ABSTRACT
Shanbhtg (1972b) showed, among other things, that for all H (in an open interval)
d X. = TJ Zo , (1'2.1.3) with U as uniformly tiist,ributctl on (0 , l ) indepcnctelitly of Ze and the distributions of X. and Zo forniirig respectively certain exponential farriilics, if arid only if X Q or - X u is exponcntially distributed, with rncail of a specific form, for each U . With a minor modification in the argument of Shanbhag, it follows that replacing, in (12.1.3), l J by with o positive and iirtlependcnt of U , one gets a ch:tracterization of a gamma distribution with index tr in place of that of an ex11oner1tia.l clist,ributioii. Esseritially ;t discrcte analogue of the lalter result, following implicitly Shanbhag's idea, has recently been given by Sapatirias (1993, 1999). [Incitlcntally, Alarnastaz (1985) has extended yet mother rcsult in Shankhag (1972b), which is not directly linked with the res~llts in our paper; for a bibliography of the literature relevaril to thc Alarnastaz rcsnlt, sec Pakcs (1992, 1994) and R;to arid Sharibhag (1994, C l i q~ t e r 61.1
Characterkm tions of Sonic Expontmtial Families 211
The purpose of the present paper is to unify and extend the aforementioned rcsldts of S h a i ~ b h g and Sapatinas, and sl-low that these results are linked with certain results of Laha and Lukacs (1960), Sllanbhag (1972~~) 1979), and Morris (1982). In view of the ohservatiorl that we havc made above coriccrning the te-miimodality in the discrete case, it follows that the result of Sapatiriits referred to is linked with a certain result on tlamage rnotlels; we also arrive at variations of this latter rcsult of the type in Shanbl-iag and Clark (1972) arid Patil and R.atnapa.rkhi (1975, 1977).
