freund61                package:VGAM                R Documentation

_F_r_e_u_n_d'_s (_1_9_6_1) _B_i_v_a_r_i_a_t_e _E_x_t_e_n_s_i_o_n _o_f _t_h_e _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:

     Estimate the four parameters of the Freund (1961) bivariate
     extension of the exponential distribution using maximum likelihood
     estimation.

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

     freund61(la="loge", lap="loge", lb="loge", lbp="loge",
              ia=NULL, iap=NULL, ib=NULL, ibp=NULL, 
              independent=FALSE, zero=NULL)

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

la,lap,lb,lbp: Link function applied to the (positive) parameters
          alpha, alpha', beta and beta', respectively (the ``'p'''
          stands for ``prime''). See 'Links' for more choices.

ia,iap,ib,ibp: Initial value for the four parameters respectively. The
          default is to estimate them all internally.

independent: Logical. If 'TRUE' then the parameters are constrained to
          satisfy alpha=alpha' and beta=beta', which implies that y1
          and y2 are independent and each have an ordinary exponential
          distribution.

    zero: An integer-valued vector specifying which linear/additive
          predictors are modelled as intercepts only. The values must
          be from the set {1,2,3,4}. The default is none of them.

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

     This model represents one type of bivariate extension of the
     exponential distribution that is applicable to certain problems,
     in particular, to two-component systems which can function if one
     of the components has failed. For example, engine failures in
     two-engine planes, paired organs such as peoples' eyes, ears and
     kidneys. Suppose y1 and y2 are random variables representing the
     lifetimes of two components A and B in a two component system. The
     dependence between y1 and y2 is essentially such that the failure
     of the B component changes the parameter of the exponential life
     distribution of the A  component from alpha to alpha', while the
     failure of the A  component changes the parameter of the
     exponential life distribution of the B  component from beta to
     beta'.

     The joint probability density function is given by

 f(y1,y2) = alpha * beta' * exp(-beta' * y2 -  (alpha+beta-beta') * y1)

     for 0 < y1 < y2, and

 f(y1,y2) = beta * alpha' * exp(-alpha' * y1 -  (alpha+beta-alpha') * y2)

     for 0 < y2 < y1. Here, all four parameters are positive, as well
     as the responses y1 and y2. Under this model, the probability that
     component A is the first to fail is alpha/(alpha+beta). The time
     to the first failure is distributed as an exponential distribution
     with rate alpha+beta. Furthermore, the distribution of the time
     from first failure to failure of the other component is a mixture
     of Exponential(alpha') and  Exponential(beta') with proportions
     beta/(alpha+beta) and alpha/(alpha+beta) respectively.

     The marginal distributions are, in general, not exponential. By
     default, the linear/additive predictors are eta1=log(alpha),
     eta2=log(alpha'), eta3=log(beta), eta4=log(beta').

     A special case is when alpha=alpha' and beta'=beta', which means
     that y1 and y2 are independent, and both have an ordinary
     exponential distribution with means 1/alpha and 1/beta
     respectively.

     Fisher scoring is used, and the initial values correspond to the
     MLEs of an intercept model. Consequently, convergence may take
     only one iteration.

_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'.

_N_o_t_e:

     To estimate all four parameters, it is necessary to have some data
     where y1<y2 and y2<y1.

     The response must be a two-column matrix, with columns y1 and y2.
     Currently, the fitted value is a matrix with two columns; the
     first column has values (alpha'+beta)/(alpha' * (alpha+beta)) for
     the mean of y1, while the second column has values
     (beta'+alpha)/(beta' * (alpha+beta)) for the mean of y2. The
     variance of y1 is

 [(alpha')^2 + 2 * alpha * beta + beta^2]/ [(alpha')^2 * (alpha + beta)^2],

     the variance of y2 is

 [(beta')^2 + 2 * alpha * beta + alpha^2]/ [(beta')^2 * (alpha + beta)^2],

     the covariance of y1 and y2 is

 [alpha' * beta' - alpha * beta]/ [alpha' * beta' * (alpha + beta)^2].

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

     T. W. Yee

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

     Freund, J. E. (1961) A bivariate extension of the exponential
     distribution. _Journal of the American Statistical Association_,
     *56*, 971-977.

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

     'exponential'.

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

     y1 = rexp(n <- 200, rate=4)
     y2 = rexp(n, rate=8)
     ymat = cbind(y1,y2)
     fit =  vglm(ymat ~ 1, fam=freund61, trace=TRUE)
     coef(fit, matrix=TRUE)
     Coef(fit)
     vcov(fit)
     fitted(fit)[1:5,]
     summary(fit)

     # y1 and y2 are independent, so fit an independence model
     fit2 = vglm(ymat ~ 1, fam=freund61(indep=TRUE), trace=TRUE)
     coef(fit2, matrix=TRUE)
     constraints(fit2)
     1 - pchisq(2*(logLik(fit)-logLik(fit2)), df=2) # p-value

