QUOTIENT MOMENTS OF THE ERLANG-TRUNCATED EXPONENTIAL DISTRIBUTION BASED ON RECORD VALUES AND A CHARACTERIZATION

Journal of Applied Mathematics & Informatics.
2014.
Jan,
32(1_2):
7-16

- Received : December 20, 2012
- Accepted : April 01, 2013
- Published : January 28, 2014

Download

PDF

e-PUB

PubReader

PPT

Export by style

Share

Article

Metrics

Cited by

TagCloud

Erlang-truncated exponential distribution is widely used in the field of queuing system and stochastic processes. This family of distribu-tion include exponential distribution. In this paper we establish some exact expression and recurrence relations satisfied by the quotient moments and conditional quotient moments of the upper record values from the Erlang-truncated exponential distribution. Further a characterization of this dis-tribution based on recurrence relations of quotient moments of record values is presented.
AMS Mathematics Subject Classification : 62G30, 62G99, 62E10.
X
is said to have Erlang-truncated exponential distribution if its probability density function (
pdf
) is of the form
and the corresponding survival function is
Therefore, in view of (1.1) and (1.2), we have
The relation in (1.3) will be used to derive some recurrence relations for the quotient moments of record values from the Erlang-truncated exponential distribution. More details on this distribution can be found in Ei-Alosey
[1]
.
Record values are found in many situations of daily life as well as in many statistical applications. Often we are interested in observing new records and in recording them: for example, Olympic records or world records in sport. Record values are used in reliability theory. Moreover, these statistics are closely connected with the occurrences times of some corresponding non homogeneous Poisson process used in shock models. The statistical study of record values started with Chandler
[9]
, he formulated the theory of record values as a model for successive extremes in a sequence of independently and identically random variables. Feller
[23]
gave some examples of record values with respect to gambling problems. Resnick
[17]
discussed the asymptotic theory of records. Theory of record values and its distributional properties have been extensively studied in the literature, for example, see, Ahsanullah
[10]
, Arnold
et al
.
[2]
,
[3]
, Nevzorov
[21]
and Kamps
[19]
for reviews on various developments in the area of records. We shall now consider the situations in which the record values (e.g. successive largest insurance claims in non-life insurance, highest water-levels or highest temperatures) themselves are viewed as ”outliers” and hence the second or third largest values are of special interest. Insurance claims in some non life insurance can be used as one of the examples. Observing successive
k
largest values in a sequence, Dziubdziela and Kopocinski
[22]
proposed the following model of
k
record values, where
k
is some positive integer.
Let {
X_{n}
,
n
≥1} be a sequence of identically independently distributed (
i.i.d
) random variables with
pdf f
(
x
) and distribution function (
df
)
F
(
x
). The
j
-th order statistics of a sample (
X
_{1}
,
X
_{2}
,...,
X_{n}
) is denoted by
X_{j:n}
. For a fix
k
≥ 1 we define the sequence
of
k
upper record times of {
X_{n}
,
n
≥ 1} as follows
The sequence
with
are called the sequences of
k
upper record values of {
X_{n}
,
n
≥ 1}.
For
k
= 1 and
n
= 1, 2,... we write
Then {
U_{n}
,
n
≥ 1} is the sequence of record times of {
X_{n}
,
n
≥ 1}. The sequence
, where
is called the sequence of
k
upper record values of {
X_{n}
,
n
≥ 1}. For convenience, we shall also take
Note that
k
= 1 we have
which are record value of {
X_{n}
,
n
≥ 1}. Moreover
Let
be the sequence of
k
upper record values then the
pdf
of
is given by
and the joint
pdf
of
and
is given by
where
Recurrence relations for single and product moments of record values from Weibull, Pareto, generalized Pareto, Burr, exponential and Gumble distribution are derived by Pawalas and Szynal
[14]
,
[15]
and
[16]
. Kumar
[4]
, Kumar and Khan
[6]
are established recurrence relations for moments of record values from exponentiated log-logistic and generalized beta II distributions respectively. And similar results for this paper have been done by Lee and Chang
[11]
,
[13]
and
[13]
, Chang
[18]
and Kumar
[5]
for exponential distribution, Pareto distribution, power function distribution, Weibull distribution and generalized Pareto distribution respectively. Kamps
[20]
investigated the importance of recurrence relations of order statistics in characterization.
In this paper, we established some explicit expressions and recurrence relations satisfied by the quotient moments and conditional quotient moments of the upper record values from the Erlang-truncated exponential distribution. A characterization of this distribution based on recurrence relations of quotient moments of record values.
Theorem 2.1.
For the Erlang-truncated exponential distribution as given in (1.1) and
1 ≤
m
≤
n
- 2,
k
= 1, 2,...,
s
= 1, 2,...
Proof.
From (1.5), we have
where
On using the equations (1.2) and (1.3) in equation (2.3), we get
(Gradshteyn and Ryzhik,
[7]
, p-346). Upon substituting this expression for
G
(
x
) in (2.2) and then integrating the resulting expression, we establish the result given in (2.1).
Theorem 2.2.
For the Erlang-truncated exponential distribution as given in (1.1) and
1 ≤
m
≤
n
- 2,
k
= 1, 2,...,
Proof
. Proof can be established on line of Theorem 2.1.
Remark 2.1.
Setting
k
= 1 in (2.1) and (2.4) we deduce the explicit expression for the quotient moments of record values from the Erlang-truncated exponential distribution.
Making use of (1.3), we can derive recurrence relations for the quotient moments of
k
upper record values.
Theorem 2.3.
For
1 ≤
m
≤
n
- 2,
k
= 1, 2,...,
r
= 0,1,2,...,
and s
= 1,2,...
Proof
. From equation (1.5) 1 ≤
m
≤
n
- 1,
r
= 0,1,2,..., and
s
= 1,2,...
where
Integrating
I
_{1}
(
x
) by parts treating
for integration and the rest of the integrand for differentiation, and substituting the resulting expression in (2.6), we get
the constant of integration vanishes since the integral in
I
_{1}
(
x
) is a definite integral. On using the relation (1.3), we obtain
and hence the result given in (2.5).
Theorem 2.4.
For
1 ≤
m
≤
n
- 2,
r,s
= 1, 2,...,
Proof
. Proof follows on the line of Theorem 2.3.
Corollary 2.5.
For m
≥ 1,
r
= 0, 1, 2,...,
and s
= 1, 2,...
Proof
. Upon substituting
n
=
m
+ 1 in (2.5) and simplifying, then we get the result given in (2.8).
Corollary 2.6.
For m
≥ 1,
r,s
= 0, 1, 2,...,
Proof
. Upon substituting
n
=
m
+ 1 in (2.7) and simplifying, then we get the result given in (2.9).
Remark 2.2.
Setting
k
= 1 in (2.5) and (2.7) we deduce the recurrence relation for the quotient moments of record values from the Erlang-truncated exponential distribution.
be a sequence of
i.i.d
continuous random variables with
df F
(
x
) and
pdf f
(
x
). Let
X
_{U(m)}
and
X
_{U(n)}
be the
m
-th and
n
-th upper record values, then the conditional
pdf
of
X
_{U(m)}
given
X
_{U(n)}
=
y
, 1 ≤
m
<
n
in view of (1.4) and (1.5), is
and the conditional
pdf
of
X
_{U(n)}
give
X
_{U(m)}
=
x
, 1 ≤
m
<
n
is
Theorem 3.1.
For the Erlang-truncated exponential distribution as given in (1.1) and
1 ≤
m
≤
n
- 2,
k
= 1, 2,...,
Proof
. From (3.2), we have
On using the (1.2) and (1.3), we have
integrating the equation (3.4), we established the result given in (3.3).
Theorem 3.2.
For the Erlang-truncated exponential distribution as given in (1.1) and
1 ≤
m
≤
n
- 2,
k
= 1, 2,...,
Proof
. Proof follows on the line of Theorem 3.1.
Making use of (1.3), we can derive recurrence relations for the quotient conditional moments of upper record values.
Theorem 3.3.
For
1 ≤
m
≤
n
- 2,
r
= 0, 1, 2,...
and s
= 1, 2,...,
Proof
. From equation (3.1), we have
where
Integrating
I
_{2}
(
x
) by parts treating
y
^{−s}
for integration and the rest of the integrand for differentiation, and substituting the resulting expression in (3.7), we get the result given in (3.6).
Theorem 3.4.
For
1 ≤
m
≤
n
- 2,
r, s
= 0, 1, 2,...,
Proof
. Proof follows on the line of Theorem 3.3.
Corollary 3.5.
For m
≥ 1,
r
= 0, 1, 2,...
and s
= 1,2,...
Proof
. Upon substituting
n
=
m
+ 1 in (3.6) and simplifying, then we get the result given in (3.9).
Corollary 3.6.
For
1 ≤
m
≤
n
- 2,
r, s
= 1, 2,...
Proof
. Upon substituting
n
=
m
+ 1 in (3.8) and simplifying, then we get the result given in (3.10).
Theorem 4.1.
Let k
≥ 1
is a fix positive integer
,
r be a non- negative integer and y be an absolutely continuous random variable with pdf
f
(
y
)
and df F
(
y
)
on the support
(0,∞),
then
if and only if
Proof
. The necessary part follows immediately from equation (2.5). On the other hand if the recurrence relation in equation (4.1) is satisfied, then on using equation (1.5), we have
Integrating the first integral on the right hand side of equation (4.2) by parts and simplifying the resulting expression, we find that
Now applying a generalization of the Müntz-Szász Theorem (Hwang and Lin,
[8]
) to equation (4.3), we get
which proves that
Devendra Kumar received M.Sc., M.Phil and Ph.D from Aligarh Muslim University Aligarh, India. Since 2012 he has been at Department of Statistics, Amity Institute of Applied Sciences Amity University, Noida, India . His research interests include Generalized Order Statistics, Order Statistics and Record values.
Department of Statistics, Amity Institute of Applied Sciences Amity University Noida, India.
e-mail: devendrastats@gmail.com

Record
;
quotient moments
;
recurrence relations
;
Erlang-truncated exponential distribution and characterization

1. Introduction

A random variable
PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

2. Relations for the quotient moment

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

3. Relation of quotient conditional expectation

Let
PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

4. Characterization

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

PPT Slide

Lager Image

5. Conclusion

In this study some exact expressions and recurrence relations for the quotient moments and conditional quotient moments of record values from the Erlangtruncated exponential distribution have been established. Further, recurrence relation of the quotient moments of record values has been utilized to obtain a characterization of the Erlang-truncated exponential distribution.
BIO

El-Alosey A.R.
(2007)
Random sum of new type of mixture of distributio
Int. J. Statist. Syst.
2
49 -
57

Arnold B.C.
,
Balakrishnan N.
,
Nagaraja H.N.
1992
A First course in Order Statistics
John Wiley and Sons
New York

Arnold B.C.
,
Balakrishnan N.
,
Nagaraja H.N.
1998
Records
John Wiley
New York

Kumar D.
(2011)
Relations for moments of k-th lower record values from exponentiated log-logistic distribution and a characterization
International Journal of Mathematical Archive
6
813 -
819

Kumar D.
(2012)
Recurrence relations for moments of record values from generalized beta II distribution and characterization.
Journal of Applied Mathematics and Informatics
(In Press)

Kumar D.
,
Khan M.I.
(2012)
Recurrence relations for moments of K-th record values from generalized beta distribution and a characterization
Seluk J. App. Math.
13
75 -
82

Gradshteyn I,S.
,
Ryzhik I.M.
2007
Tables of Integrals, Series of Products
Academic Press
New York

Hwang J.S.
,
Lin G.D.
(1984)
On a generalized moments problem II
Proc. Amer. Math. Soc.
91
577 -
580
** DOI : 10.1090/S0002-9939-1984-0746093-4**

Chandler K.N.
(1952)
The distribution and frequency of record values
J. Roy. Statist. Soc.
Ser B 14
220 -
228

Ahsanullah M.
1995
Record Statistics
Nova Science Publishers
New York

Lee M.Y.
,
Chang S.K.
(2004)
Recurrence relations of quotient moments of the exponential distribution by record values
Honam Mathematical J.
26
463 -
469

Lee M.Y.
,
Chang S.K.
(2004)
Recurrence relations of quotient moments of the Pareto dis-tribution by record values
J. Korea Soc. Math. Educ. Ser B: Pure Appl. Math.
11
97 -
102

Lee M.Y.
,
Chang S.K.
(2004)
Recurrence relations of quotient moments of the power function distribution by record values
Kangweon-Kyungki Math. J.
12
15 -
22

Pawlas P.
,
Szynal D.
(1998)
Relations for single and product moments of k-th record values from exponential and Gumbel distributions
J. Appl. Statist. Sci.
7
53 -
61

Pawlas P.
,
Szynal D.
(1999)
Recurrence relations for single and product moments of k-th record values from Pareto, generalized Pareto and Burr distributions
Comm. Statist. Theory Methods
28
1699 -
1709
** DOI : 10.1080/03610929908832380**

Pawlas P.
,
Szynal D.
(2000)
Recurrence relations for single and product moments of k-th record values from Weibull distribution and a characterization
J. Appl. Stats. Sci.
10
17 -
25

Resnick S.I.
1973
Extreme values, regular variation and point processes
Springer-Verlag
New York

Chang S.K.
(2007)
Recurrence relations of quotient moments of the Weibull distribution by record values
J. Appl. Math. and Computing
1
471 -
477

Kamps U.
(1995)
A concept of generalized Order Statistics
J. Statist. Plann. Inference
48
1 -
23
** DOI : 10.1016/0378-3758(94)00147-N**

Kamps U
(1998)
Characterizations of distributions by recurrence relations and identities for moments of order statistics. In: Balakrishnan, N. and Rao, C.R., Handbook of Statistics, Order Statistics: Theory and Methods.
North-Holland, Amsterdam
16
291 -
311

Nevzorov V.B.
1987
Records
Theory probab. Appl.
(English translation)
32

Dziubdziela W.
,
Kopocinski B.
(1976)
Limiting properties of the k-th record value.
Appl. Math.
15
187 -
190

Feller W.
1966
An introduction to probability theory and its applications, 2
John Wiley and Sons
New York

Citing 'QUOTIENT MOMENTS OF THE ERLANG-TRUNCATED EXPONENTIAL DISTRIBUTION BASED ON RECORD VALUES AND A CHARACTERIZATION
'

@article{ E1MCA9_2014_v32n1_2_7}
,title={QUOTIENT MOMENTS OF THE ERLANG-TRUNCATED EXPONENTIAL DISTRIBUTION BASED ON RECORD VALUES AND A CHARACTERIZATION}
,volume={1_2}
, url={http://dx.doi.org/10.14317/jami.2014.007}, DOI={10.14317/jami.2014.007}
, number= {1_2}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={KUMAR, DEVENDRA}
, year={2014}
, month={Jan}