PvfPoisson              package:rmutil              R Documentation

_P_o_w_e_r _V_a_r_i_a_n_c_e _F_u_n_c_t_i_o_n _P_o_i_s_s_o_n _D_i_s_t_r_i_b_u_t_i_o_n

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

     These functions provide information about the overdispersed power
     variance function Poisson distribution with parameters 'm', 's',
     and 'f': density, cumulative distribution, quantiles, and random
     generation. This function is obtained from a Poisson distribution
     as a mixture with a power variance distribution. In the limit, for
     'f=0', the mixing distribution is gamma so that it is a negative
     binomial distribution.  For 'f=0.5', the mixing distribution is
     inverse Gaussian. For 'f<0', the mixing distribution is a compound
     distribution of the sum of a Poisson number of gamma
     distributions. For 'f=1', it is undefined.

     The power variance function Poisson distribution with 'm' = mu,
     the mean, 's' = theta, and 'f' = alpha has density

 p(y) = (exp(-m((s+1)^f/s^f-s)/f) / y!) sum_{i=1}^y c_{yi}(f) m^i (s+1)^{if-y} / s^{i(f-1)}

     for y = 0, ..., where 'c_{yi}(f)' are coefficients obtained by
     recursion.

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

     dpvfpois(y, m, s, f, log=FALSE)
     ppvfpois(q, m, s, f)
     qpvfpois(p, m, s, f)
     rpvfpois(n, m, s, f)

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

       y: vector of counts

       q: vector of quantiles

       p: vector of probabilities

       n: number of values to generate

       m: scalar or vector of means

       s: scalar or vector of overdispersion parameters

       f: scalar or vector of family parameters, all < 1

     log: if TRUE, log probabilities are supplied.

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

     J.K. Lindsey

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

     'dpois' for the Poisson, 'ddoublepois' for the double Poisson,
     'dmultpois' for the multiplicative Poisson, 'dconsul' for the
     Consul generalized Poisson, 'dgammacount' for the gamma count, and
     'dnbinom' for the negative binomial distribution.

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

     dpvfpois(5,10,0.9,0.5)
     ppvfpois(5,10,0.9,0.5)
     qpvfpois(0.85,10,0.9,0.5)
     rpvfpois(10,10,0.9,0.5)

