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A characterization of the Burr type XII distribution
0893-9659 191 $3.00 + 0.00 Copyright@ 1991 Pergamon Press plc
Appl. Math. Lett. Vol. 4, No. 1, pp. 59-61, 1991 Printed in Great Britain. All rights reserved
A Characterization
of the
Burr
Type
XII
Distribution
ESSAM K. AL-HUSSAINI
University
of Assiut,
(Received
March
Egypt
1990)
Abstract. A generalization of a theorem in Galambos and Katz (1978) is made and used to characterize the Burr type XII distribution. Independence of functions of the order statistics is utilized in the characterization. 1. INTRODUCTION
The Burr type XII distribution, being a member of the Burr system [see, Johnson and Katz (197O)j has gained more attention in the last decade due to the potentiality of using it in practical situations. Among others, Papadopoulos (1978), Tadikamalla (1980), Lewis (1981), Evans and Ragab (1983), Wingo (1983), Nigm (1988) and Al-Hussaini et al. (1990) used the distribution as a life time model with inferences being made about its parameters. Khan and Khan (1987) ch aracterized the distribution by using the moments of order statistics. A random variable X is said to follow the Burr type XII distribution with parameters (c, Ic), denoted by Burr(c, k), if its distribution function is given by F(Z) with density
= 1 - (1+
q-k
I > 0, (c > 0, k > 0),
(1)
function
(2) Random variables, which are functions of order statistics, that their independence characterizes the distribution. 2. CHARACTERIZATION
are constructed
OF BURR TYPE XII
A generalization of Theorem (3.3.4) in Galambos characterize the Burr type XII distribution.
in such a way
DISTRIBUTION
and Kotz (1978) is made
a.nd used to
THEOREM (1). Suppose that g2(q,. . . ,x,) 1 0,. . . ,gn(xl,. . . ,x,) 2 0 are measurable functions. Let X1,. . . ,X, be independent random variables with common absolutely continuous distribution F(c). Suppose that Xl:,, . . . , X,:, are the order statistics of X1, . . . , X, and that 21
=
Xl:n,
52
all have finite expectations. such that 00 Q1 J Yl
...J
=92(X1:7&,*.
. ,xX*>,
If 21,. . . , 2,
*. ‘,
za
are independent,
=
!.h(Xl:n,
then
*. ‘,XruL),
there exists
a constant
A
(3)
g?(yl,...,yn)...gn(yl,...,yn)f(yn)...f(yz)dyn...dyn=A(1-F(~l))~-~,
Yn-1
forn=2,3,4,...
and all ~1, where F’(x)
PROOF: A generalized version can be utilized in proving (3).
= f(z).
of characteristic
functions
used in proving
Theorem
(3.3.4)
Typeset by A,++?-QX 59
E.K. AL-HUSSAINI
60
REMARK.
Theorem
(1) remains
true if 21 = Xltn
is replaced
by Z1 = 1 -t- Xl:,.
(2). Let X1,. . . , X, be independently identically distributed positive random according to the absolutely continuous distribution function Fx(~). Let be the order statistics of Xl,. . . ,X,. Then X1,. . . ,X, are Burr(c, k) if . . . , X,:,
THEOREM
variables Xl:n,
and only if, for c > 0, 21 =
1+ XEzn, 22 = 1 + x;:,
1 + x;:,
1 + xg,
z3 =
,
1 + xi,,
1 + xg:, ’ * . . ’ zn = 1+ x;_,:,
(4
’
are independent. PROOF:
If X is Burr(c,
k), then the random FY(Y)
If,forj=l,...,
= I-
variable
Y = Xc has the distribution y > 0.
y>-!
(1+
function (5)
n, Yj = XfIn, then
fYI,...,Y,(Yl, j=l
1 -k-l
= nlkn .
fi(l+Yj) [ j=l
,
O
If Y2
I+ zz=l+y,...,
z1=1+y1,
1 + Yn
z,=
1
l$Y,-1’
then
fZ1,...,Z,(~l,~~ * 2”) 9
where IJI = z~-1z~-2.. Therefore, fz,
,.,,, ~,(a,.
. . , zn)
=
IJI fu, (...,Y,(%l
-
1,
*. *, %1%2‘. * zn - I>,
. zE_2 zn-1. n q-
=
1 n-2 z2
. . . ~z-2
~~-1
n! k n z1-k-l(%1%2)-k-1..
-(fik+l)%;[(n-WI = n! k”% 1 Hence,
1, %1%2-
Z1, . . . , Z,, are independent, fZj(zj)
=
(~
j
+
1)
~
= Tf.**
%n > 1.
it follows from (3) that for n = 2,3,4,
(6) ...
(~)f(y,)...f(yl)dy,...dyz.
(7)
(~)f(y,)...f(y3)dyn...dy3.
(8)
7
Yl Ya
%l > 1,“‘)
%j> 1.
~,~“n-j+l’k+ll,
On the other hand, if 21, . . . , Z,, are independent,
A(1 - F(y#-l
)
for j = 1,. . . , n,
where, -
. . * %;~~+1)%-(‘“+‘) n
. (ZIZZ. * .%,)-k-1
yn-1
Similarly, B(1 - F(y#-2
= T*..
7
Ya
Substitution
Y,-1
of (8) into (7) yields co
41-
F(Yl))“_’
=
J
Yl
w
-
Y2 + 1
F(y2)y2
y1+1
(
>
f(y2) dy2.
61
A characterization of the Burr type XII distribution
Multiplying obtain
both sides by y1 + 1, differentiating F(Yl))
k(l-
with respect
= (1+
where k is a constant. Differentiating both sides of (9) with respect
Cl+ which has a solution
(9)
lV(Yl)
the differential
equation,
= 0,
of f(Y1)
where the constant Therefore, f(yl)
we
Yl)f(Yl),
to ~1, we obtain
+ (k +
~1) I'
to y1 and then simplifying,
=
41 + Yl)-(k+l),
a is such that f(y) = k(l + y~)-(~+‘),
is a density y1 > 0.
Yl > 0,
function.
REMARICS.
(1) IfXl)
. . . , X, are independently Burr(c, k) and ifXl:,, . . . ,X,:, tics, it follows from (6), that for j = 1,. . . , n, (Xc,, E 0), 1 + x;:n
-
zj = 1+ Xj&
Pareto
((Y),
are their order statis-
II% = (72 - j + 1)k.
(2) Other
distributions such as the Lomax, the Weibull-gamma mixture, the Weibullexponential mixture and the log logistic distributions can be similarly characterized as they are versions of the Burr type XII distribution with different parameters, see Tadikamalla (1980). REFERENCES
1. E.K.
AL-Hussaini,
based on censored Mathematics
M.A.
Moussa
data:
a comparative
and Statistics
2. I.G. Evans and AS.
Jaheen, Estimation
study,
presented
under the Burr type XII failure model
at Assiut
First International
Conference
of
(1990).
Hagab, Bayesian inference given a type-2 censored sample from a Burr distribution,
Commun-Statist.-Th. 3. J. Galambos
and Z.F.
Method8 A 12, 1569-1580 (1983). Characterization of Probability
and S. Katz,
Distributions,
Springer-Verlag,
pp. 51-53,
(1978).
Continuous
4. N. Johnson and S. Katz, 5. A.H.
Kahn and AI.
MetTon 45, 21-29 6. Al&da
Khan,
Moments
Univariate
7. A.M.
Nigm,
Haughton
Mifflin, pp. 30-31,
(1970).
and its characterization,
(1987).
W. Lewis, The Burr distribution
theory applications,
Distributions-f,
of order statistics from Burr distribution
Ph.D.
Prediction
Thesis,
bounds
as a general parametric
University
of North Carolina
for the Burr model,
family in survivorship of Chapel
Hill (1981).
Commun-Statist.-Th.
Methods
and reliability
A 17,
287-297
(1988). 8. A.S.
Papadopoulos,
Rel. Re-5, 369-371 9. P.R. Tadikamalla, 10. D.R.
Wingo,
Biometrical
N4L
4:1-s
The Burr distribution
as a failure model from a Bayesian approach,
IEEE
Trans.
(1978). A look at the Burr and related distributions,
Maximum
likelihood
J. 25, 77-84 (1983).
methods
Inter. Statist. Rev.
48, 337-344
for fitting the Burr type XII distribution
(1980).
to life test data,
Appl. Math. Lett. Vol. 4, No. 1, pp. 59-61, 1991 Printed in Great Britain. All rights reserved
A Characterization
of the
Burr
Type
XII
Distribution
ESSAM K. AL-HUSSAINI
University
of Assiut,
(Received
March
Egypt
1990)
Abstract. A generalization of a theorem in Galambos and Katz (1978) is made and used to characterize the Burr type XII distribution. Independence of functions of the order statistics is utilized in the characterization. 1. INTRODUCTION
The Burr type XII distribution, being a member of the Burr system [see, Johnson and Katz (197O)j has gained more attention in the last decade due to the potentiality of using it in practical situations. Among others, Papadopoulos (1978), Tadikamalla (1980), Lewis (1981), Evans and Ragab (1983), Wingo (1983), Nigm (1988) and Al-Hussaini et al. (1990) used the distribution as a life time model with inferences being made about its parameters. Khan and Khan (1987) ch aracterized the distribution by using the moments of order statistics. A random variable X is said to follow the Burr type XII distribution with parameters (c, Ic), denoted by Burr(c, k), if its distribution function is given by F(Z) with density
= 1 - (1+
q-k
I > 0, (c > 0, k > 0),
(1)
function
(2) Random variables, which are functions of order statistics, that their independence characterizes the distribution. 2. CHARACTERIZATION
are constructed
OF BURR TYPE XII
A generalization of Theorem (3.3.4) in Galambos characterize the Burr type XII distribution.
in such a way
DISTRIBUTION
and Kotz (1978) is made
a.nd used to
THEOREM (1). Suppose that g2(q,. . . ,x,) 1 0,. . . ,gn(xl,. . . ,x,) 2 0 are measurable functions. Let X1,. . . ,X, be independent random variables with common absolutely continuous distribution F(c). Suppose that Xl:,, . . . , X,:, are the order statistics of X1, . . . , X, and that 21
=
Xl:n,
52
all have finite expectations. such that 00 Q1 J Yl
...J
=92(X1:7&,*.
. ,xX*>,
If 21,. . . , 2,
*. ‘,
za
are independent,
=
!.h(Xl:n,
then
*. ‘,XruL),
there exists
a constant
A
(3)
g?(yl,...,yn)...gn(yl,...,yn)f(yn)...f(yz)dyn...dyn=A(1-F(~l))~-~,
Yn-1
forn=2,3,4,...
and all ~1, where F’(x)
PROOF: A generalized version can be utilized in proving (3).
= f(z).
of characteristic
functions
used in proving
Theorem
(3.3.4)
Typeset by A,++?-QX 59
E.K. AL-HUSSAINI
60
REMARK.
Theorem
(1) remains
true if 21 = Xltn
is replaced
by Z1 = 1 -t- Xl:,.
(2). Let X1,. . . , X, be independently identically distributed positive random according to the absolutely continuous distribution function Fx(~). Let be the order statistics of Xl,. . . ,X,. Then X1,. . . ,X, are Burr(c, k) if . . . , X,:,
THEOREM
variables Xl:n,
and only if, for c > 0, 21 =
1+ XEzn, 22 = 1 + x;:,
1 + x;:,
1 + xg,
z3 =
,
1 + xi,,
1 + xg:, ’ * . . ’ zn = 1+ x;_,:,
(4
’
are independent. PROOF:
If X is Burr(c,
k), then the random FY(Y)
If,forj=l,...,
= I-
variable
Y = Xc has the distribution y > 0.
y>-!
(1+
function (5)
n, Yj = XfIn, then
fYI,...,Y,(Yl, j=l
1 -k-l
= nlkn .
fi(l+Yj) [ j=l
,
O
If Y2
I+ zz=l+y,...,
z1=1+y1,
1 + Yn
z,=
1
l$Y,-1’
then
fZ1,...,Z,(~l,~~ * 2”) 9
where IJI = z~-1z~-2.. Therefore, fz,
,.,,, ~,(a,.
. . , zn)
=
IJI fu, (...,Y,(%l
-
1,
*. *, %1%2‘. * zn - I>,
. zE_2 zn-1. n q-
=
1 n-2 z2
. . . ~z-2
~~-1
n! k n z1-k-l(%1%2)-k-1..
-(fik+l)%;[(n-WI = n! k”% 1 Hence,
1, %1%2-
Z1, . . . , Z,, are independent, fZj(zj)
=
(~
j
+
1)
~
= Tf.**
%n > 1.
it follows from (3) that for n = 2,3,4,
(6) ...
(~)f(y,)...f(yl)dy,...dyz.
(7)
(~)f(y,)...f(y3)dyn...dy3.
(8)
7
Yl Ya
%l > 1,“‘)
%j> 1.
~,~“n-j+l’k+ll,
On the other hand, if 21, . . . , Z,, are independent,
A(1 - F(y#-l
)
for j = 1,. . . , n,
where, -
. . * %;~~+1)%-(‘“+‘) n
. (ZIZZ. * .%,)-k-1
yn-1
Similarly, B(1 - F(y#-2
= T*..
7
Ya
Substitution
Y,-1
of (8) into (7) yields co
41-
F(Yl))“_’
=
J
Yl
w
-
Y2 + 1
F(y2)y2
y1+1
(
>
f(y2) dy2.
61
A characterization of the Burr type XII distribution
Multiplying obtain
both sides by y1 + 1, differentiating F(Yl))
k(l-
with respect
= (1+
where k is a constant. Differentiating both sides of (9) with respect
Cl+ which has a solution
(9)
lV(Yl)
the differential
equation,
= 0,
of f(Y1)
where the constant Therefore, f(yl)
we
Yl)f(Yl),
to ~1, we obtain
+ (k +
~1) I'
to y1 and then simplifying,
=
41 + Yl)-(k+l),
a is such that f(y) = k(l + y~)-(~+‘),
is a density y1 > 0.
Yl > 0,
function.
REMARICS.
(1) IfXl)
. . . , X, are independently Burr(c, k) and ifXl:,, . . . ,X,:, tics, it follows from (6), that for j = 1,. . . , n, (Xc,, E 0), 1 + x;:n
-
zj = 1+ Xj&
Pareto
((Y),
are their order statis-
II% = (72 - j + 1)k.
(2) Other
distributions such as the Lomax, the Weibull-gamma mixture, the Weibullexponential mixture and the log logistic distributions can be similarly characterized as they are versions of the Burr type XII distribution with different parameters, see Tadikamalla (1980). REFERENCES
1. E.K.
AL-Hussaini,
based on censored Mathematics
M.A.
Moussa
data:
a comparative
and Statistics
2. I.G. Evans and AS.
Jaheen, Estimation
study,
presented
under the Burr type XII failure model
at Assiut
First International
Conference
of
(1990).
Hagab, Bayesian inference given a type-2 censored sample from a Burr distribution,
Commun-Statist.-Th. 3. J. Galambos
and Z.F.
Method8 A 12, 1569-1580 (1983). Characterization of Probability
and S. Katz,
Distributions,
Springer-Verlag,
pp. 51-53,
(1978).
Continuous
4. N. Johnson and S. Katz, 5. A.H.
Kahn and AI.
MetTon 45, 21-29 6. Al&da
Khan,
Moments
Univariate
7. A.M.
Nigm,
Haughton
Mifflin, pp. 30-31,
(1970).
and its characterization,
(1987).
W. Lewis, The Burr distribution
theory applications,
Distributions-f,
of order statistics from Burr distribution
Ph.D.
Prediction
Thesis,
bounds
as a general parametric
University
of North Carolina
for the Burr model,
family in survivorship of Chapel
Hill (1981).
Commun-Statist.-Th.
Methods
and reliability
A 17,
287-297
(1988). 8. A.S.
Papadopoulos,
Rel. Re-5, 369-371 9. P.R. Tadikamalla, 10. D.R.
Wingo,
Biometrical
N4L
4:1-s
The Burr distribution
as a failure model from a Bayesian approach,
IEEE
Trans.
(1978). A look at the Burr and related distributions,
Maximum
likelihood
J. 25, 77-84 (1983).
methods
Inter. Statist. Rev.
48, 337-344
for fitting the Burr type XII distribution
(1980).
to life test data,