DISTRIBUTION BASED ON RECORD STATISTICS
VASILE PREDA and ION MIERLUS-MAZILU
Single and joint moments of record statistics are derived for the exponentiated Weibull distribution. Also, for theαth moment of the kth record statistics are obtained.
AMS 2010 Subject Classification: 60-08, 62D05, 65C60.
Key words: Exponentiated Weibull distributions, record statistics.
1. INTRODUCTION
The Weibull distribution is a very popular distribution. It is called af- ter Waladdi Weibull, a Swedish physicist. He used it in 1939 to analyze the breaking strength of materials. Since then it has been widely used for mo- delling phenomena with monotone failure rates. It is not useful for modelling phenomena with non monotone failure rates.
The Weibull distribution has been shown to be useful for modelling and analyzing of life time data in medical, biological and engineering sciences. Some applications of the Weibull distribution in forestry are given in Green et al.
(1994). A great deal of research has been done on estimating the parameters of the Weibull distribution using both classical and Bayesian techniques. A good summary of this work can be found in Johnson et al. (1994). Recently, Hossain and Zimmer (2003) have discussed some comparisons of estimation methods for Weibull parameters using complete and censored samples.
Record values and the associated statistics are of interest and important in many real life applications. In industry many products fail under stress.
For example, a wooden beam breaks when sufficient perpendicular force is applied to it, an electronic component ceases to function in an environment of too high temperature, and a battery dies under the stress of time. But the precise breaking stress or failure point varies even among identical items.
Hence, in such experiments, measurements may be made sequentially and only the record values are observed. Thus, the number of measurements made is considerably smaller than the complete sample size. This “measurement
MATH. REPORTS13(63),3 (2011), 299–315
saving” can be important when the measurements are costly if the entire sample was destroyed. For more examples, see Gulati and Padgett (1994).
There are also situations in which an observation is only stored if it is a record value. These include studies in meteorology, hydrology, seismol- ogy, athletic events and mining. In recent years, there has been much work on parametric and nonparametric inference based on record values. Among others are Resnick (1987), Nagaraja (1988), Ahsanullah (1993, 1995), Arnold et al.(1992, 1998), Gulati and Padgett (1994), Raqab and Ahsanullah (2001), Raqab (2002), Preda et al. (2010a), Preda et al. (2010b) and Panaitescu et al.(2010). Sultan and Balakrishnan (1999) have developed some inferential method for the location and scale parameters of Weibull distribution. Ah- madi and Arghami (2001) discussed a comparison between the information (in Fisher’s sense) contained in a set of nupper record values with the Fisher information contained in n independent and identically distributed observa- tions from the original distribution. They showed that for the Weibull model the upper record values contain more information than the same number of independent and identically distributed observations.
The Exponentiated Weibull family is an extension of the Weibull fa- mily obtained by adding an additional shape parameter. The beauty and im- portance of this distribution lies in its ability to model monotone as well as non-monotone failure rates which are quite common in reliability and biolo- gical studies.
Proposed by Mudholkar et all (1995), the probability density function (pdf) of this distribution is
(1) f(x, γ, β) =βγ 1−e−xγβ−1
xγ−1e−xγ, x >0, β, γ >0 while the cumulative distribution function (cdf) is
(2) F(x, γ, β) = 1−e−xγβ
, x >0, β, γ >0.
Hereβ is the shape parameter. This distribution reduces to the Weibull distribution if β= 1 and to the exponential distribution if γ = 1 and β = 1.
Some of the most important features and characteristics of a distribution can be studied through moments. The moments are
µ0k= (−1)k dk (ds)kG(s)
s=0
=βΓ k
γ + 1 "
1 +
∞
X
i=1
ai
(i+ 1)−
k
γ+1
#
, k= 1,2,3, . . . ,
where G(s) is the moment generating function and theai are defined as ai = (β−1) ((β−1)−1)· · · (β−1)−i−1
i! (−1)i, i= 1,2,3, . . . . In this paper we derive exact expression for single and mixed record statistics. Further, when α/γ is a nonnegative integer, an exact expression for theαth moment of thekth record statistics is given. Also, in the general case some inequalities are derived.
2. RECORD STATISTIC
In the context of the order statistics model and reliability theory, the life length of the r-out-of-n system is the (n−r+ 1)th order statistics in a sample of size n.
Another related model is the model of record statistics defined by Chan- dler (1952) as a model for successive extremes in a sequence of independent, identically distributed (iid) random variables (r.v.’s). This model takes a cer- tain dependence structure into consideration. That is, the life-length distri- bution of the components in the system may change after each failure of the components. For this type of models, we consider the lower record statistics.
If various voltages of an equipment are considered, just the voltages less than the previous can be recorded. These recorded voltages are the lower record value sequence.
Now, let Xn, n ≥ 1, be an infinite sequence of independent and iden- tically distributed r.v.’s from an absolutely continuous distribution function F and pdf f. Let Xi,j denote the ith order statistic of the random sample X1, X2, . . . , Xj and letFi,jbe its cdf. . LetTk= min{X1, X2, . . . , Xk}, k≥1.
We say that Xj is a lower record value of this sequence if Tj < Tj−1, j ≥2.
By definition,X1 is a record value. LetL(n) = min{j :j > L(n−1), Xj <
XL(n−1) , n≥ 2, with L(1) = 1. Then XL(n), n ≥1, is called the sequence of lower record values. For more detail and references see Nagaraja (1988), Ahsanullah (1995) and Arnold et al. (1998). By the above definition, the se- quence of record statistics can be viewed as the order statistics from a sample whose size is determined by the values and the order of occurrence of the observations.
The pdf ofXL(n),n= 1,2, . . ., is given (Arnold et al., 1998) by
(3) fL(n)(x) = 1
Γ(n)[−lnF(x)]n−1f(x),
where Γ is the gamma function. Whenxis a positive integer, Γ(x) = (x−1) !.
Next, the joint pdf of any two record statistics XL(m) and XL(n), 1 ≤ m < n, can be written (Arnoldet al., 1998) as
(4)
fL(m,n)(x, y) =
−ln (F(x))m−1
ln (F(x))−ln (F(y))n−m−1
Γ(m) Γ(n−m)
f(x) F(x)f(y),
−∞< y < x <∞, where L(m, n) is defined by XL(m,n) = XL(m), XL(n) .
3. THE FIRST MOMENT OF THE RECORD STATISTICXL(n)
We first proof
Theorem 3.1. For n= 1,2, . . . ,the first moment of the record statistic XL(n) is
αn= βnΓ (β) Γ(n)
∞
X
i1...in−1=1
∞
X
j=0
(−1)j(Γ (β−j))−1
(i1. . . in−1)j! (i1+· · ·+in−1+j+ 1). Proof. By (3)
αn=E XL(n)
= Z ∞
0
xfL(n)(x)dx= 1 Γ(n)
Z ∞ 0
x[−lnF(x)]n−1f(x)dx and by (1) we get
αn=E XL(n)
= 1 Γ(n)
Z ∞ 0
−βln 1−e−xγn−1
β 1−e−xγβ−1
γxγ−1e−xγdx, i.e.,
αn= βn−1βγ Γ (n)
Z ∞ 0
−ln 1−e−xγn−1
1−e−xγβ−1
xγ−1e−xγdx.
Since
1−e−xγβ−1
=
∞
X
j=0
(−1)jΓ(β) Γ(β−j)j!e−jxγ for β >0 and
ln 1−e−xγ
=
∞
X
i=1
1 i e−ixγ,
∞
X
i=1
aiti
!r
=
∞
X
i1,...,ir=1
Ai1i2...irti1+i2+···+ir, r≥1, where
Ai1i2...ir =ai1ai2. . . air,
we have
−ln 1−e−xγn−1
=
∞
X
i1,...,in−1=1
1 i1i2. . . in−1
e−(i1+···+in−1)xγ.
Therefore, αn= βnγ
Γ(n)
∞
X
i1...in−1=1
∞
X
j=0
1 i1. . . in−1
(−1)jΓ(β)
Γ (β−j)j!×
× Z ∞
0
xγ−1e−(i1+···+in−1+j+1)xγdx and then we obtain
αn= βnΓ (β) Γ(n)
∞
X
i1...in−1=1
∞
X
j=0
(−1)j(Γ (β−j))−1
(i1. . . in−1)j! (i1+· · ·+in−1+j+ 1).
4. THE JOINT MOMENT OF THE STATISTICSXL(m) ANDXL(n)
We shall proove
Theorem4.1. Forn= 1,2, . . . , m= 1,2, . . . , m < n, the joint moment of the record statistic XL(m) and XL(n) is
αm,n = α2βn+1 Γ(m)Γ (n−m)
∞
X
i1,i2,...,im−1=1
∞
X
l=0
∞
X
j=0 n−m−1
X
s=0
∞
X
q1,...,qn−m−1−s=1
×
×
∞
X
p1,...,ps=1
1 i1i2. . . im−1
(−1)jΓ (β) Γ (β−j)j!
n−m−1 s
(−1)n−m−1−s×
× 1
p1. . . ps
1 q1. . . qn−m−1−s
∞
X
r=0
γ2(−1)r(1 +j+s+p1+· · ·+ps)r
r! (1 +γr+γ) ×
×[α(i1+· · ·+im−1+ 1 +l+q1+· · ·+qn−m−1−s)]−
2 γ+r+2
Γ 2
γ +r+ 2
. Proof. By (1) and (4) we have
fL(m,n)(x, y) = α2βn
Γ(m) Γ(n−m)A1(x, y)A2(x, y)A3(x, y),
where
A1(x, y) =
−ln 1−e−αxγm−1
xγ−1γe−αxγ
1−e−αxγ ,
A2(x, y) = 1−e−αyγβ−1
yγ−1γe−αyγ, A3(x, y) =
ln 1−e−αxγ
−ln 1−e−αyγn−m−1
. Next,
A1(x, y) =
∞
X
i1,i2,...,im−1=1
1 i1i2. . . im−1
e−α(i1+i2+···+im−1)xγxγγe−αxγ×
∞
X
l=0
e−αlxγ,
A2(x, y) =yγ−1γe−αyγ
∞
X
j=0
(−1)jΓ (β)
Γ (β−j)j!e−αjyγ, and
A3(x, y) =
n−m−1
X
s=0
n−m−1 s
×
ln 1−e−αxγn−m−1−s
−ln 1−e−αyγs
=
=
n−m−1
X
s=0
n−m−1 s
(−1)n−m−1−s
∞
X
q1,...,qn−m−1−s=1
×
×
∞
X
p1,...,ps=1
1 q1. . . qn−m−1−s
1 p1. . . ps
e−α(q1+···+qn−m−1−s)xγe−α(p1+···+ps)yγ. Now,
fL(m,n)(x, y) = α2βn+1 Γ(m)Γ(n−m)
∞
X
i1,i2,...,im−1=1
∞
X
l=0
∞
X
j=0 n−m−1
X
s=0
×
×
∞
X
q1,...,qn−m−1−s=1
∞
X
p1,...,ps=1
1 i1i2. . . im−1
×
×(−1)jΓ(β) Γ (β−j)j!
n−m−1 s
(−1)n−m−1−s 1
p1. . . ps
1 q1. . . qn−m−1−s
×
×B1(l, s, i1, i2, . . . , im−1, q1, . . . , qn−m−1−s;x)B2(j, s, p1, . . . , ps;y), where
B1(l, s, i1, i2, . . . , im−1, q1, . . . , qn−m−1−s;x) =
=xγ−1γe−α(i1+i2+···+im−1+1+l+q1+···+qn−m−1−s)xγ
and
B2(j, s, p1, . . . , ps;y) =yγ−1γe−α(1+j+s+p1,...,ps)yγ. Successively, we have
αm,n=E XL(m,n)
= Z ∞
0
x Z x
0
yfL(m,n)(x, y) dy
dx=
= α2βn+1 Γ(m)Γ (n−m)
∞
X
i1,i2,...,im−1=1
∞
X
l=0
∞
X
j=0 n−m−1
X
s=0
∞
X
q1,...,qn−m−1−s=1
×
×
∞
X
p1,...,ps=1
1 i1i2. . . im−1
(−1)jΓ (β) Γ (β−j)j!
n−m−1 s
(−1)n−m−1−s 1
p1. . . ps
×
× 1
q1. . . qn−m−1−s
Z ∞ 0
xB1(l, s, i1, i2, . . . , im−1, q1, . . . , qn−m−1−s;x)×
× Z x
0
yB2(j, s, p1, . . . , ps;y) dy
dx.
We have Z x
0
yB2(j, s, p1, . . . , ps;y) dy= Z x
0
yγγe−α(1+j+s+p1+···+ps)yγdy =
= Z xγ
0
tγ1e−α(1+j+s+p1+···+ps)tdt= 1
[α(1+j+s+p1+· · ·+ps)]1γ+1 Γ
1
γ + 1;xγ
, where Γ(a;b) is incomplete gamma function.
Now, we prefer to use the series expansion for this theorem. Thus we obtain
Z x 0
yB2(j, s, p1, . . . , ps;y) dy=
=
∞
X
r=0
(−1)r(1 +j+s+p1+· · ·+ps)r
r! xγ+rγ+1 1
r+ 1 + 1/γ. We have
αm,n = α2βn+1 Γ(m)Γ (n−m)
∞
X
i1,i2,...,im−1=1
∞
X
l=0
∞
X
j=0 n−m−1
X
s=0
×
×
∞
X
q1,...,qn−m−1−s=1
∞
X
p1,...,ps=1
1 i1i2. . . im−1
×
×(−1)jΓ (β) Γ (β−j)j!
n−m−1 s
(−1)n−m−1−s 1
p1. . . ps
1 q1. . . qn−m−1−s
×
× Z ∞
0
xB1(l, s, i1, i2, . . . , im−1, q1, . . . , qn−m−1−s;x)×
×
∞
X
r=0
(−1)r(1 +j+s+p1+· · ·+ps)r r!
1
γ +r+ 1 xγ+rγ+1dx=
= α2βn+1 Γ(m)Γ (n−m)
∞
X
i1,i2,...,im−1=1
∞
X
l=0
∞
X
j=0 n−m−1
X
s=0
∞
X
q1,...,qn−m−1−s=1
×
×
∞
X
p1,...,ps=1
1 i1i2. . . im−1
(−1)jΓ (β) Γ (β−j)j!
n−m−1 s
(−1)n−m−1−s 1
p1. . . ps
×
× 1
q1. . . qn−m−1−s
∞
X
r=0
γ2(−1)r(1 +j+s+p1+· · ·+ps)r r! (1 +γr+γ)
Z ∞ 0
xr+γ+rγ+1×
×e−α(i1+···+im−1+1+l+q1+···+qn−m−1−s)xγdx=
= α2βn+1 Γ(m)Γ (n−m)
∞
X
i1,i2,...,im−1=1
∞
X
l=0
∞
X
j=0 n−m−1
X
s=0
∞
X
q1,...,qn−m−1−s=1
×
×
∞
X
p1,...,ps=1
1 i1i2. . . im−1
(−1)jΓ (β) Γ (β−j)j!
n−m−1 s
(−1)n−m−1−s 1
p1. . . ps
×
× 1
q1. . . qn−m−1−s
∞
X
r=0
γ2(−1)r(1 +j+s+p1+· · ·+ps)r r! (1 +γr+γ)
Z ∞ 0
t
2 γ+r+2
−1×
×e−α(i1+···+im−1+1+l+q1+···+qn−m−1−s)tdt=
= α2βn+1 Γ(m)Γ (n−m)
∞
X
i1,i2,...,im−1=1
∞
X
l=0
∞
X
j=0 n−m−1
X
s=0
∞
X
q1,...,qn−m−1−s=1
×
×
∞
X
p1,...,ps=1
1 i1i2. . . im−1
(−1)jΓ (β) Γ (β−j)j!
n−m−1 s
(−1)n−m−1−s 1
p1. . . ps
×
× 1
q1. . . qn−m−1−s
∞
X
r=0
γ2(−1)r(1 +j+s+p1+· · ·+ps)r
r! (1 +γr+γ) ×
×[α(i1+· · ·+im−1+ 1 +l+q1+· · ·+qn−m−1−s)]−
2 γ+r+2
×Γ 2
γ +r+ 2
as stated.
Remark 4.1. For γ = 1, we obtain a new form for the moment of record statistic for generalized exponential considered by Raqab (2002).
Remark 4.2. Using the results from Sections 3 and 4 we can obtain exact expressions for means, variances and covariances of record statistics.
5. THEαTH MOMENT OF THE kTH RECORD STATISTICS.
SOME INEQUALITIES.
Thekth record statisticsXn(k) from the independent and identically dis- tributed. Random variables X1, X2, . . . are defined according to Dziubdziela and Kopocinski (1976) by
Xn(k)=XLk(n),Lk(n)+k−1, n= 0,1,2, . . . , k≥1, where Lk(0) = 1,Lk(n+ 1) = min
j:XLk(n),Lk(n)+k−1 < Xj,j+k−1 forn= 0,1,2, . . ..
The formula below for the αth moment of the kth record statistic was proved by Grudzien and Szynal (1983).
(5) E
Xn(k)α
= kn+1 n!
Z 1 0
F−1(t)α
[−ln(1−t)]n(1−t)k−1dt, where
(6) F−1(t) =h
−ln 1−tβ1iγ1 .
In the case wherec(α, γ) =α/γ is a nonnegative integer, the next theo- rem gives an exact expression for the αth moment of the kth record statistic.
Theorem 5.1. If c(α, γ) =α/γ is a nonnegative integer we have E Xn(k)α
= kn+1 n!
∞
X
i1,...,ic(α,γ)=1
1 i1. . . ic(α,γ)×
×
∞
X
j1,...,jn=1
1 j1. . . jn
Γ
1 +j1+· · ·+jn+i1+···+iβc(α,γ)
Γ(k) Γ
k+j1+· · ·+jn+i1+···+iβc(α,γ) . Proof. By (1), (5) and (6) we have
E Xn(k)α
= kn+1 n!
Z 1 0
h
−ln 1−t1βic(α,γ)
[−ln(1−t)]n(1−t)k−1dt.
Using series expansions as in Section 3 we get E Xn(k)α
= kn+1 n!
Z 1
0
∞
X
i=1
tβi1 i
!c(α,γ)
∞
X
j=1
tj1 j
nk−1
X
s=0
k−1 s
(−1)stsdt=
= kn+1 n!
∞
X
i1,...,ic(α,γ)=1
1 i1. . . ic(α,γ)
∞
X
j1,...,jn=1
1 j1. . . jn
k−1
X
s=0
k−1 s
(−1)s×
× Z 1
0
ts+j1+···+jn+
i1+···+ic(α,γ)
β dt=
= kn+1 n!
∞
X
i1,...,ic(α,γ)=1
1 i1. . . ic(α,γ)
∞
X
j1,...,jn=1
1 j1. . . jn
k−1
X
s=0
k−1 s
(−1)s×
× 1
s+j1+···+jn+i1+···+βic(α,γ)+1
. In terms of the gamma function we have
E Xn(k)α
= kn+1 n!
∞
X
i1,...,ic(α,γ)=1
1 i1. . . ic(α,γ)×
×
∞
X
j1,...,jn=1
1 j1. . . jn
Γ
1 +j1+· · ·+jn+ i1+···+iβc(α,γ) Γ(k) Γ
k+j1+· · ·+jn+i1+···+iβc(α,γ) . For the general case, the next theorem gives some bounds forE Xn(k)
α
. Theorem 5.2. We have
D1+D2 ≤E Xn(k)α
≤D3+D4, where
D1=
∞
X
i1,...,ic+1=1
∞
X
j1,...,jn=1
1 i1. . . ic+1
1 j1. . . jn×
×B
j1+· · ·+jn+i1+· · ·+ic+1
β + 1, k; 0, t0
, D2=
∞
X
i1,...,ic=1
∞
X
j1,...,jn=1
1 i1. . . ic
1 j1. . . jn
×
×B
j1+· · ·+jn+i1+· · ·+ic
β + 1, k;t0,1
,
D3=
∞
X
i1,...,ic=1
∞
X
j1,...,jn=1
1 i1. . . ic
1 j1. . . jn×
×B
j1+· · ·+jn+i1+· · ·+ic
β + 1, k; 0, t0
, D4=
∞
X
i1,...,ic+1=1
∞
X
j1,...,jn=1
1 i1. . . ic+1
1 j1. . . jn
×
×B
j1+· · ·+jn+i1+· · ·+ic+1
β + 1, k;t0,1
,
B(a, b; 0, t0) = Z t0
0
ta−1(1−t)b−1dt, B(a, b;t0,1) =
Z 1 t0
ta−1(1−t)b−1dt and c is the integer part of α/γ.
Proof. We have F−1(t) =h
−ln 1−tβ1i1γ
>1⇔t∈
"
e−1 e
β
,1
#
and
F−1(t) = h
−ln 1−tβ1i1γ
<1⇔t∈
"
0,
e−1 e
β# . Ifc is integer part of α/γ it then follows that
F−1(t)c+1
< F−1(t)αγ
≤ F−1(t)c
, t∈
"
0,
e−1 e
β# , and
Z t0
0
F−1(t)c+1
(−ln(1−t))n(1−t)k−1dt <
<
Z t0
0
F−1(t)αγ
(−ln(1−t))n(1−t)k−1dt≤
≤ Z t0
0
F−1(t)c
(−ln(1−t))n(1−t)k−1dt, where t0= e−1e β
.
For t ∈ [ e−1e β
,1] we get F−1(t)c
≤ F−1(t)αγ
< F−1(t)c+1
and then
Z 1 t0
F−1(t)c
(−ln(1−t))n(1−t)k−1dt≤
≤ Z 1
t0
F−1(t)αγ
(−ln(1−t))n(1−t)k−1dt <
<
Z 1 t0
F−1(t)c+1
(−ln(1−t))n(1−t)k−1. Using these inequalities, we have
Z t0
0
F−1(t)c+1
(−ln(1−t))n(1−t)k−1dt+
+ Z 1
t0
F−1(t)c
(−ln(1−t))n(1−t)k−1dt≤E Xn(k)α
<
<
Z t0
0
F−1(t)c
(−ln(1−t))n(1−t)k−1dt+
+ Z 1
t0
F−1(t)c+1
(−ln(1−t))n(1−t)k−1dt.
But
D1 = Z t0
0
(F−1(t))c+1(−ln(1−t))n(1−t)k−1dt=
= Z t0
0
(−ln(1−t1β))c+1(−ln(1−t))n(1−t)k−1dt=
= Z t0
0
∞
X
i1,...,ic+1=1
1 i1. . . ic+1
t
i1+···+ic+1
β ×
×
∞
X
j1,...,jn=1
1
j1. . . jntj1+···+jn(1−t)k−1dt=
=
∞
X
i1,...,ic+1=1
∞
X
j1,...,jn=1
1 i1. . . ic+1
1 j1. . . jn
×
× Z t0
0
t
j1+···+jn+i1+···+βic+1+1
−1(1−t)k−1dt,
i.e.,
D1 =
∞
X
i1,...,ic+1=1
∞
X
j1,...,jn=1
1 i1. . . ic+1
1 j1. . . jn
×
×B
j1+· · ·+jn+i1+· · ·+ic+1
β + 1, k; 0, t0
. Similarly,
D2 =
∞
X
i1,...,ic=1
∞
X
j1,...,jn=1
1 i1. . . ic
1 j1. . . jn
×
×B
j1+· · ·+jn+i1+· · ·+ic
β + 1, k;t0,1
, D3 =
∞
X
i1,...,ic=1
∞
X
j1,...,jn=1
1 i1. . . ic
1 j1. . . jn
×
×B
j1+· · ·+jn+i1+· · ·+ic
β + 1, k; 0, t0
, D4 =
∞
X
i1,...,ic+1=1
∞
X
j1,...,jn=1
1 i1. . . ic+1
1 j1. . . jn
×
×B
j1+· · ·+jn+i1+· · ·+ic+1
β + 1, k;t0,1
, as stated.
Remark 5.1. In the case c(α, γ) = 0, in the expressions ofD2 orD3 the sums i1+· · ·+ic ori1+· · ·+ic+1 do not occur.
Now using binomial expansions in expressions which contain incomplete forms of the beta function we get
Theorem 5.3. We have
∞
X
i1,...,ic=1
1 i1. . . ic
∞
X
j1,...,jn=1
1
j1. . . jnA(i1, . . . , ic, j1, . . . , jn)≤E Xn(k)α
≤
≤
∞
X
i1,...,ic=1
1 i1. . . ic
∞
X
j1,...,jn=1
1
j1. . . jnB(i1, . . . , ic, j1, . . . , jn),
where
A(i1, . . . , ic, j1, . . . , jn) =
=
k−1
X
s=0
(−1)s×
∞
X
ic+1=1
ts+j1+···+jn+
i1+···+ic+1
β +1
0
ic+1+s+j1+· · ·+jn+ i1+···+iβ c+1 + 1+
+ 1−ts+j1+···+jn+
i1+···+ic
β +1
0
s+j1+· · ·+jn+ i1+···+iβ c + 1
,
B(i1, . . . , ic, j1, . . . , jn) =
=
k−1
X
s=0
(−1)s×
∞
X
ic+1=1
1−ts+j1+···+jn+
i1+···+ic+1
β +1
0
ic+1+s+j1+· · ·+jn+ i1+···+iβ c+1 + 1+
+ ts+j1+···+jn+
i1+···+ic
β +1
0
s+j1+· · ·+jn+ i1+···+iβ c + 1
. Proof. We have
D1 = Z t0
0
F−1(t)c+1
(−ln(1−t))n(1−t)k−1dt=
∞
X
i1,...,ic+1=1
1 i1. . . ic+1
×
×
∞
X
j1,...,jn=1
1 j1. . . jn
k−1
X
s=0
(−1)s ts+j1+···+jn+
i1+···+ic+1
β +1
0
s+j1+· · ·+jn+i1+···+iβ c+1 + 1 and
D2= Z 1
t0
F−1(t)c
(−ln(1−t))n(1−t)k−1dt=
∞
X
i1,...,ic=1
1 i1. . . ic×
×
∞
X
j1,...,jn=1
1 j1. . . jn
k−1
X
s=0
(−1)s 1−ts+j1+···+jn+
i1+···+ic
β +1
0
s+j1+· · ·+jn+i1+···+iβ c + 1. Hence
D1+D2=
∞
X
i1,...,ic=1
1 i1. . . ic
∞
X
j1,...,jn=1
1
j1. . . jnA(i1, . . . , ic, j1, . . . , jn).
Similarly,
D3 = Z t0
0
F−1(t)c
(−ln(1−t))n(1−t)k−1dt=
=
∞
X
i1,...,ic=1
1 i1. . . ic
∞
X
j1,...,jn=1
1 j1. . . jn
k−1
X
s=0
(−1)s ts+j1+···+jn+
i1+···+ic
β +1
0
s+j1+· · ·+jn+i1+···+iβ c + 1 and
D4= Z 1
t0
F−1(t)c+1
(−ln(1−t))n(1−t)k−1dt=
∞
X
i1,...,ic+1=1
1 i1. . . ic+1
×
×
∞
X
j1,...,jn=1
1 j1. . . jn
k−1
X
s=0
(−1)s 1−ts+j1+···+jn+
i1+···+ic+1
β +1
0
s+j1+· · ·+jn+i1+···+iβ c+1 + 1 Hence
D3+D4=
∞
X
i1,...,ic=1
1 i1. . . ic
∞
X
j1,...,jn=1
1 j1. . . jn
B(i1, . . . , ic, j1, . . . , jn). Using Theorem 5.3 we can obtain an approximation degree for the αth moment of thekth record statistics.
Corollary 5.1. We have
∆ =D3+D4−(D1+D2) =
=
∞
X
i1,...,ic=1
1 i1. . . ic
∞
X
j1,...,jn=1
1 j1. . . jn
C(i1, . . . , ic, j1, . . . , jn), where
C(i1, . . . , ic, j1, . . . , jn) =
k−1
X
s=0
(−1)s×
∞
X
ic+1=1
1−2ts+j1+···+jn+
i1+···+ic
β +1
0
s+j1+· · ·+jn+i1+···+iβ c + 1−
− 1
ic+1+s+j1+· · ·+jn+i1+···+iβ c+1 + 1
! .
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Received 25 July 2009 University of Bucharest
Faculty of Mathematics and Computer Science Str. Academiei 14
010014 Bucharest, Romania and
Technical University of Civil Engineering Department of Mathematics and Computer Science
Bd. Lacul Tei 122-124 020396 Bucharest, Romania
