Recurrence relations for the moments of order statistics from doubly truncated modified Makeham distribution and its characterization

Ali A. Ismail; S.E. Abu-Youssef

doi:10.1016/j.jksus.2013.11.003

View/Download PDF

Buy Reprints

PDF

Translate this page into:

Original article

26 (

); 200-204

doi:

10.1016/j.jksus.2013.11.003

Recurrence relations for the moments of order statistics from doubly truncated modified Makeham distribution and its characterization

Ali A. Ismail^{^a,^b,*}, S.E. Abu-Youssef^{^c}

a King Saud University, College of Science, Department of Statistics and Operations Research, P.O. Box 2455, Riyadh 11451, Saudi Arabia

b Cairo University, Faculty of Economics & Political Science, Department of Statistics, Giza, Egypt

c Al-Azhar University, Faculty of Science, Department of Mathematics, Nasr City, 11884 Cairo, Egypt

*Corresponding author at: King Saud University, College of Science, Department of Statistics and Operations Research, P.O. Box 2455, Riyadh 11451, Saudi Arabia. Tel.: +966 114676335; fax: +966 114673036 ismail@ksu.edu.sa (Ali A. Ismail)

Received: 2013-7-25, Accepted: 2013-11-12, Published: 2013-07-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Available online 22 November 2013

Peer review under responsibility of King Saud University.

Abstract

In this study a general form of recurrence relations of continuous function for doubly truncated modified Makeham distribution is obtained. Recurrence relations between single and product moments of order statistics from doubly truncated modified Makeham distribution are given. Also, a characterization of modified Makeham distribution from the right and the left is discussed through the properties of order statistics.

Keywords

Modified Makeham distribution

Bathtub hazard rate function

Single and product moments

Order statistics

Recurrence relations

Characterization

Show Related Articles from PubMed

Acronyms and Notations

MMD: modified Makeham distribution
h(x): hazard rate function
f(x): density function
F(x): cumulative distribution function
b: MMD shape parameter (b > 0)
a: MMD scale parameter (a > 0)

1 Introduction

Order statistics arise obviously in many real life applications. The moments of order statistics have assumed considerable interest in recent years and have been tabulated quite widely for several distributions. For an extensive survey, see for example (Arnold et al., 1992; Hendi et al., 2006). Many authors have presented some recurrence relations satisfied by single and product moments of order statistics see for example (Khan et al., 1983a,b; Athar and Islam, 2004 ; among others). A number of recurrence relations satisfied by these moments of order statistics are available in the literature. Balakrishnan and Malik (1985) derived some identities involving the density functions of order statistics. These identities are useful in checking the computation of the moments of order statistics. Balakrishnan and Malik (1986) established some recurrence relations of order statistics from the linear-exponential distribution. Balakrishnan et al. (1988) reviewed several recurrence relations and identities for the single and product moments of order statistics from some specific distributions. Recently, Bekçi (2009) considered some recurrence relations for the moments of order statistics from uniform distribution.

Makeham distribution is an important life distribution and has been commonly used to fit actuarial data (see, Marshall and Olkin, 2007). According to Scollnik (1995), Makeham distribution of mortality represents a failure law, where the hazard rate is a mixture of non-aging failure distribution and the aging failure distribution with exponential increase in failure rates. For a description on the genesis and applications of Makeham distribution one may refer to Makeham (1860).

Feng et al. (2008) used the least squares type estimation to estimate the parameters of Makeham distribution. Aboutahoun and Al-Otaibi (2009) discussed relations between single and product moments of Makeham distribution.

The distribution presented in this paper is called modified Makeham distribution (MMD). This distribution has frequently been used to describe human mortality and to establish actuarial tables. Its hazard rate function is a product of a monotonically decreasing and a monotonically increasing function. The hazard rate function is given by

(1.1)

h (x) = \frac{b}{a} {(\frac{x}{a})}^{b - 1} \exp [{(\frac{x}{a})}^{b}],

where a is the scale parameter and b is the shape parameter. It is noted that for b < 1 the hazard rate function has a bathtub curve. This is a very desirable characteristic of the reliability models. It is important to note that the scale parameter a can be taken equal to 1 with no loss in generality. This simplifies many of the expressions.

Hazard rate function is the conditional probability that failure of the device will occur during a small interval [t;t + Δt] given that device has survived to time t. The bathtub hazard rate function is typical for the lifetime description of most technical devices. At the beginning the failure rate decreases, after a while it becomes stable, and then it increases due to natural wear out processes.

The cumulative distribution function of the MMD is of the following form:

(1.2)

F (x) = 1 - \exp [1 - \exp {(\frac{x}{a})}^{b}] .

Derivative of the cumulative distribution function (1.2) gives density function of the following form:

(1.3)

f_{X} (x) = \frac{b}{a} {(\frac{x}{a})}^{b - 1} \exp [{(\frac{x}{a})}^{b}] \exp [1 - \exp {(\frac{x}{a})}^{b}] .

For basic characteristics of this distribution see (Kosznik-Biernacka, 2006, 2007, 2011 ).

In this paper, some recurrence relations satisfied by single and product moments of order statistics from doubly truncated MMD are established. Recurrence relations for the non-truncated case are given as a special case. A characterization of this distribution based on the properties of the order statistics has also been considered.

The doubly truncated probability density function of the MMD is given by

(1.4)

f (x) = \frac{1}{P - Q} (\frac{b}{a}) {(\frac{x}{a})}^{b - 1} \exp [{(\frac{x}{a})}^{b}] \exp [1 - \exp {(\frac{x}{a})}^{b}], Q_{1} ⩽ x ⩽ P_{1} .

where

1 - P = \exp [1 - \exp \{{(\frac{P_{1}}{a})}^{b}\}]

and

1 - Q = \exp [1 - \exp \{{(\frac{Q_{1}}{a})}^{b}\}]

Let $P_{2} = \frac{1 - P}{P - Q}$ and $Q_{2} = \frac{1 - Q}{P - Q}$ .

From (1.4), we have

(1.5)

1 - F (x) = - P_{2} + (\frac{a}{b}) {(\frac{a}{x})}^{b - 1} e^{- {(\frac{x}{a})}^{b}} f (x) .

Let X_1:n ⩽ X_2:n ⩽ ⋯ ⩽ X_n:n be the order statistics from continuous distribution functions (c.d.f) F₁(x) and probability density function (p.d.f) f₁(x). The p.d.f of order statistics X_r:n and the joint p.d.f of two order statistics X_r:n and X_s:n are

\begin{matrix} f_{r : n} (x) = \frac{n!}{(r - 1)! (n - r)!} {[F_{1} (x)]}^{r - 1} {[1 - F_{1} (x)]}^{n - r} f_{1} (x), \\ f_{r, s : n} (x, y) = \frac{n!}{(r - 1)! (s - r - 1)! (n - s)!} {[F_{1} (x)]}^{r - 1} {[F_{1} (y) - F_{1} (x)]}^{s - r - 1} {[1 - F_{1} (y)]}^{n - s} f_{1} (x) f_{1} (y), \\ x < y; 1 ⩽ r < s ⩽ n . \end{matrix}

Then for any monotone continuous function ϕ(x), we have

(1.6)

α_{r : n} = E [ϕ (x)] = \frac{n!}{(r - 1)! (n - r)!} \int_{Q_{1}}^{P_{1}} ϕ (x) {[F (x)]}^{r - 1} {[1 - F (x)]}^{n - r} f (x) dx

and for any measurable joint function ϕ(x,y), we have

(1.7)

α_{r, s : n} = E [ϕ (x, y)] = \frac{n!}{(r - 1)! (s - r - 1)! (n - s)!} \int_{Q_{1}}^{P_{1}} \int_{x}^{P_{1}} ϕ (x, y) {[F (x)]}^{r - 1} {[F (y) - F (x)]}^{s - r - 1} {[1 - F (y)]}^{n - s} f (x) f (y) dydx .

Also, assume that

$μ_{r : n}^{(k)} = E (X_{r : n}^{k}), μ_{r, s : n}^{(j, k)} = E (X_{r : n}^{j} X_{s : n}^{k}), μ_{0 : n}^{(k)} = Q_{1}^{k}, k = 1, 2, \dots; n = 0, 1, \dots ., μ_{n : n - 1}^{(k)} = P_{1}^{k}, E (e^{{(X_{0 : n} / a)}^{b}}) = e^{{(Q_{1} / a)}^{b}}$ and $E (e^{{(X_{n : n - 1} / a)}^{b}}) = e^{{(P_{1} / a)}^{b}}$ .

Now, to derive recurrence relations for the moments of order statistics from modified Makeham distribution, let us consider the following theorems as shown in the next sections.

2 Recurrence relations for single moments

Theorem 1

Let X_r:n ⩽ X_r+1:n, (1 ⩽ r ⩽ n) be an order statistics, Q₁ ⩽ X_r:n ⩽ P₁, n ⩾ 1 and for any measurable function ϕ(x), then the recurrence relation for the single moments of order statistics is

(2.1)

α_{r : n} = - P_{2} α_{r : n - 1} + Q_{2} α_{r - 1 : n - 1} + \frac{a^{b}}{nb} E [X^{1 - b} e^{- {(X / a)}^{b}} ϕ^{'} (x)] .

Proof

From (1.6), we have

(2.2)

α_{r : n} - α_{r - 1 : n - 1} = (\begin{matrix} n - 1 \\ r - 1 \end{matrix}) \int_{Q_{1}}^{P_{1}} ϕ^{'} (x) {[F (x)]}^{r - 1} {[1 - F (x)]}^{n - r + 1} dx .

In view of (1.5) and (2.2), we have

(2.3)

\begin{matrix} α_{r : n} - α_{r : n - 1} = (\begin{matrix} n - 1 \\ r - 1 \end{matrix}) \int_{Q_{1}}^{P_{1}} ϕ^{'} (x) {[F (x)]}^{r - 1} {[1 - F (x)]}^{n - r} [- P_{2} + (\frac{a}{b}) {(\frac{a}{x})}^{b - 1} e^{- {(x / a)}^{b}} f (x)] dx \\ = - \frac{(n - 1)}{(n - r)} P_{2} (α_{r : n - 1} - α_{r - 1 : n - 2}) + \frac{a^{b}}{nb} E [ϕ^{'} (x) X^{1 - b} e^{- {(X / a)}^{b}}], \end{matrix}

(2.4)

But (n - r) α_{r : n} + r α_{r + 1 : n} = n α_{r : n - 1} .

(2.5)

Or (n - r) α_{r - 1 : n - 1} + (r - 1) α_{r : n - 1} = (n - 1) α_{r - 1 : n - 2} .

Now, from (2.3) and (2.5) we obtain

α_{r : n} = - P_{2} α_{r : n - 1} + (1 + P_{2}) α_{r - 1 : n - 1} + \frac{a^{b}}{nb} E [ϕ^{'} (x) X^{1 - b} e^{- {(X / a)}^{b}}] .

Since 1 + P₂ = Q₂, then the theorem is proved. □

Notation 1

For the non-truncated case, by putting Q₁ = 0, P₁ = ∞, Q₂ = 0, P₂ = 1, we get $α_{r : n} = - α_{r : n - 1} + \frac{a^{b}}{nb} E [X^{1 - b} e^{- (X / a)} ϕ^{'} (x)] .$

Special cases:

Putting ϕ(x) = x^b, then we obtain the simple recurrence relations between single moments of order statistics
(2.6)
$μ_{r : n}^{(b)} = - P_{2} μ_{r : n - 1}^{(b)} + Q_{2} μ_{r - 1 : n - 1}^{(b)} + \frac{a^{b}}{n} E (e^{- {(X / a)}^{b}}) .$
Putting $ϕ (x) = e^{{(x / a)}^{b}}$ , we obtain
(2.7)
$E (e^{{(X_{r : n} / a)}^{b}}) = - P_{2} E (e^{{(X_{r : n - 1} / a)}^{b}}) + Q_{2} E (e^{{(X_{r - 1 : n - 1} / a)}^{b}}) + \frac{1}{n},$
If we put r = 1 and n = 1 in (2.6) and (2.7), then we get
(2.8)
$μ_{1 : 1}^{(b)} = - P_{2} P_{1}^{b} - Q_{2} Q_{1}^{b} + a^{b} E (e^{- {(X / a)}^{b}}),$ where $E (e^{- {(X / a)}^{b}}) = \int_{e^{{(\frac{Q_{1}}{a})}^{b}}}^{e^{{(\frac{P_{1}}{a})}^{b}}} \ln u e^{1 - u} du$ , which can be easily computed numerically.

and
(2.9)
$E (e^{{(X_{r : n} / a)}^{b}}) = - P_{2} e^{{(P_{1} / a)}^{b}} + Q_{2} e^{{(Q_{1} / a)}^{b}}) + 1 .$
Putting r = 1, n = n − r in (2.7), we obtain
(2.10)
$E (e^{{(X_{1 : n - r} / a)}^{b}}) = - P_{2} e^{{(P_{1} / a)}^{b}}) + Q_{2} e^{{(Q_{1} / a)}^{b}}) + \frac{1}{n - r} .$
By putting r = 1, n = 2 in (2.10), we get
(2.11)
$E (e^{{(X_{1 : 2} / a)}^{b}}) = - P_{2} e^{{(P_{1} / a)}^{b}}) + Q_{2} e^{{(Q_{1} / a)}^{b}}) + 1 .$

3 Recurrence relations for product moments

Theorem 2

Let X_r:n ⩽ X_r+1:n,r = 1, 2, … , n − 1 be an order statistics from a random sample of size n, and for r < s > n, then the recurrence relation for the product moments of order statistics is

(3.1)

α_{r, s : n} - α_{r, s - 1 : n} = \frac{- {nP}_{2}}{n - s + 1} (α_{r, s : n - 1} - α_{r, s - 1 : n - 1}) + \frac{a^{b}}{(n - s + 1) b} E [ϕ^{'} (x, y) . Y^{1 - b} . e^{- {(Y / a)}^{b}}],

where

ϕ^{'} (x, y) = \frac{\partial ϕ (x, y)}{\partial y}

Proof

From (1.7), we have

(3.2)

\begin{matrix} α_{r, s : n} - α_{r, s - 1 : n} = \\ \frac{n!}{(r - 1)! (s - r - 1)! (n - s + 1)!} \int_{Q_{1}}^{P_{1}} \int_{x}^{P_{1}} ϕ^{'} (x, y) {[F (x)]}^{r - 1} {[F (y) - F (x)]}^{s - r - 1} {[1 - F (y)]}^{n - s + 1} f (x) dydx . \end{matrix}

But

(3.3)

[1 - F (y)] = - P_{2} + (\frac{a}{b}) {(\frac{a}{y})}^{b - 1} e^{- {(y / a)}^{b}} f (y) .

Now, by substituting from (3.3) in (3.2), then

(3.4)

\begin{matrix} α_{r, s : n} - α_{r, s - 1 : n} = \frac{n!}{(r - 1)! (s - r - 1)! (n - s + 1)!} \\ \times \int_{Q_{1}}^{P_{1}} \int_{x}^{P_{1}} ϕ^{'} (x, y) {[F (x)]}^{r - 1} {[F (y) - F (x)]}^{s - r - 1} {[1 - F (y)]}^{n - s} [- P_{2} + (\frac{a}{b}) {(\frac{a}{y})}^{b - 1} e^{- {(y / a)}^{b}} f (y)] f (x) dydx . \end{matrix}

Hence, the theorem is proved. □

Notation 2

For the non-truncated case, by putting Q₁ = 0, P₁ = ∞, Q₂ = 0,P₂ = 1, we obtain $α_{r, s : n} - α_{r, s - 1 : n} = - \frac{n}{n - s + 1} (α_{r, s : n - 1} - α_{r, s - 1 : n - 1}) + \frac{a^{b}}{(n - s + 1) b} E [ϕ^{'} (x, y) . Y^{1 - b} . e^{- {(Y / a)}^{b}}] .$ Special cases:

If we choose ϕ(x,y) = x^jy^b, then we get
(3.5)
$μ_{r, s : n}^{(j, b)} - μ_{r, s - 1 : n}^{(j, b)} = - \frac{{nP}_{2}}{n - s + 1} (μ_{r, s : n - 1}^{(j, b)} - μ_{r, s - 1 : n - 1}^{(j, b)}) + \frac{a^{b}}{n - s + 1} . E (X^{j} e^{- {(Y / a)}^{b}}),$ which is the recurrence relation between product moments of order statistics for the doubly truncated MMD
If we choose ϕ(x,y) = y^b and s = r + 1 in above theorem and noting that $μ_{r, r + 1 : n}^{(0, b)} = μ_{r + 1 : n}^{(b)} and μ_{r, r : n}^{(j, b)} = μ_{r : n}^{(j + b)},see (Khan et al., 1983b),$ we get
(3.6)
$μ_{r + 1 : n}^{(b)} - μ_{r : n}^{(b)} = - \frac{{nP}_{2}}{n - r} (μ_{r + 1 : n - 1}^{(b)} - μ_{r + 1 : n - 1}^{(b)}) + \frac{a^{b}}{n - r} . E (e^{- {(Y / a)}^{b}}),$ which is the recurrence relation between single moments of order statistics for the doubly truncated MMD.

Notation 3

for the non-truncated case, if Q₁ = 0, P₁ = ∞, Q₂ = 0,P₂ = 1, then we have

(3.7)

μ_{r, s : n}^{(j, b)} = μ_{r, s - 1 : n}^{(j, b)} - \frac{n}{n - s + 1} (μ_{r, s : n - 1}^{(j, b)} - μ_{r, s - 1 : n - 1}^{(j, b)}) + \frac{a^{b}}{n - s + 1} . E (X^{j} e^{- {(Y / a)}^{b}}) .

4 Characterization of MMD

The unconditional p.d.f. of (s − r)th order statistics in a sample of size (n − r)th order statistics is given by

(4.1)

f (x_{s : n} | x_{r : n} = x) = \frac{(n - r)!}{(s - r - 1)! (n - s)!} \frac{{[F (y) - F (x)]}^{s - r - 1} {[1 - F (y)]}^{n - s} f (y)}{{[1 - F (x)]}^{n - r}}, x ⩽ y,

where f(x_s:n∣x_r:n = x) is the conditional p.d.f. of X_s:n given X_r:n = x and the sample drawn from

\frac{f (y)}{[1 - F (x)]}, x ⩽ y

on the left truncated at x.

Therefore, for the left truncated at x, it can be seen that $Q_{1} = x, Q = F (x), P_{1} = \infty, P = 1, Q_{2} = 1 and P_{2} = 0 .$ Also, if the parent distribution truncated from the right at y, then $f (x_{r : n} | x_{s : n} = y) = \frac{(s - 1)!}{(r - 1)! (s - r - 1)!} \frac{{[F (x)]}^{r - 1} {[F (y) - F (x)]}^{s - r - 1} f (x)}{{[F (y)]}^{s - 1}}, x ⩽ y .$ Thus in the case of right truncation at x, we have $Q_{1} = 0, Q = 0, P_{1} = x, P = F (x), Q_{2} = \frac{1}{F (x)} and P_{2} = \frac{1 - F (x)}{F (x)} .$ In this section we obtain a characterization of MMD via the properties of order statistics as shown from the following theorems.

Theorem 3

If F(x) is a distribution function, 0 < F(x) < 1, then X is modified Makeham if and only if $E (e^{{(X_{r + 1 : n} / a)}^{b}} | X_{r : n} = x) = \frac{(n - r) e^{{(x / a)}^{b}} + 1}{n - r} .$

Proof

The necessary condition is proved from (2.10) by using the properties of left truncation. To prove the sufficient condition, we have $\frac{n - r}{{[1 - F (x)]}^{n - r}} \int_{x}^{\infty} e^{{(y / a)}^{b}} {[1 - F (y)]}^{n - r - 1} f (y) dy = \frac{1}{n - r} + e^{{(x / a)}^{b}} .$ That is,

(4.2)

(n - r) \int_{x}^{\infty} e^{{(y / a)}^{b}} {[1 - F (y)]}^{n - r - 1} f (y) dy = {[1 - F (x)]}^{n - r} [\frac{1}{n - r} + e^{{(x / a)}^{b}}] .

Differentiating (4.2) with respect to X, we obtain

- (n - r) e^{{(x / a)}^{b}} {[1 - F (x)]}^{n - r - 1} f (x) = - (n - r) {[1 - F (x)]}^{n - r - 1} [\frac{1}{n - r} + e^{{(x / a)}^{b}}] f (x) + {[1 - F (x)]}^{n - r} e^{{(x / a)}^{b}} (\frac{b}{a}) {(\frac{x}{a})}^{b - 1},

which gives

h (x) = \frac{b}{a} {(\frac{x}{a})}^{b - 1} \exp [{(\frac{x}{a})}^{b}] .

Then, the theorem is proved. □

Without loss of generality, when a = b = 1, we have $\frac{n - r}{{[1 - F (x)]}^{n - r}} \int_{x}^{\infty} e^{y} {[1 - F (y)]}^{n - r - 1} f (y) dy = \frac{1}{n - r} + e^{x} .$ That is, $(n - r) \int_{x}^{\infty} e^{y} {[1 - F (y)]}^{n - r - 1} f (y) dy = {[1 - F (x)]}^{n - r} [\frac{1}{n - r} + e^{x}] .$ Differentiating the above equation with respect to X, we obtain $\begin{matrix} - (n - r) e^{x} {[1 - F (x)]}^{n - r - 1} f (x) = - (n - r) {[1 - F (x)]}^{n - r - 1} [\frac{1}{n - r} + e^{x}] f (x) \\ + {[1 - F (x)]}^{n - r} e^{x} \end{matrix},$ which gives $\frac{f (x)}{1 - F (x)} = e^{x} .$ The solution of above differential equation is

$f (x) = e^{x} e^{1 - e^{- x}}$ , which is the standard modified Makeham distribution.

Theorem 4

If F(x) is any distribution function, 0 < F(x) < 1, then X is modified Makeham if and only if $E (e^{{(X_{1 : n} / a)}^{b}} | X_{2 : n} = x) = - \frac{1 - F (x)}{F (x)} e^{{(x / a)}^{b}} + \frac{1}{F (x)} + 1 .$

Proof

The sufficient condition is proved from (2.11) by using the properties of right truncation. To prove the necessity, we have $\frac{1}{F (x)} \int_{0}^{x} e^{{(y / a)}^{b}} f (y) dy = - \frac{1 - F (x)}{F (x)} e^{{(x / a)}^{b}} + \frac{1}{F (x)} + 1 .$ Or

(4.3)

\int_{0}^{x} e^{{(y / a)}^{b}} f (y) dy = - [1 - F (x)] e^{{(x / a)}^{b}} + 1 + F (x) .

Differentiating (4.3) with respect to x, we get

$e^{{(x / a)}^{b}} f (x) = f (x) e^{{(x / a)}^{b}} - [1 - F (x)] e^{{(x / a)}^{b}} (\frac{b}{a}) {(\frac{x}{a})}^{b - 1} + f (x)$ , which gives $h (x) = (\frac{b}{a}) {(\frac{x}{a})}^{b - 1} e^{{(\frac{x}{a})}^{b}} .$ Thus, the theorem is proved. □

5 Conclusion

In this study, general relations of order statistics for any continuous measurable function for doubly truncated MMD had been established. Further, a characterization of MMD based on the properties of the order statistics had been obtained. The recurrence relations for the single and product moments of order statistics are important in the theory of order statistics. The moments of order statistics were obtained by some other moments of order statistics. Through the properties of order statistics, the characterization results from left and right of MMD distribution were given.

References

Aboutahoun A.W., Al-Otaibi N.M., . Recurrence relations between moments of order statistics from doubly truncated Makeham distribution. Compt. Appl. Math.. 2009;28(3):277-290.
[Google Scholar]
Arnold B.C., Balakrishnan N., Nagaraja H.N., . A First Course in Order Statistics. New York: Wiley; 1992.
Athar H., Islam H., . Recurrence Relations for single and product moments of generalized order statistics from a General Class of Distribution. Metron. 2004;LXII(3):327-337.
[Google Scholar]
Balakrishnan N., Malik H.J., . Order statistics from the linear-exponential distribution, part I: increasing hazard rate case. Commun. Statist. Theor. Meth.. 1986;15(1):179-203.
[Google Scholar]
Balakrishnan N., Malik H.J., . Some general identities involving order statistics. Commun. Statist. Theor. Meth.. 1985;14(2):333-339.
[Google Scholar]
Balakrishnan N., Malik H.J., Ahmed S.E., . Recurrence relations and identities for moments of order statistics, II: specific continuous distributions. Commun. Statist. Theor. Meth.. 1988;17(8):2657-2694.
[Google Scholar]
Bekçi M., . Recurrence relations for the moments of order statistics from uniform distribution. Sci. Res. Essays. 2009;4(11):1302-1305.
[Google Scholar]
Feng X., He G., Abdurishit, . Estimation of parameters of the Makeham distribution using the least squares method. Math. Comput. Simul.. 2008;77:34-44.
[Google Scholar]
Hendi M.I., Abu-Youssef S.E., Alraddadi A.A., . Order Statistics from Doubly Truncated Gompertz Distribution and its Characterizations. Egypt. Statist. J.. 2006;50(1):21-31.
[Google Scholar]
Khan A.H., Parvez S., Yaqub M., . Recurrence relations between moments of order statistics. Naval Res. Logist. Quart.. 1983;30:419-441. Corrections, 32, 1985, 693
[Google Scholar]
Khan A.H., Parvez S., Yaqub M., . Recurrence relations between product moments of order statistics. J. Statist. Plan. Inf.. 1983;8:175-183.
[Google Scholar]
Kosznik-Biernacka S., . Rodzina rozkladow Makehama. W Slupsku, PAP; 2006.
Kosznik-Biernacka S., . Makeham’s generalised distribution. Computat. Meth. Sci. Technol.. 2007;13(2):113-120.
[Google Scholar]
Kosznik-Biernacka S., . Sums of independent random variables of the modified Makeham distributions. Computat. Meth. Sci. Technol.. 2011;17(1-2):35-40.
[Google Scholar]
Makeham W.M., . On the law of mortality and the construction of annuity tables. Assur. Magaz. J. Inst. Actuaries (Lond.). 1860;8:301-310.
[Google Scholar]
Marshall A.W., Olkin I., . Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families. New York: Springer-Science; 2007.
Scollnik D.P.M., . Simulating random variates from Makeham’s distribution and from others with exact or nearly Log-Concave densities. Trans. Soc. Actuaries. 1995;47:409-437.
[Google Scholar]