Some remarks on mixtures of distributions : (paper 35th international statistical institute conference, belgrado 1965)
W. Molenaar · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · 1965
We shal.ldenote by ,!e the random variable with distribution function F 8 1 ) Random variables are underlinedo 1--This is a special case of TEICHER 9 s definition o~ m-paremeter • m.ixturesoIn this report the distribution f'\1nction H is al.ways one-• • • d:i.mensional 9 though F 8 may have more than one pa.ramateroThe extra 8 under the sign '' /l'' is convenient in this caseo Sometimes we shall 'Write F 8 ~ H 1 though we could have incl.uded the constant c in the distribution f:.mctionHo The symbol 8 is also used between nam~s of distributionso Several well-known mixtures are listed in Appendix 1e Two examples of' mixtures are • Binomi~ • (n,p) ./}Poisson ( ll) a: Poisson ( l!p); Poisson (kP) Poisson (A)• Neyman A(A,P)o • We shall call a mixture non-trivial if neither H nor a11.F 8 are degenerate distribution f'tJnetionso For degenerate H the mixt 1 1re is just one F 8 ; any distribution H could be written as a mixture by It is obvious that for example the chare.cteristicf'1;inction and the moments about zero ( if existing) are mix-l.inearwith respect to (wQrot•) F 8 ~ H, ioeo i:f" ♦ 8 is the characteristic :f'l.;anctiono:f" F 8 and 'P of F 8 /J H 1 then ♦ (t) • J + 8 (t) d.H(e); (3) k E.:a.Definition 2o If theJ ran~om variable x has distribution fl1nction F and the non-negative integer-valued• J;"andom.vari.ab1ez.. has distribution t\mction G, the 9-seneral;ilz,ed :f-q.is~r~b"4-ti,?n, with a.istribution f\lnction denoted by F0-9 is the distribution of 2) GURLAND uses F.AH and xJ.80We have preferred to use another notation, more suggestive of the \.... .:..derlying idea and avoiding the possibly confusing suggestion of syr1011etry0 3) .c denotes the unit stepfunctiono -2 -'s examples, where F is infinitely divisible, g(f(z)) is al.ways a generating function 9 but this is not correct in a.ll' cases with F(O-) • O and G(O-) • Oo A co11nterexample is f'(z) • pz + q