On Moment Generating Function of Generalized Order Statistics from Erlang-Truncated Exponential Distribution ()
1. Introduction
A random variable X is said to have Erlang-truncated exponential distribution if its probability density function (pdf) is of the form
(1)
and the corresponding distribution function 
is
(2)
where
.
For more details on this distribution and its applications one may refer to [1].
[2] introduced and extensively studied the generalized order statistics
. The order statistics, sequential order statistics, Stigler’s order statistics, record values are special cases of
. Suppose 
are n 
from an absolutely continuous distribution function 
with the corresponding probability density function 
. Their joint pdf is
(3)
for
,
and 
is a positive integer.
Choosing the parameters appropriately, models such as ordinary order statistics (
), k-th record values
, sequential order statistics
, order statistics with non-integral sample size
, Pfeifer’s record values
and progressive type II censored order statistics 
are obtained [2, 3].
The marginal 
of the r-th
, 
, 
, is
(4)
and the joint 
of 
and
, 
, is
(5)
where
and
.
[4-6] have established recurrence relations for moment generating functions of record values from Pareto and Gumble, power function and extreme value distributions.
Recurrence relations for marginal and joint moment generating functions of 
from power function distribution are derived by [7]. [8,9] have established recurrence relations for conditional and joint moment generating functions of 
based on mixed population, respectively. [10] has established explicit expressions and some recurrence relations for moment generating function of gos from Gompertz distribution.
In the present study, we establish exact expressions and some recurrence relations for marginal and joint moment generating functions of gos from Erlang-truncated exponential distribution. Results for order statistics and record values are deduced as special cases and a characterization of this distribution is obtained by using the conditional expectation of function of gos.
2. Relations for Marginal Moment Generating Functions
Note that for Erlang-truncated exponential distribution defined in (1).
. (6)
The relation in (6) will be exploited in this paper to derive exact expressions and some recurrence relations for the moment generating functions of 
from the Erlang-truncated exponential distribution.
Let us denote the marginal moment generating functions of 
by 
and its j-th derivative by
.
We shall first establish the explicit expression for
. Using (4) and (6), we have when 
, (7)
where
. (8)
On expanding 
binomially in (8), we get when 
, (9)
where
On substituting for 
from (2) in (9), we have
(10)
Now on substituting for 
from (10) in (7) and simplifying, we obtain when 
(11)
When
, we have
Since (11) is of the form 
at
, therefore, we have
(12)
Differentiating numerator and denominator of (12) 
times with respect to
, we get
On applying L’ Hospital rule, we have
(13)
But for all integers 
and for all real numbers x, we have [11]
(14)
Therefore,
(15)
Now on substituting (14) in (13), we find that
(16)
Differentiating 
with respect to t and evaluating at
, we get the mean of the r-th 
when 
(17)
and when 
that
(18)
as obtained by [12] for exponential distribution at
.
Special Cases
1) Putting
, 
in (11) and (17), the explicit formula for marginal moment generating function and mean of order statistics from Erlang-truncated exponential distribution can be obtained as
and
where
.
2) Setting 
in (16) and (18), the results for upper records from Erlang-truncated exponential distribution may be obtained in the form
and
as obtained by [13] for exponential distribution at 
.
A recurrence relation for marginal moment generating function for 
from 
(1) can be obtained in the following theorem.
Theorem 2.1 For the distribution given in (1) and for 
(19)
Proof [10] has shown that for a positive integer
,
(20)
On substituting for 
from (6) in (20) and simplifying the resulting expression, we find that
(21)
Differentiating both the sides of (21) j times with respect to t, we get
The recurrence relation in (19) is derived simply by rewriting the above equation.
At 
in (19), we obtain the recurrence relations for moments of 
from Erlang-truncated exponential distribution in the form
(22)
Remark 2.1 Putting
, 
in (19) and (22), we can get the relations for marginal moment generating function and moments of order statistics for Erlang-truncated exponential distribution as
and
Remark 2.2 Setting 
and 
in (19) and (22), relations for record values can be obtained as
and
for 
Remark 2.3 At
, 
in (22), the result for single moments of gos obtained by [2] for exponential distribution is deduced.
3. Relations for Joint Moment Generating Functions
Before coming to the main results we shall prove the following Lemmas.
Lemma 3.1 For the Erlang-truncated exponential distribution as given in (1) and non-negative integers a, b and c with
,
(23)
where
(24)
Proof From (24), we have
, (25)
where
. (26)
On substituting for 
from (2) in (26), we get
.
Upon substituting this expression for 
in (25) and then integrating the resulting expression, we establish the result given in (23).
Lemma 3.2 For the distribution as given in (1) and any non-negative integers a, b and c,
(27)
(28)
where 
is as given in (24).
Proof Expanding 
binomially in (24) after noting that
, we get
Making use of Lemma 3.1, we establish the result given in (27).
When
, 
as
so after applying L’Hospital rule and (15), (28) can be proved on the lines of (16).
Theorem 3.1 For Erlang-truncated exponential distribution as given in (1) and for 
(29)
(30)
Proof From (5), we have
(31)
upon using the relation (6). Now expanding 
binomially in (31), we get
Making use of Lemma 3.2, we establish the relation given in (30).
Special Cases
1) Putting
, 
in (30), the explicit formula for the joint moment generating function of order statistics of the Erlang-truncated exponential distribution can be obtained as
where
.
2) Putting 
in (30), we deduce the explicit expression for joint moment generating function of upper k record values for Erlang-truncated exponential distribution in view of (29) and (28) in the form
Differentiating 
and evaluating at
, we get the product moments of 
when 
(32)
and when 
that
(33)
and for 
.
Making use of (6), we can derive the recurrence relations for joint moment generating function of 
from (5).
Theorem 3.2 For the distribution given in (1) and for 
(34)
Proof [10] has shown that for 
and a fixed positive integer 
Differentiating both the sides of (34) 
times with respect to 
and then 
times with respect to
, we get
which, when rewritten gives the recurrence relation in (26).
At 
in (34), we obtain the recurrence relations for product moments of gos from Erlang-truncated exponential distribution in the form
(35)
One can also note that Theorem 2.1 can be deduced from Theorem 3.2 by letting 
tends to zero.
Remark 3.1 Putting
, 
in (34) and (35), we obtain the recurrence relations for joint moment generating function and product moments of order statistics for Erlang-truncated exponential distribution in the form
and
as obtained by [14] for exponential distribution at 
and
.
Remark 3.2 Substituting 
and
, in (34) and (35), we get recurrence relations for joint moment generating function and product moments of upper k record values for Erlang-truncated exponential distribution.
4. Characterization
Let 
be
, then the conditional 
of 
given 
, in view of (4) and (5), is
(36)
Theorem 4.1 Suppose
, for all 
be a distribution function of the random variable X and
, 
, then
(37)
if and only if
.
Proof From (36), we have
(38)
By setting 
from (2) in (38)we obtain
(39)
where
.
Again by setting 
in (39), we get
and hence the relation given in (37).
To prove sufficient part, we have from (36) and (37)
(40)
where
.
Differentiating (40) both the sides with respect to
, we get
or
where
and
Therefore,
which proves that
