E L S E V I E R Statistics & Probability Letters 23(1995)157-164

Contorted uniform and Pareto distributions

B o y a n Dimi trov a, *, Elart v o n Col lani b

a Department of Science and Mathematics, GM1 Engineering and Management Institute, 1700 West Third Avenue Flint, M1 48504-4898, USA

b Institut flit Angevandte Mathematik und Statistik, Wiirzburg University, Sanderring 2, D-97070 Wfirzburg, Germany

Received January 1994; revised March 1994

Abstract

Random variables having the multiplicative almost-lack-of-memory property are introduced. An explicit representation of distributions having this property is derived and some of their properties are studied. These distributions appear to be extensions of the uniform and Pareto distributions.

Keywords: Multiplicative lack of memory; Uniform distribution; Pareto distribution; Characterization theorems; Exponential growth rate and random lifetime.

1. Introduction

It is well known (see Galambos and Kotz, 1978, p.19) that the equation

P{X~ <~ uVlXl <~ v } = P { X l ~< u}, 0 ~< u, 0 ~< v ~< 1 ( la )

characterizes the uniform distribution on the interval [0, 1] of a random variable Xl. Also the equation

P{X2 >~ uvlX2 >>- v } = P { X 2 >1 v}, u,v > 1 ( lb)

characterizes the Pareto distribution

< x } = / 0 1 x < 1, Fx2(x) = P{X2 - x - s , x >/ 1, ( 2 )

where V > 0 is an arbitrary parameter. We say, Eqs. ( la ) and ( lb ) express the multiplicative lack of memory (MLM) property of random variables having either the uniform or Pareto distribution.

We set the question: if release the conditions in the MLM property, e.g. requiring it to be true just for a discrete set of values of v, how does it affect the above classes of uniform and Pareto distributions?

*Corresponding author.

0167-7152/95/$9.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0167-7 152( 94 )00 1 0 8 - K

158 B. Dimitrov, E. yon Collanil Statistics & Probability Letters 23 (1995) 157-164

o o First we require the MLM property to be valid for an infinite sequence of distinct values {V~}n= 1 for the parameter v in the condition, where the MLM property holds for any value of the other parameter u. In this case we say, the r.v. X has "the multiplicative almost lack of memory" (briefly MALM) property. Then we look for conditions that imply this property. We derive the form of probability distributions with MALM property, and discuss its equivalent representations in terms of corresponding p.d.f.'s and functions of independent r.v.'s. Since the uniform and Pareto distributions belong to the MALM family of probability distributions, we consider also their identification and possible generalizations, that are called contorted distributions.

We investigate random variables with the MALM property, following an idea introduced by Chukova and Dimitrov (1992) and discussed in Chukova et al. (1993), where the additive almost lack of memory property is worked out.

2. The MALM property and the corresponding distributions

We start our considerations with a definition that gives formal generalization of the MLM property. Through- out the paper only non-negative random variables (r.v.) are in consideratiom

Definition 1. (i) A r.v. X1 is said to have the multiplicative almost lack of memory property of type 1 V (x~ (MALM1) iff there exist a sequence of numbers { n}n=l, 0 < v, ~< 1, vn ~ Vm for n ~ m, such that

P{X1 <~ uv~ [X, <~ v,} = P{X, <~ u} for a l l u > / 0 .

(ii) A r.v. X2 is said to have the multiplicative almost lack of memory property Of type 2 (MALM2) iff there exists a sequence {v~}~,, v~ /> 1, v, /> vm for n ~ m, such that

P{X2 >>- uv, lX2 >- vn} : P{X2 >- u} for all u i> 1.

The requirements of Definition 1 need determining the sequence {v~},~ l in each of the two cases. We say, the r.v. X has the MLM property at the point vn, if for the fixed value of v. an equation of Definition 1 holds. The next definition just simplifies the notations of such cases.

Definition 2. (i) We say, a r.v. X1 belongs to the class Kl(b), iff for a given real number b E (0 , ! ) it is true that

P{X, <~ ubl X , ~ b}=P{X , <~ u} for a l lu /> 0 ;

(ii) A random variable )(2 belongs to the class K2(b), iff for a given real number b > 1 it is true that

P{X2 >1 ublX2 >I b}=P{X2 >. u} for a l l u ~> 1.

For any Xl E K,(b) the following holds.

Lemma 1. (i) X, E K,(b) implies P{X, <~ 1} = 1, ( i i))(2 E K2(b) implies P{X2 >I 1} = 1.

Proof. For )(1 E K,(b) there exists a number v < 1 (at least it is v = b) with

P{X, <~ uvlX, <~ v}=P{Xl <~ u} for a l lu /> O.

Hence, for u ~> 1 we obtain

B. Dimitrov, E. yon Collanil Statistics & Probability Letters 23 (1995)"157-164 159

P{x~ < u} = P{X~ < uv. x , < v} = 1. P{X1 <<. v}

Analogously (ii) is proved for Xz E Kz(b). []

Next we prove the following fundamental result.

Lemma 2. (i) IfX~ E Ki(b), then X~. have MALM(1)-property with the sequence {v,},~ 1 = {b"},~ 1, i = 1,2.

Proof . (i) We give the proof for the case i = 1, where b < 1. It is true that )(1 E Kl(b) is equivalent to

P{X1 <~ ub} = P{XI <~ ublY, <~ b} • P{X~ <<. b} = P{X, ~ u} • P{X, <<. b} .

We have to show that

P{X1 <~ u . b"lYa <~ b n} =P{X1 ~< u} (3)

holds for all u /> 0. Let u >/ 1. Then the following holds:

P{X~ <~ ub" IX~ <<. b"} = P{X~ <~ ub",X~ <~ b"} = 1. (4) P(X1 ~<b"}

Hence (3) follows from Lemma l(i). It remains to show that (3) holds also for 0 < u < 1 which is proved by induction. Eq. (3) follows for n = 0 because of (4), and, for n = 1, by definition of Kl(b). Assume that (3) holds

for some n > 1, and observe that

P{X1 <~ b n} =P{X1 <<. bn- lblXl <~ b}P{Xl <~ b} = P { X l <~ bn-1}p{xl <<. b}

. . . . . [P{X, < 6}1".

Then for 0 ~< u < 1 it is true that

P{Xl <<. (ub)b") _ P{X1 <<. ub n+'lXl <<. 6 "+1 } = P{X1 <<. b n+~} -

P{X1 <<. ub}P{Xl <<. b n}

P{X~ < b}e{x, < b.}

= P{X1 <. ublX1 ~< b} = P{Xl <~ u} ,

proving the (i = 1 ) part o f the lemma. The proof of the case (i = 2) o f Lemma 2 is performed analogously, by essentially interchanging the direction of inequality signs ~< by />. Hence, it is omitted here. []

N e x t we derive the following representation theorem for random variables having the MALM property.

Theorem 1. (i) A r.v. X1 belongs to the class Kl(b) with given b E (0, 1) /ff its cumulative distribution function (c.d. f) Fx,(X) : P{XI <~ X } has the form

0 orx<.O,

Fxl(X) = ~ [ ~ l + ( 1 -- c¢1)F1 f o r b n+l < x ~ b n, n = 0 , 1 , 2 . . . . . (5)

1 f o r x >1 1 .

Here Fl ( t ) is the c.d.f o f a r. v. concentrated on the interval ( b, 1] with probability 1, i.e. Fl ( b ) = 0, F I (1 ) = 1, and ~1 E (0, 1 ) is an arbitrary parameter.

160 B. Dimitrov, E. yon Collanil Statistics & Probability Letters 23 (1995)157-164

(ii) A r.v. X2 belongs to the class K2(b) with given b > 1 iff its survival function Gx2(X)= P{X2 >1 x} is represented by the equation

1 f o r x < 1 Gx2(X) = ' (6)

~n[~2 + (1 - ~2)Gz(xb-m)] f o r b m ~< x < b re+l, m = 0, 1,2 . . . .

Here Gz(t) is the survival function of a r.v. with range on the interval [1,b), i.e. G2(1) = 1. G2(b) = O, and ~2 E (0, 1) is an arbitrary parameter.

Proof. We give the proof for the case (ii), b > 1. The result for b < 1 can be proven analogously. In view of Lemma 1 it is sufficient to consider u > 1. For any given u > 1 there exists an unique non-negative integer mu, and an unique real number tu E [1,b), such that u E [bm",b'nu+l), is fulfilled and u = b'n"t,.

First we prove, that if X2 has a survival function given by (6), then it has the MALM2 property with sequence o f constants v, = b", n = 0, 1,2 . . . .

We have, by assumption

Gx2(ub) = P{X2 ~ ub} = P{X2 >1 bm"+ltu} = 0C~n"+l[~2 q- (1 -- ~2)G2(ub-m")],

and P{X2 ~> b} = ~2. Thus,

P{X2 >1 ub [)(2 >>- b } - Gx2(ub) - ~ ' u [ ~ z + ( 1 - ~2)G2(ub-"u)] = P{Xz >>- u} . O~ 2

Hence, by Lemma 2, X2 has the MALM2 property. Next we prove, that if X2 has MALM2 property wi th constants v, = b n, n = 0, 1,2 . . . . . then its c.d.£ is

determined by a survival function o f the form (6). Using the MALM2 property and the unique representation u = b'nut,, where tu E [1,b), we obtain

Gx2(u) = P{X2 >1 u} = P{X2 >>- bm"tu} = P{X2 >i b'n~tu,X2 >1 b rn" }

= e{x2 >1 bmut~ IX2 >1 b~u}e{X2 >>- b.u} = e{X2 >1 t~}l"{X2 t> b "u} = ~g'~Cx2(t~), (7)

where ~2 = P{X2 >>. b}. Introduce the conditional survival function

62(t) := P{x2/> t lX2 < b} = P{ t ~ x2 < b} _ _ P{X2 >t t } - t " { X 2 >. b} _ 6 x 2 ( t ) - ~ 2 P{X2 < b} 1 - P { X z >1 b} 1 - ~2

Hence, for t 6 [1,b), we have

Gxz(t ) = 0~ 2 -q- ( 1 - ~ 2 ) G z ( t ) •

Thus, with (7), we finally obtain the representation

Gxz(u )=~u[~2+(1 -~2)G2(ub -m") for u E [bm",bm"+l). []

With Theorem 1 we immediately derive statements for the corresponding probability densities (p.d.f.) o f r.v. 's having the MALM property.

Corollary 1. A continuous r.v. X1 has the M A L M property iff its p.d.f has either the form

0 ~ f o r u < Ooru > 1,

f x l (u ) = ~n(1 - ~x) f o r u E (bn+l,bn), n = O, 1 . . . . . 0 < b < 1 ,

B. Dimitrov, E. yon Collanil Statistics & Probability Letters 23 (1995)157 164 161

o r

c¢)f2(~b f o r u < 1, 0 -m) f o r u E (bin, bin+l), rn fxz(U) = ~m(1 - = 0, 1 . . . . . b > 1 .

Here f i are the p.d.f, o f the c.d.f. Fi, i = 1,2, presented in (5) and (6). In order to express the p.d.f. 's o f discrete MALM distributions, we introduce the following notations: For

a fixed positive number b ¢ 1 and any other positive x by n(x, b) is denoted the integer part o f the ratio (lnx)/(lnb). The integers n(x,b) are non-negative, i f simultaneously b and x are either both less, or both larger than 1. Then the representation x = bn(X'b)y holds either with y E (b, 1] or with y E [1,b), and is equivalent to either x E (b"+l ,b n] when b < 1 or to x E [bn, b n+l ) i f b > 1, respectively. Now the following is also the conclusion of Theorem 1.

Corollary 2. A discrete r.v. X possesses the M A L M property iff the set of its possible values has the form

X = { x : x = Xm,k = bein(xm'b)yk} ,

where ej = 1, e2 = -1 , and the numbers Yk, k = 0,1,2 . . . . are within the interval (b, 1] ( / f b < 1) and within [1,b) ( / f b > 1), and its p.d.f is given by the equation

f xi(Xm,k ) = ~n(Xm.k'6)(1 _ ot)f i(Xm, kb--Em(Xm.k,b)) .

Here f i(Yk), k = 0, 1,2 . . . . . i = 1,2 are the discrete p.d.f, o f the c.d.f. Fi, i ~- 1,2, presented in (5) and (6), respectively.

Random variables having the MALM property can be represented as functions o f two specially selected and independent r.v.'s. The following decomposition theorem holds.

Theorem 2. A r.v. X, belongs to the class Ki(b) iff there exist two independent r.v.'s Yi and Zi such that X~ = YibZi,i = 1,2, is fulfilled.

Here Y1 with probability 1 is located on the interval (b, 1 ], Yz with probability 1 takes its values within the interval [1, b), and Zi have geometric distribution P { Zi = n} = (1 - ~i )~n, n = O, 1, 2 . . . .

Proof. Let X/ be a r.v. represented by either Eq. (5) or (6). Let X,* = Yib zi be as stated in the theorem. We show that for i = 1,2 the Laplace-Stieltjes transforms of X/ and X/* (i = 1,2) are identical.

Using the properties o f conditional expectation we obtain

go go

~Oxp(s) = E[e -sbzir'] = Z E[e-sbziYi l zi = n] • P{Zi = n} = (1 - ~i) Z ~nq~r'(sbn)" (8)

n=O n=O

(i) For )(1 we have 0 < b < 1 and

f. = y ~ e-SUdFxt (ub -n) q~xl(s) = Ee -~xl ~ e-~"dFx,(u) = (1 ~ I ) Z b-~ +1

= J b n + l n=0

go f b I go = (1 -- ~ I ) Z ~ ~ e-Sb"tdF,(t) = (1 -- cq)Zot~(pl(sbn), n=0 n=0

where ~01( • ) denotes the Laplace-Stieltjes transform of Fl . Compare the last expression with (8) and confirm the statement. Analogous calculations also give ¢p2(s) = q~x*(S). []

162 B. Dimitrov, E. yon Collanil Statistics & Probability Letters 23 (1995) 157-164

From the Laplace-Stieltjes transform (8) we immediately get expressions for the expected value and other moments of X'.

Corollary 3. (i) fiX,. E Ki(b) and #k = f(b, 1] XkdFi(x)' then

e [ x / k ] = 1 - o,i 1 - oqb kirk' k = 1,2 . . . .

Moreover, the moments of order k o f X2 exist i f and only i f 0~2 b k < 1. From Theorem 2 we obtain the following interpretation of the r.v.'s that have MALM property. Xi are resulting o f two independent random experiments: the first one, described by inteoer valued random

variable Zi, determines the segment (b "+l, b ~] or [b", b n+l ), respectively, where X,. will be realized. The second one, described by the random variable Yi (with Y1 E (b, 1] and Y2 E [1,b) respectively) determines what actual value bnt within the predetermined segment will occur, accordino to the event {Yi = t}.

A simple example. Consider a phenomenon with exponential growth rate and random lifetime (e.g. overall interest in financial business). Let n denote the number of growth periods and b be the growth rate. Then at the beginning of the nth period, there is an overall growth rate of b n. After completion of the nth growth period there will be an overall growth rate of b n+l. The probability to survive a given growth period is ~i.

Suppose that the lifetime of the phenomenon terminates at random during the nth period. Then the overall growth rate X adopts a value between b n and b n+l. Its actual value is given by the outcome of the second random experiment described by the random variable Yi and representing the conditional distribution of the accumulated growth rate given that it is terminated during the same period.

3. Examples

As remarked at the beginning, each distribution from Kl(b) can be considered as an extension of the uniform distribution on [0, 1] for which the MALM1 property is satisfied with arbitrary b E (0, 1). Similarly, each distribution from K2(b) can be interpreted as an extension of the Pareto distribution (2). Hence, we first give representations of the uniform and the Pareto distribution as elements of Kl(b) and KE(b), respectively.

3.1. Contortion of the uniform distribution

Take an arbitrary constant b, 0 < b < 1. Define

{ 1 ~ - for t ~< b , F l ( t ) = t - b forb < t ~< 1,

for t~> 1.

(a) For ~t = b, obtain with (5) the uniform distribution on [0, 1]:

Fx l (u )=bn b + ( 1 - b ) ~ = u .

(b) Let • ~ b, and Fl(t) be given by (9). Then obtain

[ 1 - b)J Fx1(u)=~" ~ + ( 1 - ~ ) V , ~ forb "+l < u ~< b",

(9)

B. Dimitrov, E. yon Collani/ Statistics & Probability Letters 23 (1995) 157-164 163

which is a continuous distribution with density

f x I (u) = { O(~)n ll -°tb elsewheref°ruE(bn+"bn]' n = 0, 1,2, . . .

Hence, )(1 being a r.v. on (0, 1], and having the MALM1 property is not uniformly distributed. We say, X has contorted uniform distribution. Fig. 1 shows the p.d.f, o f such distributions for b = 0.8; ~ = 0.2; ~ = 0.8 and ~ = 0.9.

3.2. Contortion of the Pareto distribution

Take an arbitrary constant b, 1 < b < o¢. Define the survival function

1 f o r t < 1,

t - r _ b - r G2(t) = --i --b---Y for 1 ~< t < b , (10)

O for t >~ b .

( a ) For • : b -~, by (6) and G2 from (10), obtain the survival function

[ u-rbmr-b-r] Gx2(U)=(b-r) m b - r + ( 1 - b - r ) 1-~-b--~ = u - r ' f o r b m ~< u < b m+l.

Thus, in this case the r.v. )(2 C K2(b) has the Pareto distribution over [1, c¢). A complete collection o f properties for Pareto distribution is given by Arnold (1983), and the above

construction is not there. Also the representation )(2 = Y2b zz of Pareto distributed r.v. )(2 in terms of an arbitrary b > 1, and two independent r.v.'s Y2 and Z2 (in correspondence with Theorem 2) is a new property. It is gained from the class of MALM distributions.

(b) Let ~ # b - r and G2(t) be given by (10). Then obtain the following survival function o f X2:

[ u-~'b(m+l'r-1]"~ , b m b-+l Gr,..,(u) : 0~ m c t + ( 1 - ~ ) for ~< u < I

)(2 has the MALM2 property, and is not Pareto distributed. Its density is given by

"VU-r- 1 b(m+l)r [b m, b m+l fx2(U) = cxm(1 -- C¢)" br_ 1 for u E ) , m = 0, 1,2 . . . .

0 elsewhere.

We say )(2 has contorted Pareto distribution. (c) Let the survival function G2(t) be defined, for b > 1, by the equation

1 f o r t < 1, e-~t(x-1) _ e-2(b-l)

G2(t)---- 1 - - e - 2 ( b - 1 ) for 1 ~< t < b ,

0 for t >~ b .

Then the r.v. )(2 defined by G2(t) and (6) has density function

{ cxm(1 -- ~))'e-2<ub-n-l) for u E [bm, b re+l) fx2(u) : bm[1 - e -'~(b-1)] ' m = 0, 1,2 . . . .

0 elsewhere,

164 B. Dimitrov, E. yon Collani/Statistics & Probability Letters 23 (1995) 157-164

possesses the MALM2 property for any ~ E (0, 1), and is not Pareto distributed.

3.3. Discrete extension of Pareto distribution

Next we use the results of Corollaries 2 and 3 for constructing discrete integer-valued distributions with MALM2 properties.

(a) Let b > 1 be fixed integer and

P { Y 2 = k } = I for some f i x e d k E { 1 , 2 . . . . . b - l } .

Then, according to Corollary 2 we have

P{X =kb m } = ~ m ( 1 - c t ) fo rm-- -0 ,1 ,2 . . . .

Hence, )(2 has geometric distribution with parameter ct over the set {1,kb, kbZ,kb 3 . . . . }. (b) Let b = 3 and let

I ql = P { Y 2 = 1 } = P { Y 2 = 2 } = q 2 = ~.

Then for an arbitrary value of ~, we obtain from Corollary 3

P { X 2 = 3 m } = P { X 2 - - 2 . 3 ' n } = ~ ' n ( 1 - c t ) / 2 , m = 0 , 1 , 2 . . . .

This distribution obviously is not geometric and not Pareto. The r.v. )(2 adopts high values very seldom with small probability.

We suggest that each discrete distribution from the class MALM2 is a discrete generalized Pareto distribu- tion. The distribution in 3.3(a) is likely a competitor for the discrete integer-valued Pareto distribution, having its conventional sense.

Possible areas of application of contorted distributions considered here is expected to be the same as that of the original uncontorted distributions, i.e. reliability, quality analysis and control, social sciences physics, finan- cial mathematics and others. We believe that the additional probabilistic, physical, analytical and computational properties of MALM distributions should extend the scope of its applications.

Acknowledgements

The authors are very thankful to the unknown referee for his helpful comments and suggestions that shorten and improve the initial version of this paper.

References

Arnold, B.C. (1983), Pareto Distributions (International Co-operative Publishing House, Burtonsville, MD). Chukova, S. and B. Dimitrov (1992), On the distributions having the almost lack-of-memory property, J. Appl. Probab.

29, 691~598. Chukova, S., B. Dimitrov and Z. Khalil (1993), A characterization of probability distributions similar to the exponential,

Canad. J. Statist. 21, 269-276. Galambos, J. and S. Kotz (1978), Characterization of Probability Distributions, Lecture Notes in Mathematics No. 675

(Springer, Berlin).