting different values for the
     'ishape' argument if numerical problems are encountered or failure
     to convergence occurs. Even if convergence occurs try perturbing
     the initial value to make sure the global solution is obtained and
     not a local solution. The algorithm may fail if the estimate of
     the shape parameter is too close to unity.

_N_o_t_e:

     Fisher scoring is used, however, convergence is usually very slow.
     This is a good sign that there is a bug, but I have yet to check
     that the expected information is correct. Also, I have yet to
     implement Type-I right censored data using the results of Gupta
     and Kundu (2006).

     Another algorithm for fitting this model is implemented in
     'expexp1'.

_A_u_t_h_o_r(_s):

     T. W. Yee

_R_e_f_e_r_e_n_c_e_s:

     Gupta, R. D. and Kundu, D. (2001) Exponentiated exponential
     family: an alternative to gamma and Weibull distributions,
     _Biometrical Journal_, *43*, 117-130.

     Gupta, R. D. and Kundu, D. (2006) On the comparison of Fisher
     information of the Weibull and GE distributions, _Journal of
     Statistical Planning and Inference_, *136*, 3130-3144.

_S_e_e _A_l_s_o:

     'expexp1', 'gamma2.ab', 'weibull'.

_E_x_a_m_p_l_e_s:

     # A special case: exponential data
     y = rexp(n <- 1000)
     fit = vglm(y ~ 1, fam=expexp, trace=TRUE, maxit=99)
     coef(fit, matrix=TRUE)
     Coef(fit)

     # Ball bearings data (number of million revolutions before failure)
     bbearings = c(17.88, 28.92, 33.00, 41.52, 42.12, 45.60,
     48.80, 51.84, 51.96, 54.12, 55.56, 67.80, 68.64, 68.64,
     68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92,
     128.04, 173.40)
     fit = vglm(bbearings ~ 1, fam=expexp(iscale=0.05, ish=5),
                trace=TRUE, maxit=300)
     coef(fit, matrix=TRUE)
     Coef(fit)   # Authors get c(shape=5.2589, scale=0.0314)
     logLik(fit) # Authors get -112.9763

     # Failure times of the airconditioning system of an airplane
     acplane = c(23, 261, 87, 7, 120, 14, 62, 47,
     225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14,
     71, 11, 14, 11, 16, 90, 1, 16, 52, 95)
     fit = vglm(acplane ~ 1, fam=expexp(ishape=0.8, isc=0.15),
                trace=TRUE, maxit=99)
     coef(fit, matrix=TRUE)
     Coef(fit)   # Authors get c(shape=0.8130, scale=0.0145)
     logLik(fit) # Authors get log-lik -152.264

