Note on discrete $\alpha$-unimodality
Citation for published version (APA):
Steutel, F. W. (1987). Note on discrete $\alpha$-unimodality. (Memorandum COSOR; Vol. 8722). Technische Universiteit Eindhoven.
Document status and date: Published: 01/01/1987 Document Version:
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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science
COS OR-memorandum 87-22 Note on discrete
a-unimodality F. W. Steutel
Eindhoven University of Technology
Department of Mathematics and Computing Science P.O. Box 513
5600 MB Eindhoven The Netherlands
Eindhoven, september 1987 The Netherlands
NOTE ON DISCRETE a-UNIMODALITY
Abstract. An interpretation is given of Abouammoh's (1987) discrete analogue of the (one-dimensional) definition of a-unimodality by Olshen and Savage. The con-cepts are of some interest in the context of self decomposability.
1. Discrete a-monotonicity
Since in all unimodality concepts considered here the modes
are
at zero, some confusion may be avoided at very little loss of generality by restricting ourselves to distributions on the half-line, i.e., by considering monotonicity rather than unimodality. We shall especially be interested in distributions on IV 0 := {O, 1,2, ... }.It is well known that a nonnegative random variable (IV) X has a monotone density if and only if
d
X::;: UY • (1)
where U and Y
are
independent, Y is nonnegative and U is uniformly distributed on (0,1). Olshen and Savage (1970), among other things, define a one-dimensional IV to have an a-unimodal distribution (or, to be a-a-unimodal, for short) ifd
X
=
Ullay , (2)where a > 0. Of course, if Y in (2) is nonnegative, then X is said to be a-monotone.
Recently, Abouammoh (1987) has given an analogue of a-unimodality for distributions on the integers, i.e. of a-monotonicity at both sides of zero. It turns out that his definition is essen-tially equivalent to one that is quite similar to the definition in (2).
For any IN o-valued IV Y let Py denote its probability generating function (pgf). Then in Steutel
and Van Ham (1979) the multiplication u 0 Y is defined (in distribution) by P,. flY(Z)
=
Py(l - u + uz) .2
-Definition. An IV -valued or X is said to be (discrete) a~monotone if
Ii
X = Ulla 0 Y , (3)
where U and Y
are
independent, Y is IVo~valued and U is uniformly distributed on (0,1); equivalently1
Px (z) = a
J
Pr (1 - v + vz )vD-1dvo
Relation (4) can be put in the form
1
Px(Z) = a(1-z)-a
J
P y(w)(1 - w)a-1dw , z(4)
(5)
which is equivalent to relation (2.6) of Abouammoh (1987) in terms of characteristic functions. In the following proposition our definition is seen to be essentially equivalent to the, somewhat artificial. defining inequalities of Abouammoh (1987) for discrete a-unimodality (Le., a~ monotonicity at both sides of zero).
Proposition 1. A distribution (P,,)O' on IVo is a~monotone if and only if
< n+a
P,,+l - n+l p" (n
=
0,1,2, ... ) .00 00
Proof. Writing Px(z)
=
LP"Z" and Py(z) = Lq,.z", and usingo 0
equivalent to (4) we obtain, by equating the coefficients of
z,.:
r(n+a) 00 k!PI!
=
a n !f
r(k+a+l) qi;(6)
(7)
for n = 0,1, ... From (7) the inequalities (6) follow, since the qi;
are
nonnegative. Conversely, the existence of nonnegative qi; satisfying (7) easily follows from (6).Corollary. If (P,.)O' is a~monotone. then it is a/~monotone for every a' > a.
3
-Remark. The concept of a-mono tonicity is not restricted to probability distributions; through (6) or (7) it extends to arbitrary sequences of nonnegative numbers.
2. Application
In Steutel and Van Ham (1979) it is proved that a (infinitely divisible) distribution (P,,)O' is self decomposable if and only if the (nonnegative) sequence (r,,)O' defined by
"
(n + l)P,.+l
=
LPl:r,.-.\: (n = 1,2 •... ) • (8)1:=00
is monotone (i.e. nonincreasing). A similar result holds for the Levy measure of a self-decomposable density.
By Jurek (see Jurek (1985» and O'Connor (1979) the concept of self-decomposability was generalized and related to a weaker form of unimodality of the corresponding Levy measures. It turns out that their definition is the special case for a
=
2 of the more general concept of0.-self decomposability introduced by Hansen (1987). For distributions on /No this definition reads: a distribution (P,,)O' on /No with pgf P is called a-self decomposable if for every c e (0,1) a pgf
Pc exists such that a-I
P(z)
=
pc (1- C + cz)Pc(z)Hansen proves the following general result (and an analogous result for densities).
Proposition 2 (Hansen). An infinitely divisible distribution (p,,)O on /No is a-self decomposable if and only if (r,,)O' as defined by (8) is a-monotone.
Remark;. Some confusion arises from the ambiguity of what exactly should be called the Levy measure ({r,,)O' or (r,./ (n + 1»0) and the fact that (rll) is 2-monotone if and only if (rll I (n + 1» is I-monotone.
References
Abouammoh, A.M. (1987), On generalized discrete unimodality, Statistica Neerl. 41, nr .... ,
4
-Jurek, Z.J. (1985), Relations between the s -self decomposable measures and self decomposable measures, Ann. Prob.
li.
592-608.O'Connor, T.A. (1979), Infinitely divisible distributions with unimodal spectral functions, Ann. Prob. 1. 494499.
Olshen, R.A. and L.J. Savage (1970), A generalized unimodality, J. Appl. Prob. 1.21/34.
Steutel, F.W. and K. van Ham (1979). Discrete analogues of self-decomposability and stability, Ann. Prob. 1. 893-899.
F.W. Steutel
Dept of Mathematics and Computing Science Eindhoven University of Technology
P.O. Box 513 5600 MB Eindhoven The Netherlands.
