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Applied Mathematical Sciences, Vol. 8, 2014, no. 150, 7475 - 7485
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.49749
Bayesian Estimation of Rayleigh Distribution
Based on Generalized Order Statistics
Chansoo Kim
Department of Applied Mathematics
Kongju National University, Gongju, Korea
Keunhee Han
Department of Applied Mathematics
Kongju National University, Gongju, Korea
(corresponding author)
c 2014 Chansoo Kim and Keunhee Han. This is an open access article
Copyright distributed under the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Abstract
Exact expressions for single and product moments of generalized order statistics are derived when the underlying population is assumed to
have a Rayleigh distribution. Bayesian estimators and highest posterior
density (HPD) credible intervals are obtained for the scale parameter
and reliability function based on generalized order statistics. We also
derive the Bayes predictive estimator and HPD prediction interval for
an independent future observations.
Keywords : Bayes estimation, Generalized order statistics, HPD credible
interval
1
Introduction
Let F be an absolutely continuous distribution function with density function
f . Let n be a natural number and k > 0, m1 , · · · , mn−1 ∈ R be given P
constants
n−1
such that γi = k + n − i + Mi > 0 for all 1 ≤ i ≤ n − 1, where Mi = j=i
mj .
Suppose Y (1, n, m, k), · · · , Y (n, n, m, k) are the first n of the generalized order
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Chansoo Kim and Keunhee Han
statistics based on F . The concept of generalized order statistics is introduced
by Kamps (1995). The joint probability density function f1,··· ,n (y1 , · · · , yn ) is
given by
f1,··· ,n (y1 , · · · , yn ) = k
n−1
Y
j=1
!
γj
n−1
Y
!
[F̄ (yi )]mi f (yi )
k−1
F̄ (yn )
f (yn ), (1.1)
i=1
where F −1 (0) ≤ y1 < · · · < yn < F −1 (1).
If mi = 0 and k = 1, then Y (r, n, m, k) reduces to the ordinary rth order
statistic and (1.1) is the joint probability density function of n order statistics
Y1 ≤ · · · ≤ Yn . If mi = −1 and k = 1 then (1.1) is the joint probability density
function of the first n upper record value of a sequence {Yn , n ≥ 1} of independent and identically distributed random variables with distribution function F .
Also, if mi = Ri , 1 ≤ i ≤ n − 1, k = Rn + 1 and Ri is natural number, (1.1) is
the joint probability density function of (Y1R1 , · · · , YnRn ) based on progressive
type II censored sample with censoring scheme (R1 , · · · , Rn ). Therefore, the
notion of generalized order statistics provides a unified approach to some distributional and moment properties of ordered random variables. The results
obtained in this paper can be generalized to the Rayleigh distribution based on
progressive type II censored sample in Wu et al. (2006). Kim and Han (2009)
considered the estimation of the scale parameter of Rayleigh distribution under
general progressive censoring.
For various distributional properties and estimation of generalized order
statistics, Ashanullah (1996, 2000), Habibullah and Ahsanullah (2000) and
Malinowsska et al. (2006) considered the minimum variance unbiased estimator of location and scale parameters of uniform, exponential, Pareto II and
Burr XII distributions, respectively. Recently, Abushal and Al-Zaydi (2012)
considered the prediction of mixture of Rayleigh distribution using MCMC
algorithm.
Let Y be a random variable whose probability density function(p.d.f) is
given by
y2
y
(1.2)
g(y) = 2 exp − 2 ,
σ
2σ
and its corresponding cumulative distribution function(c.d.f) is
y2
G(y) = 1 − exp − 2 .
2σ
(1.3)
The Rayleigh distribution is widely used in communication engineering and
life testing of electro vacum devices and is a special case of the two parameter
Weibull distribution.
Bayesian estimation of Rayleigh distribution under GOS
7477
Let X = Y /σ. Then the random variable X has a standard form of the
Rayleigh distribution with c.d.f and p.d.f are written by respectively
2
x
,
F (x) = 1 − exp −
2
f (x) = xF̄ (x),
(1.4)
where F̄ (x) = 1 − F (x).
In this paper, our main object is to describe the variance, covariance and
Bayes estimation procedures for the scale parameter σ of the Rayleigh distribution based on generalized order statistics assuming the natural conjugate prior.
The rest of this paper is organized as follows. In Section 2, a brief description
of generalized order statistics and some distributional properties of the generalized order statistics from Rayleigh distribution are considered. In Section
3, we suggest Bayes estimators and highest posterior density (HPD) credible
intervals for the scale parameter and the reliability function of Rayleigh distribution using generalized order statistics. Also, we consider the Bayesian two
sample prediction problems.
2
Variance and Covariance of Generalized Order Statistics
Let Y (1, n, m, k), · · · , Y (n, n, m, k), (k ≥ 1, m is a real number), be the first
n of the generalized order statistics from {Yn , n ≥ 1}. Then X(r, n, m, k) =
Y (r, n, m, k)/σ, r = 1, · · · , n are the sequence of generalized order statistics
from {Xn , n ≥ 1}. Let
vrr = V ar[X(r, n, m, k)],
ers = E[X(r, n, m, k)X(s, n, m, k)],
vrs = Cov[X(r, n, m, k), X(s, n, m, k)], r, s = 1, · · · , n, r < s.
Reverting to the observations, we have
V ar[Y (r, n, m, k)] = σ 2 vrr ,
Cov[Y (r, n, m, k), X(s, n, m, k)] = σvrs .
(2.1)
(2.2)
We consider the variance and covariance of generalized order statistics from
the c.d.f. F (x). Let Xr ≡ X(r, n, m, k) and Xs ≡ X(s, n, m, k) be the rth and
sth generalized order statistics from c.d.f. (1.4), respectively.
The marginal density function of the rth generalized order statistics when
m1 = · · · = mn−1 = m and Xr based on the distribution function F and density
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Chansoo Kim and Keunhee Han
function f is obtained by integrating out x1 , · · · , xr−1 and xr+1 , · · · , xn from
(1.1) as (for details, refer to Kamps (1995))
fr,n,m,k (x) =
where cr =
Qr
j=1
cr
r−1
(Fθ (x)),
[F̄ (x)]γr −1 fθ (x)gm
Γ(r)
(2.3)
γj , γj = k + (n − j)(m + 1) and
1
1 − (1 − x)m+1 , m 6= −1,
m+1
= −ln(1 − x), m = −1.
gm (x) =
By substituting F̄ (x) and f (x) from (1.4) in (2.3), the marginal density function fr,n,m,k (x) is given by
r−1
1
x
m+1
r−1
γr x2
(m + 1)x2
× exp −
1 − exp −
.
2
2
cr
fr,n,m,k (x) =
Γ(r)
(2.4)
From (2.4), the qth moment of the rth generalized order statistics is given
by
r−1 Z ∞
γr x2
1
cr
q+1
E[X(r, n, m, k) ] =
x exp −
Γ(r) m + 1
2
0
r−1
2
(m + 1)x
× 1 − exp −
dx
2
r−1 X
r−1
Γ q+2
cr
1
r−1
l
2
(−1)
=
.
q+2
l
Γ(r) m + 1
2
2 γr−l
l=0
2
q
Therefore, the first and second moments are obtained by letting q = 1 and
q = 2, respectively, as follows:
cr
E[X(r, n, m, k)] =
Γ(r)
1
m+1
r−1 r
r−1
π X
1
r−1
l
(−1)
3 , (2.5)
l
2 l=0
[γr−l )] 2
and
cr
E[X(r, n, m, k) ] =
Γ(r)
2
1
m+1
r−1 X
r−1
2
r−1
l
(−1)
.
2
l
[γ
r−l ]
l=0
(2.6)
Bayesian estimation of Rayleigh distribution under GOS
7479
Lemma 2.1 For 1 ≤ r < s ≤ n, let Xr and Xs be the rth and sth generalized
order statistics from the c.d.f in (1.4). Then
r−1
1
cs
E[X(r, n, m, k)X(s, n, m, k)] =
Γ(r)Γ(s − r) m + 1
r−1 s−r−1
X
X
r−1
s−r−1
l1 +l2
×
(−1)
l1
l2
l1 =0 l2 =0
h
i
a
2 −b2l2 + (al1 ,l2 + bl2 )22 F1 21 , 2, 32 ; − lb1l,l2
2
×
,
al1 ,l2 b2l2 (al1 ,l2 + bl2 )2
(2.7)
where al1 ,l2 = (m + 1)(s − r + l1 − l2 ), bl2 = γs−l2 and 2 F1 denotes the hypergeometric function.
Proof To find the covariance between X(r, n, m, k) and X(s, n, m, k), r < s,
we consider the joint probability density function fr,s (u, v) of X(r, n, m, k)
and X(s, n, m, k). The joint p.d.f. fr,s (u, v) of X(r, n, m, k) and X(s, n, m, k)
is given by
fr,s (u, v) =
γs −1
m
cs
f (u)f (v)
F̄ (u) (gm (F (u)))r−1 F̄ (v)
Γ(r)Γ(s − r)
× [gm (F (v)) − gm (F (u))]s−r−1 ,
(2.8)
where F −1 (0) < u < v < F −1 (1).
By substituting F̄ (x) and f (x) from (1.4) in (2.8) and after some algebraic
computations, the product moment of X(r, n, m, k) and X(s, n, m, k) is given
by
Z ∞Z v
uvfr,s (u, v)dudv
E[U V ] =
0
0
r−1 X
r−1 s−r−1
X
cs
1
r−1
l1 +l2
=
(−1)
l1
Γ(r)Γ(s − r) m + 1
l1 =0 l2 =0
r
Z ∞
r v
av 2
1 π
a
s−r−1
×
− exp −
+ 3
erf
v
l2
a
2
2
a2 2
0
r−1 X
r−1 s−r−1
X
cs
1
r−1
l1 +l2
=
(−1)
l1
Γ(r)Γ(s − r) m + 1
l1 =0 l2 =0
2
s−r−1
×
−
l2
al1 ,l2 (al1 ,l2 + bl2 )2
2
1
3 al1 ,l2
+
, 2, ; −
,
2 F1
al1 ,l2 b2l2
2
2
bl 2
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Chansoo Kim and Keunhee Han
√ Rz 2
where erf (z) = 2/ π 0 e−t dt which is the integral of the Gaussian distribution and 2 F1 denotes a hypergeometric function (see Abramowitz and Stegun
(1970)). Hence the product moment between X(r, n, m, k) and X(s, n, m, k)
is given by (2.7).
According to Lemma 2.1, (2.1) and (2.2), the variance of rth generalized
order statistics and covariance between Y (r, n, m, k) and Y (s, n, m, k), r < s
are obtained by respectively
2 π
2
,
V ar(Y (r, n, m, k)) = σ br,m 2ar,2 − br,m ar, 3
2
2
and
Cov(Y (r, n, m, k), Y (s, n, m, k)) = σ 2 (ers − 4bs,m br,m as,2 ar,2 ) ,
where
bi,m
ai,j
i−1
ci
1
, i = r, s,
=
Γ(i) m + 1
i−1
X
1
3
i−1
l
=
(−1)
, i = r, s; j = 2, ,
j
l
[γi−l ]
2
l=0
and ers is given by (2.7).
Remark If we take m = 0 and k = 1 then X(r, n, m, k) coincides with the rth
order statistics from the sample of size n. In this case, γi−l = n − i + l + 1,
and bi,0 = ci /Γ(i), i = r, s.
3
3.1
Bayes Estimation
Bayes estimators
Let m̃ = (m1 , · · · , mn−1 ) and Y = (Y (1, n, m̃, k), · · · , Y (n, n, m̃, k)) be the
first n of the generalized order statistics of {Yn , n ≥ 1} from the Rayleigh
distribution given by (1.2). Then by substituting G(y) and g(y) from (1.2)
and (1.3) in (1.1), the likelihood function L(θ; Y) is proportional to
!
Pn−1
2n
2
2
(m
+
1)y
+
ky
1
i
i
n
exp − i=1
.
(3.1)
L(σ; Y) ∝
σ
2σ 2
It is easy to obtain the maximum likelihood estimator (MLE) of σ which is
s
Pn−1
2
2
i=1 (mi + 1)yi + kyn
σ̂M =
.
2n
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Bayesian estimation of Rayleigh distribution under GOS
By the invariance property of the MLE, the MLE of the reliability function
R(t) with fixed t > 0 is given by
t2
.
R̂M = exp −
2σ̂M
Since the σ is a random variable, we consider the natural conjugate family
of prior distributions for the σ used in Fernández (2000). It is
2α+1
1
β
π(σ) ∝
exp − 2 ,
σ
2σ
σ > 0,
(3.2)
where α > 0 and β > 0. This density is known as the square-root inverted
gamma distribution. When β = 0 in (3.2), π(σ) reduces to a general class
of improper priors. When α = 0 and β = 0, an improper prior for σ is the
Jeffreys (1961) prior. Note that if λ = 1/σ 2 , the density function of λ has a
gamma distribution with parameters α and β/2.
By combining (3.1) with (3.2), the posterior density of σ is given by
n+α 2n+2α+1
2
2
+
β
+
ky
(m
+
1)y
1
i
n
i
i=1
π(σ|Y) =
n+α−1
2
Γ(n + α)
σ
" n−1
#!
1 X
× exp − 2
(mi + 1)yi2 + kyn2 + β
.
2σ i=1
Pn−1
(3.3)
It is well known that the Bayes estimator of σ under squared error loss is
the posterior mean. It follows from (3.3) that the Bayes estimator of σ is the
posterior mean
s
n+α
Pn−1
2
2
i=1 (mi + 1)yi + kyn + β
σ̂B = E[σ|Y] =
2
1
Γ n+α− 2
×
.
(3.4)
Γ(n + α)
Another problem of interest is to estimating the reliability function R(t)
1/2
with fixed t > 0. Let R = exp (−t2 /2σ 2 ) and substituting σ = (t2 (−2 log R)−1 )
into (3.3), the posterior probability density function of R = R(t) is given by
1
π(R|Y) =
Γ(n + α)
×R
"P
n−1
i=1 (mi
+ 1)yi2 + kyn2 + β
t2
Pn−1
2 +β
(mi +1)yi2 +kyn
i=1
−1
t2
.
#n+α
(− log R)n+α−1
(3.5)
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Chansoo Kim and Keunhee Han
From (3.5), the Bayes estimator of R(t) is given by
#n+α
" P
n−1
2
2
+
β
+
ky
(m
+
1)y
i
n
i
i=1
.
R̂B = E[R(t)|Y] =
P
n−1
2
2
t + i=1 (mi + 1)yi + kyn2 + β
The highest posterior density (HPD) estimation is another popular method
used by the Bayesian perspective. This parameter estimation is based on the
maximum likelihood principle; hence it leads to the mode of the posterior
density or HPD estimator. Since the posterior density (3.3) is unimodal, the
HPD estimator of σ, denoted by σ̂H , is obtained by
sP
n−1
2
2
i=1 (mi + 1)yi + kyn + β
.
σ̂H =
2(n + α) + 1
From (3.5), the HPD estimator of R(t), denoted by R̂H is
!
(n + α − 1)t2
.
R̂H = exp − Pn−1
2
2 + β − t2
+
ky
(m
+
1)y
i
n
i
i=1
Remark If we take mi = Ri , i = 1, · · · , n−1 and k = Rn +1 then X(r, n, m, k)
coincides with the rth progressive type II order statistics from the sample of
size n. In this case, the Bayes estimators of the scale parameter σ and the
reliability function R(t) with fixed t > 0 are given by respectively,
v "
#
u
n
1
X
u1
Γ
n
+
α
−
2
σ̃B = t β +
(Ri + 1)yi2
,
2
Γ(n
+
α)
i=1
and
Pn
n+α
(Ri + 1)yi2 + β
i=1
R̃B = 2 Pn
.
t + i=1 (Ri + 1)yi2 + β
3.2
HPD credible intervals
A set C ⊂ (0, ∞) of the form C = {σ : π(σ|y) ≥ cα }, where cα is the
largest constant such that P (σ ∈ C|y) = 1 − α, is called a 100(1 − α)%
HPD credible set for σ. Hence, a 100(1 − α)% HPD credible interval choose
(cl , cu ) consist of all values of σ with π(σ|y) > cα . Due to the unimodality
of
R cu(3.3), the 100(1 − α)% HPD credible interval (cl , cu ) for σ must satisfy
π(σ|y)dσ = 1 − α and π(cu |y) = π(cl |y).
cl
With some algebra, the 100(1 − α)% HPD credible interval (cl , cu ) for σ is
given by the simultaneous solution of the equations
γ(v1 , n + α) − γ(v2 , n + α) = 1 − α,
Bayesian estimation of Rayleigh distribution under GOS
and
cu
cl
7483
2(n+α)+1
= exp(v1 − v2 ),
P
Pn−1
2
2
2
2
2
where v1 = [ n−1
i=1 (mi + 1)yi + kyn + β]/(2cl ), v2 = [
i=1 (mi + 1)yi + kyn +
β]/(2c2u ), and γ(·, ·) denotes the incomplete gamma function defined by
Z x
1 a−1 −z
γ(a, x) =
z e dz.
0 Γ(a)
Similarly, the 100(1 − α)% HPD credible interval (dl , du ) for R(t) must
satisfy the following two equations:
γ(−δlogdl , n + α) − γ(−δlogdu , n + α) = 1 − α,
and
logdu
logdl
n+α−1
=
dl
du
δ−1
,
P
2
2
2
where δ = [ n−1
i=1 (mi + 1)yi + kyn + β]/t .
3.3
Prediction of future observations
Let Y = (Y (1, n, m̃, k), · · · , Y (n, n, m̃, k)) be the first n of the generalized
order statistics of {Yn , n ≥ 1} from the Rayleigh distribution given by (1.2)
where m̃ = (m1 , · · · , mn−1 ). Two sample schemes are used in which informative sample is the first n of generalized order statistics and X(1) < · · · < X(n)
are the order statistics of a future sample. Let Xs , s = 1, · · · , n be the order
statistics in a sample of size n with same distribution.
The p.d.f of the sth (1 ≤ s ≤ n) order statistics is
s−1
xs
(n − s + 1)x2s
x2s
N!
exp −
1 − exp − 2
f (xs |σ) =
(s − 1)!(n − s)! σ 2
2σ 2
2σ
(3.6)
for xs > 0.
From (3.6) and (3.3), the predictive distribution of Xs given Y is
Z ∞
f (xs |Y) =
f (xs |σ)π(σ|Y)dσ,
0
!n+α s−1
n−1
X
X
2n!(n + α)
s−1
2
2
l
(−1)
=
(mi + 1)yi + kyn + β
l
(s − 1)!(n − s)! i=1
l=1
!−(n+α+1)
n−1
X
×xs (n − s + l + 1)x2s +
(mi + 1)yi2 + kyn2 + β
(3.7)
i=1
7484
Chansoo Kim and Keunhee Han
for xs > 0.
Under squared error loss, the Bayes predictive estimator of Xs is the expectation of the predictive distribution. Therefore, from (3.8) it follows
p
s−1
X
n! π2
3
s−1
l
(n − s + l + 1)− 2 σ̂B
(−1)
X̂s = E(Xs |Y] =
l
(s − 1)!(n − s)! l=1
where σ̂B is the Bayes estimator of θ given by (3.4).
According to (3.6), the predictive reliability function of xs is given by
n+α
Pn−1
2
2
2n!(n + α)
i=1 (mi + 1)yi + kyn + β
Rs (t|Y) =
(s − 1)!(n − s)!
s−1
X
1
s−1
l
×
(−1)
n+α
l
(n − s + l + 1)wl,t
l=0
n+α
Pn−1
2
2
.
where wl,t =
i=1 (mi + 1)yi + kyn + β + (n − s + l + 1)t
TheR100(1−α)% HPD credible interval (cl , cu ) for Xs should simultaneously
c
satisfy clu f (xs |y)dxs = 1 − α and f (cl |y) = f (cu |y). Therefore, after some
algebra, the 100(1 − α)% HPD credible interval (cl , cu ) for Xs is given by the
simultaneous solution of the equations
s−1 (
j
−(n+α)
s−1 (−1)
X
l
n!
(n − s + j + 1)c2l
1+
s!(n − s)! j=0 n − s + j + 1
V
−(n+α) )
(n − s + j + 1)c2u
− 1+
=1−α
V
and
cl Ψ (V, n + α + 1; cl ) = cu Ψ (V, n + α + 1; cu )
where V =
Pn−1
2
2
i=1 (mi +1)yi +kyn +β
and Ψ(a, b, c) =
Ps−1
j=0 (−1)
j
s−1
l
(1+
(n − s + j + 1)c2 /a)−b .
References
1 Abramowitz, M. and Stegun, I. A. (1970). Handbook of Mathematical
Functions, Dover Publications, New Yorks.
2 Ahsanullah, M. (1996). Generalized order statistics from two parameter
uniform distribution, Communications in Statistics: Theory and Methods,
25, 2311-2318.
Bayesian estimation of Rayleigh distribution under GOS
7485
3 Abushal, T.A. and Al-Zaydi, A.M. (2012). Prediction based on generalized
order statistics from a mixture of Rayleigh distribution using MCMC
algorithm, Open Journal of Statistics, 2, 356-367.
4 Ahsanullah, M. (2000). Generalized order statistics from exponential distribution, Journal of Statistical Planning and Inference, 85, 85-91.
5 Fernández, AJ. (2000). Bayesian inference from type II doubly censored
Rayleigh data, Statistics and Probability Letters, 48, 393-399.
6 Habibullah, M. and Ahsanullah, M. (2000). Estimation of parameters of
a Pareto distribution by generalized order statistics, Communications in
Statistics: Theory and Methods,29, 1597-1609.
7 Malinowska, I., Pawlas, P. and Szynal, D. (2006). Estimation of location
and scale parameters for the Burr XII distribution using generalized order
statistics, Linear Algebra and Its Applications, 417, 150-162.
8 Kim, C. and Han K. (2009). Estimation of the scale parameter of the
Rayleigh distribution under general progressive censoring, Journal of the
Korean Statistical Scociety, 38, 239-246.
9 Kamps, U. (1995). A concept of generalized order statistics, B. G. Teubner
Stuttgart, 25, 2311-2318.
10 Wu, S-J., Chen, D-H. and Chen, S-T. (2006). Bayesian inference for
Rayleigh distribution under progressive censored sample, Applied Stochastic Models in Business and Industry, 22, 269-279.
Received: September 7, 2014; Published: October 26, 2014