ghyper.types            package:SuppDists            R Documentation

_K_e_m_p _a_n_d _K_e_m_p _g_e_n_e_r_a_l_i_z_e_d _h_y_p_e_r_g_e_o_m_e_t_r_i_c _t_y_p_e_s

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

     Generalized hypergeometric types as given by Kemp and Kemp

_T_w_o-_w_a_y _t_a_b_l_e:

     The basic representation is in terms of a two-way table:

        x    k-x    k
       a-x  b-k+x  N-k
        a     b     N

     and the associated hypergeometric probability P(x)=choose(a, x)
     choose(b, k-x) / choose(N, k).

     The types are classified according to ranges of a, k, and N.

_K_e_m_p _a_n_d _K_e_m_p _t_y_p_e_s:

     Minor modifications in the definition of three of the types have
     been made to avoid numerical difficulties. Note, J denotes a
     nonnegative integer.

       [Classic]                         
                                         0<a, 0<N, 0<k
                                         integers: a, N, k.
                                         max(0,a+k-N) <= x <= min(a,k)
       [IA(i)] (Real classic)            at least one noninteger parameter
                                         0<a, 0<N, 0<k, k-1<a<N-(k-1)
                                         integer: k
                                         0 <= x <= k
       [IA(ii)] (Real classic)           at least one noninteger parameter
                                         0<a, 0<N, 0<k, a-1<k<N-(a-1)
                                         integer: a
                                         0 <= x <= a
                                         Interchanging a and k transforms this to type IA(i)
       [IB]                              
                                         0<a, 0<N, 0<k, a+k-1<N, J < (a,k) < J+1
                                         integer: 0 <= J
                                         non-integer: a, k
                                         0 <= x ...
                                         NOTE: Kemp and Kemp specify -1<N.
                                         No practical applications for this distribution.
       [IIA] (negative hypergeometric)   
                                         a<0, N<a-1,0<k
                                         integer: k
                                         0 <= x <= k
                                         NOTE: Kemp and Kemp specify N<a and  N!=(a-1)
       [IIB]                             
                                         a<0, -1<N<k+a-1, 0<k, J < (k,k+a-1-N) < J+1
                                         non-integer: k
                                         integer: 0 <= J
                                         0 <= x ...
                                         This is a very strange distribution.  Special calculations were used.
                                         Note: No practical applications.
       [IIIA] (negative hypergeometric)  
                                         0<a,N<k-1,k<0
                                         integer: a
                                         0  <=  x  <=  a
                                         Interchanging a and k transforms this to type IIA
                                         NOTE: Kemp and Kemp specify N<k and N != k-1
       [IIIB]                            
                                         0<a,-1<N<a+k-1,k<0, J<(a,a+k-1-N)<J+1
                                         non integer: a
                                         integer: 0 <= J
                                         0  <=  x ...
                                         Interchanging a and k transforms this to type IIB
                                         Note: No practical applications
       [IV] (Generalized Waring)         
                                         a<0,-1<N, k<0
                                         0  <=  x  ...

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

     Bob Wheeler bwheeler@echip.com

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

     Kemp, C.D., and Kemp, A.W. (1956). Generalized hypergeometric
     distributions. Jour. Roy. Statist. Soc. B. 18. 202-211. 39. 
     887-895.

