World Applied Sciences Journal 7 (10): 1331-1334, 2009
ISSN 1818-4952
© IDOSI Publications, 2009
Corresponding Author: Dr. R.U. Khan, Department of Statistics and Operations Research, Aligarh Muslim University,
Aligarh-202 002, India
1331
Recurrence Relations for Single and Product Moments of Record Values
from Gompertz Distribution and A Characterization
R.U. Khan and Benazir Zia
Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh-202 002, India
Abstract: In this study we give some recurrence relations satisfied by single and product moments of upper
record values from Gompertz distribution. Further a characterization of the Gompertz distribution based on
conditional expectation of function of record values is presented.
AMS subject classification: 62G30 • 62G99 • 62E10
Key words: Record • single moments • product moments • recurrence relations • Gompertz distribution •
characterization • conditional expectation
INTRODUCTION
A random variable X is said to have a Gompertz
distribution if its probability density function (pdf) is of
the form
x x
f(x) e exp (e 1) , 0 x , , 0
α α
β
=β - - ≤ ≤∞αβ>
α
(1.1)
and the cumulative distribution function (cdf) is of the
form
x
F(x) exp (e 1)
α
β
= - -
α
(1.2)
where
F(x) 1 F(x) = -
The Gompertz distribution in (1.1) was introduced
by Benjamin Gompertz (1825). This distribution is
applicable as a model for surviving distributions which
has an increasing hazard rate for the life of the creatures
and systems. Prentice and Elshaarawi (1973) have used
this model in their studies, Elandt-Johnson and Johnson
(1980) have shown that this distribution is widely used
in actuarial works. Garg et al. (1970) studied the
maximum likelihood estimates of the parameters of the
Gompertz distribution.
Let X
1
, X
2
,… be a sequence of independent and
identically distributed random variables with cdf F(x)
and pdf f (x). Let Y
n
= max (min) {X
1
, X
2
,…, X
n
}, n =
1,2,…. We say X
j
is an upper (lower) record value of
this sequence if Y
j
>(<)Y
j -1
, j ≥2. By definition, X
1
is an
upper as well as a lower record value. The indices at
which the upper record values occur are given by
record times {U
(n)
, n ≥1}, when U
(n)
= min {j|j>U
(n-1)
,
X
j
>U
(n-1)
}, n ≥2 with U
(1)
= 1 (Ahsanullah, 1995).
Chandler (1952) formulated the theory of record
values arising from a sequence of independently
identically distributed continuous random variables and
has now spread in various directions. Interested readers
may refer to the works Glick (1978), Nevzorov (1987),
Resnick (1987), Arnold and Balakrishnan (1989) and
Arnold et al. (1992, 1998).
In this paper, we established some recurrence
relations satisfied by the single and product moments of
upper record values from the Gompertz distribution in
(1.1). A characterization of this distribution has also
been obtained on using the conditional expectation of
record values. Similar results for modified Weibull
distribution have been derived by Sultan (2007).
We shall denote
(r) r
n U(n)
E(X ), r,n 1,2, , μ = = L
(r,s) r s
m,n U(m) U(n)
E(X X ),1 m n 1andr,s 1,2, , μ = ≤ ≤ - = L
(r,0) r (r)
m,n U(m) m
E(X ) ,1 m n 1andr 1,2, , μ = =μ ≤ ≤ - = K
(0,s) s (s)
m,n U(n) n
E(X ) ,1 m n 1ands 1,2, μ = =μ ≤ ≤ - = L
RELATIONS FOR SINGLE MOMENTS
First of all, we may note that for the Gompertz
distribution in (1.1)