Random sum of mid truncated Lindley distribution 1

Journal of Advanced Research in Statistics and Probability

Vol. 2, Issue. 1, 2010, pp. 27-36 Online ISSN: 1943-2399

Random sum of mid truncated Lindley distribution M.M. Mohie El-Din, Abd El-Moneim A.M. Teamah, Abd El-Nasser M. Salem, Ahmed M.T. Abd El-Bar∗ Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt.

Abstract. The concept of random sum of distributions in actuarial science is used to represent such variables as the total amount of claims or losses payable by an insurer. In this note we are concerned with the random sum SN and sums Sn , where every constituent follows the mid truncated Lindley distribution. We derive the probability density function (pdf) of mid truncated distribution and used Lindley distribution as illustrative example. The pdf of random sum of mid truncated lindley distribution is discussed. Finally, the pdf of sum of mid truncated Lindley distribution is obtained. Keywords: Lindley distribution; Mid truncated distribution; Unit step function; Random sum. Mathematics Subject Classification 2010: 62G30, 62E10.

1

Introduction

Some practical problems can be analyzed by reference to sum of independent random variables in which the number of random variables in the sum is also a random variable. The random sum has widely used in various areas of applied probability, such as modeling the aggregate claims in an insurance portfolio, for example, let X1 , X2 , ... be independent and identically distributed random variables with the common pdf, f (x), and let SN =

N X

Xi , S0 = 0,

i=1

where N and Xi are independent. Suppose Xi denotes the amount of an insurance claime or loss and N denotes the number of claims or losses in a given reference period. Then SN denotes the total amount of claims or losses of the insurance portfolio payable by an

∗

Correspondence to: Ahmed M.T. Abd El-Bar, Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt. Email: ahmed− [email protected] † Received: 2 August 2010, accepted: 11 October 2010. http://www.i-asr.com/Journals/jarsp/

27

c °2010 Institute of Advanced Scientific Research

28

Random sum of mid truncated Lindley distribution

insurer in that reference period. Sums of independent random variables play an important role in solving engineering problems. Goovaertes and Kaas [2] derived a recursive scheme, involving Panjer’s recusion to compute the distribution of random sum of integer claims when the number of summands follows a generalized Poisson distribution. Nadarajah and Kotz [6] discussed the moments of truncated t and F distributions. Teamah and ElAlosey [8] discussed the random sum of mixtures of exponential distributions. Teamah and Saif El-Naser [9] studied the random sum of Lagrange random variables. Teamah and Abd El-Bar [10] derived the probability density function of random sum of mixtures of sum of bivariate exponential distributions.

2

Mid truncated Lindley distribution

The Lindley distribution specified by the pdf: f (x) =

p2 (1 + x) e−px , x > 0, p > 0, p+1

(2.1)

was introduced by Lindley [4, 5]. The corresponding cumulative distribution function (cdf) is: p + 1 + px −px e , x > 0, p > 0. (2.2) p+1 Ghitany et al. [1] studied the mathematical properties of Lindley distribution, the properties studied include: moments, cumulants, characteristic function, failure rate function, mean residual life function, mean deviations, Lorenz curve, stochastic ordering, entropies, a symptotic distribution of the extreme order statistics, distributions of sums, products and ratios, maximum likelihood estimation and simulation schemes. Ghitany et al. [1] showed that in many ways the Lindley distribution is better model than one based on the exponential distribution. Sankaran [7] used (2.1) as a mixing model for the Poisson parameter to generate a mixed Poisson distribution known as the discrete Poisson-Lindley distribution. The purpose of this note is to study the random sum and sum of mid truncated Lindley distribution. Mid truncated Lindley distribution is stuided in section 2. Random sum of mid truncated Lindley distribution is discussed in section 3. Finally, section 4 gives the pdf of sum of mid truncated Lindley distribution. The truncated distributions mean deleting some values of the domain of the random variables. Truncation may be classified into two classes such as: single truncation from one, left or right, side of a domain A (x) = {−∞ < x < ∞} of probability function (pf) p (x) and double truncation from both sides of the domain. This means deleting some values of the domain A (x) of p (x), but in this section we are interested of the distribution of mid truncated random variables defined by the following definition. F (x) = 1 −

Definition 2.1. Let Y be a random variable with pdf f (y), define X as a corresponding mid truncated of the random variable Y with pdf g (x). Then the following relation of pdf of mid truncated random variable can be defined as:

M.M. Mohie El-Din et al.

29

 1 f (x)   , −∞ < x < α,  α   2 R   f (x) dx  g (x) =

−∞

1 f (x)   , β < x < ∞.   2 R∞   f (x) dx  

(2.3)

β

Using equations (2.1) , (2.2) and (2.3), one can write the mid truncated Lindley distribution as given by the following pdf

g (x) =

 (1 + x) e−px 1 p2   , 0 < x < α,   2 (p + 1) Rα p2   −px dx  (1 + x) e  p+1 0

1 p2 (1 + x) e−px   , β < x < ∞.  2  ∞  2 (p + 1) R p  −px  (1 + x) e dx  β p+1

for 0 < α < β < ∞. Thus, the pdf of mid truncated Lindley distribution is:  1 p2 (1 + x) e−px   µ ¶ , 0 < x < α,   p + 1 + pα −pα  2 (p + 1) 1− e g (x) = p+1    1 p2 (1 + x) e−px   , β < x < ∞. 2 (p + 1 + pβ) e−pβ

3

(2.4)

Random sum of mid truncated Lindley distribution

In this section, we derive the pdf of random sum SN of mid truncated Lindley distribution. The calculations of next theorem use several functions, including the unit step function, u (t) , defined by ½ 0, t < 0 u (t) = 1, t ≥ 0, the shifted unit step function, u (t − a) , defined by ½ u (t − a) =

0, t < a 1, t ≥ a,

the properties of the Lapace transformation of the these functions can be found in N P Kuhfittig [3]. Further, let SN = Xi , where every constituent follows mid truncated i=1

30

Random sum of mid truncated Lindley distribution

Lindley distribution. Denote: µ=

1 2

p2 µ ¶, p + 1 + pα −pα (p + 1) 1 − e p+1

and

1 p2 . 2 ((p + 1 + pβ) e−pβ ) The next theorem provides the exact expression for the pdf of random sum SN of mid truncated Lindley distribution. ξ=

Theorem 3.1. The pdf of SN is: n µ ¶ i µ ¶ i−m X X µi − m¶ n n−i i X i i−m m fSN (t) = q p1 µ ξ i 1 m k m=0 i=0 k=0 k µ ¶ i−m−k X X µi − m − k ¶ k l × (−1) (−1)h (1 + α)h l h l=0 h=0 m µ ¶ X m × (1 + β)m−r e−α(l+h)p e−βmp r r=0

×

u (t − [α (l + h) + mβ]) e−p(t−[α(l+h)+mβ]) (t − [α (l + h) + mβ])i+k+r−1 , (i + k + r − 1)! (3.1)

for 0 < t < ∞, 0 < α < β < ∞. Proof. Using definition of Laplace transformation of shifted unit step function, one can write the Laplace transformation of mid truncated Lindley distribution given by (2.4) as follows:     1 p2 (1 + x) e−px µ ¶ (3.2) L {g (x)} = L  p + 1 + pα −pα 2 (p + 1)   1− e p+1 ¾ 2 1 p (1 + x) e−px + , 2 (p + 1 + pβ) e−pβ where

   

p2 (1 + x) e−px ¶ µ  p + 1 + pα −pα  2 (p + 1)     e 1− p+1 ( ) 1 1 − e−α(s+p) 1 − (1 + α) e−α(s+p) p2 µ ¶ + ,(3.3) p + 1 + pα −pα 2 (s + p) (s + p)2 (p + 1) 1 − e p+1

L

=

   1

M.M. Mohie El-Din et al.

31

for 0 < x < α, and ¾ 1 p2 (1 + x) e−px L 2 (p + 1 + pβ) e−pβ ( ) 1 p2 e−β(s+p) (1 + β) e−β(s+p) = + , 2 (p + 1 + pβ) e−pβ (s + p)2 (s + p) ½

(3.4)

for β < x < ∞. From relations (3.3) and (3.4) into (3.2), one obtain: L {g (x)} =

p2 µ ¶ p + 1 + pα −pα (p + 1) 1 − e p+1 ( ) 1 − e−α(s+p) 1 − (1 + α) e−α(s+p) × + (s + p) (s + p)2 ( ) p2 e−β(s+p) (1 + β) e−β(s+p) 1 . + + 2 (p + 1 + pβ) e−pβ (s + p)2 (s + p) 1 2

(3.5)

Let Xi , i = 1, 2, ..., N be independent and identically distributed random variables. Each Xi , i = 1, 2, ..., N has mid truncated Lindley distributed with pdf as given in relation (2.4) and Laplace transformation of the pdf given in relation (3.5). Assume the random N P sum SN = Xi , S0 = 0, N is a non-negative integer valued random variable has a i=1

binomial distribution with pmf as follows: µ ¶ n k n−k Pr (N = k) = p q , k = 0, 1, 2, ..., n, p1 + q1 = 1, k 1 1 with probability generating function (pgf): PN (t) = (q1 + tp1 )n .

(3.6)

The Laplace transformation of the pdf of random sum of mid truncated Lindley random variable is given by the following relation: L {fSN (t)} = fS∗N (s) = PN {L {g (x)}} ,

(3.7)

where PN (t) is the pgf of binomial random variable. Substituting from relations (3.5)

32

Random sum of mid truncated Lindley distribution

and (3.6) into (3.7) we have:

fS∗N (s) =

=

=

=

=

   



1 p2 µ ¶ q1 + p1  2  p + 1 + pα −pα   (p + 1) 1 − e p+1 ( ) 1 − e−α(s+p) 1 − (1 + α) e−α(s+p) × + s+p (s + p)2 ( )!)n 1 p2 e−β(s+p) (1 + β) e−β(s+p) + + 2 (p + 1 + pβ) e−pβ (s + p)2 s+p ( ( ) n µ ¶ X n n−i i 1 − e−α(s+p) 1 − (1 + α) e−α(s+p) + q p1 µ 2 i 1 s+p (s + p) i=0 ( ))i e−β(s+p) (1 + β) e−β(s+p) +ξ + (s + p) (s + p)2 )m ( n µ ¶ i µ ¶ −β(s+p) −β(s+p) X n n−i i X i e (1 + β) e q p1 µi−m ξ m + (s + p) i 1 m (s + p)2 m=0 i=0 ( )i−m 1 − e−α(s+p) 1 − (1 + α) e−α(s+p) × + s+p (s + p)2 n µ ¶ i µ ¶ X n n−i i X i q p1 µi−m ξ m i 1 m m=0 i=0 !i−m−k à !k à µ ¶ i−m X i−m 1 − e−α(s+p) 1 − (1 + α) e−α(s+p) × s+p k (s + p)2 k=0 ! à !m−r à r m µ ¶ X m e−β(s+p) (1 + β) e−β(s+p) × r (s + p) (s + p)2 r=0 µ ¶ µ ¶ n i i−m X X µi − m¶ µ 1 ¶i−m+k n n−i i X i i−m m q p1 µ ξ k s+p i 1 m m=0 i=0 k=0 k µ ¶ i−m−k X X µi − m − k ¶ k l −α(s+p)l × (−1) e (−1)h (1 + α)h e−α(s+p)h l h l=0

h=0

¶m+r m µ ¶µ X m 1 × e−β(s+p)r (1 + β)m−r e−β(s+p)(m−r) . r s+p

(3.8)

r=0

Now we can find the pdf of random sum SN by using the inverse Laplace transformation

M.M. Mohie El-Din et al.

33

of relation (3.8) as follows: © ª fSN (t) = L−1 fS∗N (s) n µ ¶ ∞ n−m X X X µn − m¶ n = h (n) ψ n−m ξ m m k n=0 m=0 k=0 k µ ¶ n−m−k X X µn − m − k ¶ k l × (−1) (−1)h (1 + α)h l h l=0

×

m µ ¶ X m r=0

r

h=0

(µ (1 + β)m−r e−α(l+h)p e−βmp × L−1

1 s+p

¶n+k+r

) e−s(l+h)α e−smβ

, ((3.9))

where

(µ −1

L

(

1 s+p

e−ηs

−1

= L = =

)

¶n+k+r e

−s(l+h)α −smβ

e

)

(s + p)n+k+r

½ ¾ u (t − η) −pt −1 (n + k + r − 1)! e L (n + k + r − 1)! sn+k+r u (t − η) e−pt tn+k+r−1 |t→t−η (n + k + r − 1)! =

u (t − η) e−p(t−η) (t − η)n+k+r−1 , (n + k + r − 1)!

((3.10))

where η = α (l + h) + mβ. Thus, the result of theorem follows by substituting (3.10) into (3.9) .

4

Sum of mid truncated Lindley distribution

In this section we study the pdf of sum Sn of mid truncated Lindley distribution. The pdf of Sn is provided in the following theorem. n P Let Sn = Xi , denote i=1

µ=

1 2

p2 µ ¶, p + 1 + pα −pα (p + 1) 1 − e p+1

and ξ=

p2 1 . 2 ((p + 1 + pβ) e−pβ )

34

Random sum of mid truncated Lindley distribution

Theorem 4.1. The pdf of Sn is ¶ m µ ¶ k µ ¶ n µ ¶ n−m µ X n n−m m X n − m X m X k (−1)l µ ξ k r l m r=0 m=0 k=0 l=0 µ ¶ n−m−k X n−m−k (−1)h (1 + α)h (1 + β)m−r e−α(l+h)p e−βmp × h

fsn (s) =

h=0

×

(s − [α (l + h) + mβ])n+k+r−1 e−p(s−[α(l+h)+mβ]) , (n + k + r − 1)!

(4.1)

for 0 < s < ∞. Proof. Let X1 , X2 , ..., Xn be independent and identically distributed random variables. Each Xi , i = 1, 2, ..., n has mid truncated Lindley distribution with pdf as given in relation (2.4) and characteristic function given as follows: ª © CXi (t) = E eitXi à ! 1 − (1 + α) e−α(p−it) 1 − e−α(p−it) + = µ p − it (p − it)2 à ! (1 + β) e−β(p−it) e−β(p−it) + +ξ . p − it (p − it)2 Assume Sn =

n P i=1

(4.2)

Xi , the sum of n-mid truncated Lindley random variables. The char-

acteristic function of Sn is given by: CSn (t) = (CXi (t))n ! ( à 1 − (1 + α) e−α(p−it) 1 − e−α(p−it) = µ + p − it (p − it)2 à !)n (1 + β) e−β(p−it) e−β(p−it) +ξ + p − it (p − it)2 à !n−m n µ ¶ X n n−m m 1 − (1 + α) e−α(p−it) 1 − e−α(p−it) + = µ ξ p − it m (p − it)2 m=0 !m à (1 + β) e−β(p−it) e−β(p−it) . (4.3) + × p − it (p − it)2 By using the inversion formula for characteristic function, we get the pdf of the random

M.M. Mohie El-Din et al.

35

variable Sn as follows: f (s) =

1 2π

Z∞ e−its CSn (t) dt −∞

 à !n−m µ ¶ Z∞ −α(p−it) −α(p−it) n n−m m  1 1 − (1 + α) e 1 − e = µ ξ e−its +  2π m p − it (p − it)2 m=0 −∞ !m ) à (1 + β) e−β(p−it) e−β(p−it) + dt × p − it (p − it)2 ¶ m µ ¶ n µ ¶ n−m µ X n n−m m X n − m X m = µ ξ m k r m=0 r=0 k=0  !k à à !n−m−k  1 Z∞ 1 − (1 + α) e−α(p−it) 1 − e−α(p−it) −its × e  2π p − it (p − it)2 −∞ à !r à !m−r ) e−β(p−it) (1 + β) e−β(p−it) × dt p − it (p − it)2 ¶ m µ ¶ n µ ¶ n−m µ X n n−m m X n − m X m = µ ξ k r m r=0 m=0 k=0 n−m−k k µ ¶ X µn − m − k ¶ X k l (−1)h (1 + α)h (1 + β)m−r (−1) × h l l=0 h=0 ∞  ¶n+k+r  µ  1 Z 1 ×e−α(l+h)p e−βmp e−it(s−η) dt ,  2π  p − it n X

−∞

where η = θ + mβ, and θ = α (l + h) . The result of the theorem follows by elementary integration of the integral above.

Acknowledgments The authors would like to thank the Editor-in-Chief, the associate editor and the referees for carefully reading the paper and for their great help in improving the paper. References [1] M. E. Ghitany, B. Atieh, S. Nadarajah. Lindley distribution and its application. Mathematics and Computers in Simulation, 2008, 78: 493-506. [2] M.J. Goovaertes, R. Kaas. Evaluating compound generalized Poisson distributions recursively. Astin Bulletin, 1991, 21: 193-198.

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Random sum of mid truncated Lindley distribution

[3] P.F. Kuhfittig. Introduction to Laplace transform. Mathematical Concepts and Methods in Science and Engineering, (1978). [4] D.V. Lindley. Fiducial distributions and Bayes’theorem. Journal of the Royal Statistical Society, Series B, 1958, 20: 102-107. [5] D.V. Lindley. Introduction to probability and statistics from a Bayesian viewpoint, part II: Inference, Cambridge University Press, New York, 1965. [6] S. Nadarajah, S. Kotz. Moments of truncated t and F distributions. Port Econ J, 2008, 7: 63-73. [7] M. Sankaran. The discrete Poisson-Lindley distribution. Biometrics, 1970, 26: 145-149. [8] A.A. Teamah, A.R. El- Alosey. Random sum of mixtures of exponential distributions. Journal of Applied Mathematics, 2004, 16: 237-247. [9] A.A. Teamah, H.H. Saif El-Naser. Random sum of Lagrange random variables. 32nd International Conference of Statistics, Computer Science and it’s Application, Egypt, 2007. [10] A.A. Teamah, A.T. Abd El-Bar. Random sum of mixtures of sum of bivariate exponential distributions. Journal of Mathematics and Statistics, 2009, 4: 270-275.