Tamsui Oxford Journal of Information and Mathematical Sciences 30 (2014) 1-21
Aletheia University
Relations for Moments of Lower Generalized
Order Statistics from Exponentiated Inverted
Weibull Distribution*
Devendra Kumar†
Department of Statistics, Amity Institute of Applied Sciences
Amity University, Noida, India
Received February 13, 2013, Accepted April 21, 2014.
Abstract
In this paper we consider exponentiated inverted Weibull
distribution. Exact expressions and some recurrence relations for single
and product moments of lower generalized order statistics are derived.
Further the results are deduced for moments of order statistics and
lower record values and characterization of this distribution has been
considered on using conditional expectation of function of lower
generalized order statistics and a recurrence relation for single moments.
The first four moments, mean and variance of order statistics and record
values are computed for various values of parameters.
Keywords and Phrases: Lower generalized order statistics, Order statistics,
Lower record values, Exponentiated inverted Weibull distribution, Single and
product moments, Recurrence relations and characterization.
*
†
2010 Mathematics Subject Classification. Primary 62G30, 62E10
E-mail: [email protected]
Devendra Kumar
2
1. Introduction
Kamps [14] introduced the concept of generalized order statistics (gos) . It is
known that ordinary order statistics, sequential order statistics, Stigler’s order
statistics and upper record values are special cases of gos . In this article we will
consider the lower generalized order statistics (l gos) defined as follows:
Let n N , k  1, m   , be the parameters such that
 r  k  (n  r ) (m  1)  0 for all 1  r  n .
Then X  (1, n, m, k ), , X  (n, n, m, k ) are called l gos from an absolutely
continuous distribution function (df ) F (x) with the probability distribution
function ( pdf ) f (x) if their joint pdf has the form
 n1
k 
 j 1

  n1

m
k 1


f ( xn )
j   [ F ( xi )] f ( xi )  [ F ( x n )]
 i 1


for F 1 (1)  x1  x 2    x n  F 1 (0) .
The marginal pdf of r  th l gos , X  (r , n, m, k ) , is
C
f X  ( r ,n,m,k ) ( x)  r 1 [ F ( x)] r 1 f ( x) g mr 1 ( F ( x))
(r  1)!
(1.1)
(1.2)
and the joint pdf of X  (r , n, m, k ) and X  (s, n, m, k ) , 1  r  s  n , is
C s 1
r 1
f X  ( r ,n,m,k ), X  ( s,n,m,k ) ( x, y) 
[ F ( x)]m f ( x) g m
( F ( x))
(r  1)!( s  r  1)!
 [hm ( F ( y))  hm ( F ( x))]s r 1[ F ( y)] s 1 f ( y) , x  y ,
where
r
C r 1    i ,
i 1
1

x m1 ,

hm ( x)   m  1

 ln x ,
and
m  1
m  1
(1.3)
Relations for Moments of Lower Generalized Order
3
g m ( x)  hm ( x)  hm (1) , x [0,1) .
We shall also take X  (0, n, m, k )  0 . If m  0 , k  1 , then X  (r , n, m, k )
reduces to the (n  r  1)  th order statistic, X nr 1:n from the sample
X 1 , X 2 , , X n and when m  1 , then X  (r , n, m, k ) reduces to the r  th lower
k record value (Pawlas and Szynal [9]). The work of Burkschat et al. [8] may also
refer for l gos .
Recurrence relations for single and product moments of l gos from the
inverse Weibull distribution are derived by Pawlas and Syznal [9]. Ahsanullah [7]
and Mbah and Ahsanullah [2] characterized the uniform and power function
distributions based on distributional properties of l gos , respectively. Khan and
Kumar [10, 11 & 12], Kumar and Khan [3] and Kumar [4, 5] have established
recurrence relations for moments of l gos from exponentiated Pareto,
exponentiated gamma and generalized exponential, exponentiated log-logistic,
exponentiated Kumaraswamy and J- shaped distributions.
In this study we obtain some recurrence relations for single and product
moments of l gos from exponentiated inverted Weibull distribution. The relations
for order statistics and lower records are deduced from the relations derived.
Further, the distribution has been characterized on using conditional expectation
of function of l gos and a recurrence relation for single moment of l gos .
A random variable X is said to have exponentiated inverted Weibull
distribution (Flaih et al., [1]) if its pdf is of the form
f ( x)   x (  1) e  x

,
x  0 , ,   0
(1.4)
and the corresponding df is
F ( x)  e  x

, x  0 , ,   0 .
(1.5)
It is easy to see that
x  1 f ( x)   F ( x)
(1.6)
The inverted Weibull and inverted exponential distributions are considered as
special cases of the exponentiated inverted Weibull distribution when   1 and
Devendra Kumar
4
  1,   1 , respectively. More details on this distribution can be found in Flaih
et al., [1].
2.
Relations for single moments
In this section, we have derived the exact expressions for l gos fom exponentiated
inverted Weibull distribution. Further these results are used to evaluate the first
four moments, mean and variance of order statistics and record values are
presented in table 2.1, 2.2, 2.3 and table 2.4. We shall first establish the exact
expression for E[ X * j (r, n, m, k )] . Using (1.2), we have when m  1
E[ X * j (r , n, m, k )] 

C r 1  j
x [ F ( x)] r 1 f ( x) g mr 1 ( F ( x))dx

0
(r  1)!
C r 1
(r  1)!
I j ( r  1, r  1) ,
(2.1)
where

b
I j (a, b)   x j [ F ( x)]a f ( x) g m
( F ( x))dx .
0
 1

b
On expanding g m
( F ( x))  
{1  ( F ( x)) m 1}
 m 1

(2.2)
b
binomially in (2.2), we
get when m  1
I j ( a, b) 
1
(m  1)b
b
 (1)
u 0
u
b   j
  x [ F ( x)]au ( m1) f ( x)dx .
u  0
Making the substitution t   ln F ( x) in (2.3), we find that
I j ( a, b) 
 j/
b
b 
u 0
 

(1) u   t  j /  e [ au ( m1)1]t dt
b 
u 0
(m  1)
(2.3)
Relations for Moments of Lower Generalized Order

 j/
b
b 
u 0
 
(1  j /  )
(1) u  
b 
u [a  u (m  1)  1]1 j / 
(m  1)
5
.
(2.4)
Now substituting for I j ( r  1, r  1) from (2.4) in (2.1) and simplifying, we
obtain when m  1
E[ X
*j
(r , n, m, k )] 
 j /  C r 1
r 1
 r  1 (1  j /  )

1 j / 
 u  ( r u )
(1) u 
r 1 
(r  1)!(m  1)
u 0
,
  j , and j  0,1,2,
(2.5)
and when m  1 that
E[ X * j (r , n, m, k )] 
Since (2.5) is of the form
E[ X
*j
r 1
0
0
as
 r  1
  0 .
 u 
 (1)u 
u 0
0
at m  1 , therefore, we have
0
 1 [k  (n  r  u )(m  1)](1 j /  )

(r , n, m, k )]  A  (1) 
,
(m  1) r 1
u 0
 u 
r 1
u r
(2.6)
where
A
 j /  C r 1
(r  1)!
(1  j /  ) .
Differentiating numerator and denominator of (2.6) (r  1) times with respect
to m , we get
r 1
 r  1 (1  j /  )(2  j /  ) (r  1  j /  )(n  r  u ) r 1

E[ X * j (r , n, m, k )]  A  (1) u  ( r 1) 
(r  1)![k  (n  r  u )(m  1)]r  j / 
 u 
u 0
Devendra Kumar
6
r 1
u  r  1 (1 

 A  (1) 
 u 
u 0
j /  )(2  j /  )(r  1  j /  )(r  n  u ) r 1
(r  1)![k  (n  r  u )(m  1)]r  j / 
On applying L’ Hospital rule, we have
lim E[ X * j (r , n, m, k )]  A
(1  j /  )(2  j /  )(r  1  j /  )
(r  1)!k r  j / 
m  1
r 1
 r  1
(r  n  u ) r 1 .
  (1)u 
u


u 0
(2.7)
But for all integers n  0 and for all real numbers x , we have Ruiz [13]
n
n
i 0
 
 (1)i  i ( x  i) n  n!.
(2.8)
Now substituting (2.8) in (2.7) and simplifying, we find that
E[ X * j (r , n,1, k )]  E[( X Lj (r ) ) k ] 
( k ) j / 
( r  j /  ) .
(r  1)!
(2.9)
Special cases
Putting m  0 , k  1 in (2.5), the exact expression for the single
moments of order statistics of the exponentiated inverted Weibull distribution can
be obtained as
i)
E [ X njr 1:n ]   j / 
That is
r 1
 r  1
(1  j /  )

.
C r:n  (1) u 
1 j / 
 u  (n  r  1  u )
u 0
.
Relations for Moments of Lower Generalized Order
E [ X rj:n ]  
j/
7
nr
 n  r  (1  j /  )

,
C r:n  (1) u 
1 j / 
 u  (r  u )
u 0
where
Cr :n 
n!
.
(r  1)!(n  r )!
ii) Putting k  1 in (2.9), we get the exact expression for the single moments
of lower records for the exponentiated inverted Weibull distribution can be
obtained as
E[ X
*j
E[ X Lj ( r ) ] 
(r , n,1,1)] 
 j/
(r  1)!
( r  j /  ) .
Recurrence relations for single moments of l gos from (1.6) can be obtained
in the following theorem.
Theorem 2.1 For the distribution given in (1.4) and for 2  r  n , n  2 and
k  1,2, ,
E[ X * j   1 (r , n, m, k )] 
 r
j 1
{E[ X * j 1 (r  1, n, m, k )]  E[ X * j 1 (r , n, m, k )]} .(2.10)
Proof From (1.2) and (1.6), we have
E [ X * j   1 (r , n, m, k )] 
 C r 1

(r  1)! 0
x j [ F ( x)] r g mr 1 ( F ( x))dx .
Integrating by parts treating x j for integration and the rest of the integrand
for differentiation, we get
E [ X * j   1 (r , n, m, k )] 
 r  Cr 2

x j 1[ F ( x)] r 1 1 f ( x) g mr 2 ( F ( x))dx


j  1  (r  2)! 0
Devendra Kumar
8


C r 1  j 1
x [ F ( x)] r 1 f ( x) g mr 1 ( F ( x))dx 

(r  1)! 0

and hence the result.
Remark 2.1 Setting m  0 , k  1 in (2.10), we obtain a recurrence relation for
single moments of order statistics of the exponentiated inverted Weibull
distribution in the form
E [ X njr1:1n ] 
 (n  r  1)
j 1
{E [ X njr12:n ]  E [ X njr11:n ]} .
That is
E [ X rj:n1 ]  E [ X rj11:n ] 
j 1
E [ X rj1:n1 ] .
 (r  1)
Remark 2.2 Putting m  1 , in Theorem 2.1, we get a recurrence relation for
single moments of lower k record values from exponentiated inverted Weibull
distribution in the form
E [( X Lj (r) 1 ) k ] 
 k
j 1
{E [( X Lj (r11) ) k ]  E [( X Lj (r1) ) k ]} .
Remark 2.3 Putting   1 , in (2.10), we get the recurrence relation for single
moments of l gos from inverted Weibull distribution.
Remark 2.4 For   1 and   1 , the relation in (2.10), gives the recurrence
relation for single moments of l gos from inverted exponential distribution.
Table 2.1: First four moments of order statistics
n
1
2
3
4
r
1
1
2
1
2
3
1
j  1,   2
j 1 ,   4
j 1 ,   3
 1
 2
 3
 1
 2
 3
 1
 2
 3
1.77245
1.03828
2.50663
0.86746
1.37992
3.06998
0.78506
4.44288
2.60258
6.28319
2.17439
3.45896
7.69530
1.96785
5.44140
3.18750
7.69530
2.66308
4.23634
9.42478
2.41011
1.35412
1.00215
1.70608
0.89708
1.21229
1.95298
0.84236
2.31024
1.70976
2.91071
1.53050
2.06827
3.33194
1.43713
2.64456
1.95718
3.33194
1.75198
2.36758
3.81412
1.64510
1.22542
0.99356
1.45727
0.91717
1.14634
1.61274
0.87598
1.78577
1.44789
2.12365
1.33656
1.67054
2.35020
1.27654
1.97628
1.60235
2.35020
1.47915
1.84875
2.60093
1.41272
Relations for Moments of Lower Generalized Order
5
2
3
4
1
2
3
4
5
n
r
1
1
1
2
1
2
3
1
2
3
4
1
2
3
4
5
2
3
4
5
n
1
2
3
4
5
1.11465
1.6452
3.54491
0.73458
0.98699
1.30615
1.87123
3.96333
2.79402
4.12390
8.88577
1.84131
2.47401
3.27403
4.69048
4.44288
3.42196
5.05072
10.88280
2.25514
3.03003
4.00985
5.74464
12.16734
1.06127
1.36332
2.14953
0.80740
0.98218
1.17989
1.48560
2.31551
1
1
2
1
2
3
1
2
3
4
1
2
3
4
5
2.07263
2.66253
4.19798
1.57683
1.91818
2.30430
2.90135
4.52213
1.04074
1.25195
1.73300
0.84915
0.98327
1.12693
1.33530
1.83243
j  2,   3
j 1 ,   5
1.51664
1.82444
2.52546
1.23745
1.43290
1.64224
1.94590
2.67035
1.67844
2.01907
2.79488
1.36946
1.58576
1.81744
2.1535
2.95522
j  2,   4
 1
 2
 3
 1
 2
 3
 1
 2
 3
1.16423
0.99111
1.33735
0.93096
1.11141
1.45032
0.89788
1.03019
1.19264
1.53621
0.87610
0.98501
1.09795
1.25576
1.60632
1.55698
1.32546
1.78850
1.24502
1.48635
1.93958
1.20078
1.37772
1.59497
2.05445
1.17165
1.31731
1.46834
1.67939
2.14821
1.68850
1.43742
1.93958
1.35019
1.61190
2.10342
1.30222
1.49410
1.72970
2.22799
1.27062
1.42858
1.59237
1.82125
2.32968
2.67894
1.10533
4.25255
0.85158
1.61282
5.57242
0.73962
1.18747
2.03816
6.75050
0.67411
1.00164
1.46622
2.41946
7.83326
11.39232
4.70046
18.08418
3.62140
6.85858
23.69698
3.14527
5.04978
8.66738
28.70685
2.86671
4.25953
6.23517
10.28885
33.31134
14.92816
6.15934
23.69698
4.74537
8.98728
31.05183
4.12147
6.61709
11.35748
37.61661
3.75645
5.58156
8.17038
13.48221
43.65021
1.77245
1.03828
2.50663
0.86746
1.37992
3.06998
0.78506
1.11465
1.64520
3.54491
0.73458
0.98699
1.30615
1.87123
3.96333
4.44288
2.60258
6.28319
2.17439
3.45896
7.69530
1.96785
2.79402
4.12390
8.88577
1.84131
2.47401
3.27403
4.69048
9.93459
5.44140
3.18750
7.69530
2.66308
4.23634
9.42478
2.41011
3.42196
5.05072
10.88280
2.25514
3.03003
4.00985
5.74464
12.16734
j  2,   5
r
1.81061
2.32593
3.66727
1.37749
1.67569
2.01299
2.53456
3.95045
9
j 3,  5
j 3,  4
 1
 2
 3
 1
 2
 3
 1
 2
 3
1.48919
1.01338
1.96500
0.88357
1.27301
2.31100
0.81791
1.08054
1.46548
2.59283
0.77665
0.98297
1.22689
1.62454
2.83491
2.92626
1.99130
3.86123
1.73622
2.50147
4.54111
1.60720
2.12326
2.87967
5.09492
1.52612
1.93154
2.41084
3.19223
5.57059
3.44152
2.34193
4.54111
2.04193
2.94192
5.34070
1.89020
2.49712
3.38673
5.99203
1.79484
2.27164
2.83534
3.75432
6.55146
3.62561
1.15370
6.09752
0.84886
1.76337
8.26460
0.72093
1.23264
2.29409
10.25477
0.64793
1.01295
1.56218
2.78204
12.12296
22.10725
7.03468
37.17981
5.17594
10.75218
50.39362
4.39590
7.51606
13.98829
62.52873
3.95075
6.17650
9.52541
16.96354
73.92003
29.96423
9.53483
50.39362
7.01548
14.57353
68.30367
5.95821
10.18730
18.95977
84.75163
5.35485
8.37164
12.91077
22.99244
100.19143
2.21816
1.07422
3.36210
0.85628
1.51009
4.28811
0.75646
1.15573
1.86445
5.09599
0.69694
0.99454
1.39752
2.17573
5.82606
7.45768
3.61163
11.30372
2.87890
5.07708
14.41705
2.54331
3.88570
6.26846
17.13324
2.34319
3.34376
4.69861
7.31503
19.58780
9.51170
4.60636
14.41705
3.67183
6.47543
18.38785
3.24380
4.95591
7.99495
21.85216
2.98857
4.26471
5.99272
9.32977
24.98275
Devendra Kumar
10
n
r
1
2
1
1
2
1
2
3
1
2
3
4
1
2
3
4
5
3
4
5
Table 2.2
n
r
1
1
1
2
1
2
3
1
2
3
4
1
2
3
4
5
2
3
4
5
j 4  5
 1
 2
 3
4.59084
1.18856
7.99312
0.84894
1.86780
11.05578
0.71093
1.26298
2.47263
13.91684
0.63339
1.02110
1.62580
3.03718
16.63675
36.69518
9.50034
63.89002
6.78572
14.92958
88.37024
5.68257
10.09516
19.76400
111.23898
5.06277
8.16179
12.99522
24.27652
132.97960
50.75537
13.14051
88.37024
9.38575
20.65003
122.23034
7.85992
13.96324
27.33681
153.86152
7.00263
11.28908
17.97450
33.57835
183.93231
Variance of order statistics
 3
 4
 1
 2
 3
 1
 2
0.845299
0.101025
1.341841
0.046827
0.143173
1.758289
0.030050
0.061176
0.179519
2.130021
0.022215
0.036962
0.074080
0.212453
2.471673
6.055111
1.777181
9.611947
1.278970
2.580839
12.59516
1.079927
1.771471
3.257430
15.25798
0.969231
1.451593
2.183041
3.864856
17.70528
7.934462
2.328786
12.59516
1.675936
3.381845
16.50432
1.415116
2.321295
4.268414
19.99357
1.270057
1.902145
2.860582
5.064378
23.20055
0.270796
0.051119
0.382994
0.026259
0.065825
0.469050
0.017719
0.031510
0.077821
0.541621
0.013524
0.020170
0.036179
0.088204
0.60553
1.253906
0.506195
1.773301
0.387997
0.668256
2.171860
0.338296
0.493823
0.795319
2.507822
0.310027
0.420808
0.577078
0.903953
2.803821
 5
 3
1.535717
0.619974
2.171860
0.475195
0.818463
2.659943
0.414332
0.604799
0.974076
3.071446
0.379719
0.515395
0.706762
1.107078
3.434015
 1
 2
 3
0.133759
0.031081
0.176495
0.016883
0.037778
0.207572
0.011722
0.019249
0.043090
0.232889
0.009099
0.012725
0.021396
0.047607
0.254646
0.502073
0.234456
0.662498
0.186145
0.292234
0.779139
0.165327
0.225148
0.335741
0.874155
0.153356
0.196234
0.254818
0.371879
0.955784
0.590488
0.275754
0.779139
0.218917
0.343698
0.916324
0.194423
0.264785
0.394868
1.028091
0.180365
0.230799
0.299698
0.437368
1.124051
Relations for Moments of Lower Generalized Order
Table 2.3
First four moments of lower records
j  1,   2
r
1
2
3
4
5
 1
1.77245
0.88623
0.66467
0.55389
0.48466
1
2
3
4
5
j 1 ,   4
j 1 ,   3
 2
 3
 1
 2
 3
 1
 2
 3
2.50663
1.25331
0.93999
0.78332
0.68541
3.06998
1.53499
1.15124
0.95937
0.83945
1.35412
0.90275
0.75229
0.66870
0.61298
1.70608
1.13739
0.94782
0.84251
0.77230
1.95298
1.30198
1.08499
0.96443
0.88406
1.22542
0.91906
0.80418
0.73716
0.69109
1.45727
1.09296
0.95634
0.87664
0.82185
1.61274
1.20955
1.05836
0.97016
0.90953
j  2,   3
j  1,   5
r
11
j  2,  4
 1
 2
 3
 1
 2
 3
 1
 2
 3
1.16423
0.93138
0.83825
0.78236
0.74324
1.33735
1.06988
0.96289
0.89870
0.85376
1.45032
1.16025
1.04423
0.97461
0.92588
2.67894
0.89298
0.59532
0.46303
0.38586
4.25255
1.41752
0.94501
0.73501
0.61251
5.57242
1.85747
1.23831
0.96313
0.80261
1.77245
0.88623
0.66467
0.55389
0.48466
2.50663
1.25331
0.93999
0.78332
0.68541
3.06998
1.53499
1.15124
0.95937
0.83945
j 3,  4
j 2,  5
j 3,  5
r
 1
 2
 3
 1
 2
 3
 1
 2
 3
1
2
3
4
5
1.48919
0.89352
0.71481
0.61950
0.55755
1.96500
1.17900
0.94320
0.81744
0.73570
2.31100
1.38660
1.10928
0.96137
0.86524
3.62561
0.90640
0.56650
0.42488
0.34521
6.09752
1.52438
0.95274
0.71455
0.58057
8.26460
2.06615
1.29134
0.96851
0.78691
2.21816
0.88726
0.62108
0.49687
0.42234
3.36210
1.34484
0.94139
0.75311
0.64014
4.28811
1.71524
1.20067
0.96054
0.81646
r
1
2
3
4
5
Table 2.4
j 4  5
 1
 2
 3
4.59084
0.91817
0.55090
0.40399
0.32320
7.99312
1.59862
0.95917
0.70339
0.56272
11.05578
2.21116
1.32669
0.97291
0.77833
Variance of lower records
 3
r
 1
 2
1
2
3
4
5
0.845299
0.078022
0.029380
0.015870
0.010116
1.341841
0.123864
0.046647
0.025187
0.016063
 4
 3
1.758289
0.162318
0.061107
0.033005
0.021048
 1
 2
0.270796
0.041559
0.017965
0.010485
0.007055
0.382994
0.058748
0.025404
0.014822
0.009973
 5
 3
0.469050
0.071979
0.031114
0.018160
0.012205
 1
 2
0.133759
0.026051
0.012147
0.007413
0.005144
0.176495
0.034357
0.016043
0.009778
0.006794
 3
0.207572
0.04042
0.018864
0.011505
0.007986
Devendra Kumar
12
3. Relations for product moments
Theorem 3.1 For the given exponentiated inverted Weibull distribution in (1.3)
and n  2 , m   , 1  r  r  1  n ,
E [ X *i (r , n, m, k ) X * j   1 (r  1, n, m, k )] 
 r 1
j 1
{E [ X *i  j 1 (r , n, m, k )]
 E [ X *i (r, n, m, k ) X * j 1 (r  1, n, m, k )]}
(3.1)
and for 1  r  s  n , s  r  2 and i, j  0 ,
E [ X *i (r , n, m, k ) X * j   1 (s, n, m, k )]

 s
j 1
{E [ X *i (r , n, m, k ) X * j 1 ( s  1, n, m, k )]
 E [ X *i (r, n, m, k ) X * j 1 (s, n, m, k )]} .
(3.2)
Proof From (1.3), we have
E[ X *i (r, n, m, k ) X * j   1 (s, n, m, k )]

 i
Cs 1
r 1
x [ F ( x)]m f ( x) g m
( F ( x)) I ( x)dx ,

0
(r  1)!( s  r  1)!
where
I ( x)   y j   1[hm ( F ( y))  hm ( F ( x))]s  r 1[ F ( y)] s 1 f ( y)dy
x
0
(3.3)
Relations for Moments of Lower Generalized Order
13
   y j [hm ( F ( y))  hm ( F ( x))]s r 1[ F ( y)] s dy
x
0
upon using the relation in (1.6). Integrating now by parts treating y j for
integration and the rest of the integrand for differentiation, we obtain when
s  r  1 that

x
 r 1  x j 1

I ( x) 
[ F ( x)] r 1   y j 1[ F ( y )] r 1 1 f ( y )dy 

0
j 1 

  r 1

and when s  r  1 that
I ( x) 
 s  ( s  r  1) x j 1
y [hm ( F ( y))  hm ( F ( x))]s r 2 [ F ( y)] s m f ( y)dy


0
j 1   s
x
0 y

[hm ( F ( y))  hm ( F ( x))]s  r 1[ F ( y)] s 1 f ( y)dy  .

j 1
Upon substituting the above expressions for I (x) in (3.3), we have, after
simplifications, the recurrence relations (3.1) and (3.2).
Remark 3.1 Setting m  0 , k  1 , in (3.2), we obtain recurrence relations for
product moments of order statistics for the exponentiated inverted Weibull
distribution in the form
E [ X ni r 1:n X njs1:1n ] 
 (n  s  1)
{E [ X ni r 1:n X njs12:n ]  E [ X ni r 1:n X njs11:n ]} .
j 1
That is
E[ X ri:n1 X sj:n ]  E[ X ri 11:n X sj:n ] 
i 1
E[ X ri 1:n1 X sj:n ] .
 (r  1)
Devendra Kumar
14
Remark 3.2
Setting m  1 and k  1, in Theorem 3.1, we get the recurrence
relations for product moments of lower k record values from exponentiated
inverted Weibull distribution in the form
E [( X Li ( r ) ) k ( X Lj (s) 1 ) k ] 
 k
j 1
{E [( X Li ( r ) ) k ( X Lj (s11) ) k ]  E [( X Li ( r ) ) k ( X Lj (s1) ) k ]} .
Remark 3.3 Putting   1 , in (3.1) and (3.2), we deduce the recurrence relations
for product moments of l gos from inverted Weibull distribution.
Remark 3.4 For   1 and   1 , the relations in (3.1) and (3.2) give the
recurrence relations for product moments of l gos from inverted exponential
distribution.
Remark 3.5 At s  0 , Theorem 3.1 can be reduced to Theorem 2.1.
4. Characterization
This Section, contains characterization of exponentiated inverted Weibull
distribution by using conditional expectation of function of l gos and a recurrence
relation for single moments of l gos .
Let X * (r , n, m, k ) , r  1,2,, n be l gos from a continuous population with df
F (x) and pdf f (x) , then the conditional pdf of X * ( s, n, m, k ) given
X * (r , n, m, k )  x , 1  r  s  n , in view of (1.2) and (1.3), is
f X * ( s , n , m, k ) | X * ( r , n , m, k ) ( y | x ) 
Cs 1
[ F ( x)]m  r 1
( s  r  1)!Cr 1
 [hm ( F ( y))  hm ( F ( x))]s  r 1[ F ( y)] s 1 f ( y) .
(4.1)
Theorem 4.1 Let X be a non-negative random variable having an absolutely
continuous distribution function F (x) with F (0)  0 and 0  F ( x)  1 for all
x  0 , then
Relations for Moments of Lower Generalized Order
E[ ( X  (s, n, m, k )) | X  (l , n, m, k )  x]  e  x

  l j 
 , l  r, r  1


1
j 1  l  j

15
s l
  
(4.2)
if and only if
F ( x)  e

x 
,
x  0 , ,  , 0 ,
where

 ( x)  e  y .
Proof
From (4.1), we have
E[ ( X  ( s, n, m, k )) | X  (r , n, m, k )  x] 
x
 0 e  y

Cs 1
( s  r  1)!Cr 1 (m  1) s  r 1
  F ( y )  m1 
1  
 
  F ( x)  
s r 1
 F ( y) 


 F ( x) 
 s 1
f ( y)
dy .
F ( x)
(4.3)

F ( y ) e  y
By setting u 
from (1.4) in (4.3), we obtain

F ( x) e  x  
E[ ( X  ( s, n, m, k )) | X  (r , n, m, k )  x] 
 e  x

1 
s
0 u
C s 1
( s  r  1)!C r 1 (m  1) s r 1
(1  u m1 ) s r 1 du .
Again by setting t  u m 1 in (4.4), we get
E[ ( X  ( s, n, m, k )) | X  (r , n, m, k )  x] 
Cs 1
( s  r  1)!Cr 1 (m  1) s  r
(4.4)
Devendra Kumar
16
k 1
 n s 1
 x   1 m1
e
t
(1  t ) s r 1 dt
0


C s 1 e
 x  
C r 1 (m  1) s r
e
 x  
 k 1


 n  s
 m 1

 k 1


 n  r
 m 1

 r j 
s r 
  
j 1  r  j

 1 
and hence the relation in (4.2).
To prove sufficient part, we have from (4.1) and (4.2)
x  y  
e
[( F ( x)) m1
s r 1 0
(m  1)
C s 1
( s  r  1)!C r 1

 ( F ( y)) m1 ] s r 1
 [ F ( y)] s 1 f ( y)dy  [ F ( x)] r 1 H r ( x) ,
where
H r ( x)  e
 x  
s r 
  
 r j 
j 1  r  j
.
 1 
Differentiating (4.5) both the sides with respect to x , we get
Cs1[ F ( x)]m f ( x)
( s  r  2)!Cr 1 (m  1) sr 2

x
0

e  y [( F ( x)) m1  ( F ( y)) m1 ]sr 2
 [ F ( y)] s 1 f ( y)dy  H r ( x)[ F ( x)] r 1   r 1H r ( x)[ F ( x)] r 1 1 f ( x)
or
(4.5)
Relations for Moments of Lower Generalized Order
17
 r 1H r 1 ( x)[ F ( x)] r 2  m f ( x)
 H r ( x)[ F ( x)] r 1   r 1H r ( x)[ F ( x)] r 1 1 f ( x) ,
where
H r ( x)   x (  1) e  x

e  x

H r 1 ( x)  H r ( x) 
 r 1
s r 
 r j 
j 1 
r j
  
,
 1 
 r j 

    1  .
j 1  r  j

s r 
Therefore,
H r ( x)
f ( x)
  x (  1)

F ( x)  r 1[ H r 1 ( x)  H r ( x)]
which proves that
F ( x)  e  x

x  0 , ,   0 .
,
Theorem 4.2 Let X be a non-negative random variable having an absolutely
continuous distribution function F (x) with F (0)  0 and 0  F ( x)  1 for all
x  0 , then
E [ X * j   1 (r , n, m, k )] 
 r
j 1
{E[ X * j 1 (r  1, n, m, k )]  E[ X * j 1 (r , n, m, k )]} (4.6)
if and only if

F ( x)  e  ( / x) ,
x  0 , ,   0 .
Devendra Kumar
18
Proof The necessary part follows immediately from equation (2.10). On the other
hand if the recurrence relation in equation (4.6) is satisfied, then on using equation
(1.2), we have
Cr 1  j   1
x
[ F ( x)] r 1 f ( x) g mr 1 ( F ( x))dx

(r  1)! 0

 (r  1) Cr 1

 r Cr 1

 C r 1
j 1

(r  1)! 0

j  1 (r  1)! 0

j  1 (r  1)! 0
r 2
x j 1[ F ( x)] r m f ( x) g m
( F ( x))dx
r 1
x j 1[ F ( x)] r 1 f ( x) g m
( F ( x))dx
r 2
x j 1[ F ( x)] r f ( x) g m
( F ( x))
 g ( F ( x)) 

 (r  1)[ F ( x)]m  r m
dx .
F ( x)


r 1
h( x)  [ F ( x)] r g m
( F ( x))
Let
(4.7)
(4.8)
and

 g ( F ( x)) 
h( x)  [ F ( x)] r f ( x) g mr  2 ( F ( x))(r  1)[ F ( x)]m  r m
.
F ( x)


Thus,
Cr 1  j   1
r 1
x
[ F ( x)] r 1 f ( x) g m
( F ( x))dx

0
(r  1)!

 C r 1

j  1 (r  1)! 0
x j 1h( x)dx .
(4.9)
Now integrating RHS in (4.9) by parts and using the value of h(x) from (4.8), we
get
 j
C
Cr 1  j   1

r 1
r 1
x
[ F ( x)] r 1 f ( x) g m
( F ( x))dx   r 1  x [ F ( x)] r g m ( F ( x))dx

(r  1)! 0
(r  1)! 0
which reduces to
Relations for Moments of Lower Generalized Order
C r 1  j
r 1
x [ F ( x)] r 1 g m
( F ( x)){ F ( x)  x  1 f ( x)} dx  0 .

(r  1)! 0
19
(4.10)
Now applying a generalization of the Müntz-Szász Theorem (Hwang and Lin, [6])
to equation (4.10), we get
f ( x)
  x (  1)
F ( x)
which proves that
F ( x)  e  x

,
x  0 , ,   0 .
Computations of the moments and their Applications
The exact and explicit expressions for the single moments of order statistics and
record statistics allow us to evaluate the mean and variance of all order statistics
and record statistics. Table 2.1, 2.2 and Table 2.3, 2.4 present the mean and
variance of order statistics and record statistics of exponentiated inverted Weibull
distribution respectively.
5. Conclusion
This paper deals with the l gos from the exponentiated inverted Weibull
distribution. Some explicit expressions and recurrence relations for single and
product moments are derived. Characterizing results of this distribution has been
obtained by using conditional expectation of function of l gos and a recurrence
relation for single moments of l gos . Special cases are also deduced.
Acknowledgments The author appreciates the comments and remarks of the
referees which improved the original form of the paper.
References
Devendra Kumar
20
[1]
A. Flaih, H. Elsalloukh, Mendi, and M. Milanova, The Exponentiated
Inverted Weibull Distribution. Applied Mathematics & Information Sciences,
6 (2012), 167-171.
[2]
A. K. Mbah and M. Ahsanullah, Some characterization of the power
function distribution based on lower generalized order statistics. Pakistan J.
Statist., 23 (2007), 139-146.
[3]
D. Kumar and R. U. Khan, Moments of lower generalized order statistics
from exponentiated log-logistic distribution and characterization. Journal of
Advanced Research in Statistics and Probability, 3 (2011), 48-57.
[4]
D. Kumar, Explicit expressions for moments of lower generalized order
statistics from exponentiated Kumaraswamy distribution and its
characterization. J. Appl. Probab. Statis., 6 (2011), 61-72.
[5]
D. Kumar, Relations for moments of lower generalized order statistics from
a family of J- shaped distributions and its characterization. J. Appl. Probab.
Statis., 7 (2012), 71-86.
[6]
J.S. Hwang and G. D. Lin, On a generalized moments problem II. Proc.
Amer. Math. Soc., 91 (1984), 577-580.
[7]
M. Ahsanullah, Characterization of the uniform distribution by dual
generalized order statistics. Comm. Statist. Theory Methods, 33 (2004),
2921-2928.
[8]
M. Burkschat, E. Cramer and U. Kamps, Dual Generalized order statistics.
Metron, LXI (2003), 13-26.
[9]
P. Pawlas and D. Szynal, Recurrence relations for single and product
moments of lower generalized order statistics from the inverse Weibull
distribution, Demonstratio Math., XXXIV (2001), 353-358.
[10] R. U. Khan and D. Kumar, On moments of lower generalized order statistics
from exponentiated Pareto distribution and its characterization. Appl. Math.
Sci., 4 (2010), 2711-2722.
Relations for Moments of Lower Generalized Order
21
[11] R. U. Khan and D. Kumar, Expectation Identities of lower generalized order
statistics from generalized exponential distribution and its characterization,
Math. Method. Statis., 20 (2011), 150-157.
[12] R. U. Khan and D. Kumar, Lower generalized order statistics from
exponentiated gamma distribution and its characterization, ProbStats Forum,
4 (2011), 12-24.
[13] S. M. Ruiz, An algebraic identity leading to Wilson’s theorem, Math. Gaz.,
80 (1996), 579-582.
[14] U. Kamps, A Concept of Generalized Order Statistics, B. G. Teubner
Stuttgart, 1995.