Doubly Truncated Power- Hazard Rate Distribution via Generalized
Order Statistics
M. I. KHAN
Department of Mathematics, Faculty of Science,
Islamic University of Madinah,
SAUDI ARABIA
Abstract: - The paper highlights the moments characteristics of the doubly truncated power hazard rate
distribution via generalized order statistics. The particular cases and several deductions are explained. The
characterization result has also deliberated. Additionally, some numerical computations through R software are
listed.
Key-Words: - Single and product moments, Truncation, and Characterization.
Received: July 13, 2021. Revised: April 15, 2022. Accepted: May 17, 2022. Published: June 8, 2022.
1 Introduction
The behavior of any probability distribution depends
on its hazard functions. Several hazard functions are
available to deal with the different data. The power
hazard function has one of them to receive attention
among researchers. The power hazard function has
suggested by [1]. This model is adaptable to befit all
classical structures, including increasing, constant,
and decreasing.
The hazard function , probability density
function and cumulative density function
for the power hazard rate distribution
are stated respectively as observes
, , and (1)
, , (2)
, (3)
where and are scale and shape parameters.
The is still getting a lot of attention by
several authors due to its flexible properties of
hazard rate function . The model given in
this article generalizes various important
distributions, (see, Weibull, exponential, Rayleigh,
and linear failure rate distribution). More detail
information, see [2].
1.1 Doubly Truncated Power Hazard Rate
Distribution
This sub-section describes the formulation of doubly
as follows
For stated and
and
.
The of doubly is
, ,
(4)
and the of (4) is
, (5)
(6)
where
,
,
.
The doubly truncated distributions have a significant
contribution in many domains of science such as
hydrology, economics, biology, cosmology
engineering psychology, etc. ([3-4]). After a
detailed search, we notice that the moment
properties of doubly truncated remain
unknown, which is the theme of the findings.
1.2 Generalized Order Statistics
This sub-section reviews some basic definitions of
generalized order statistics .
The has been reported in literature by [5]. It is
a well-developed model for ascendingly ordered
random variables ). This concept has become an
indispensable tool in the field of mathematical and
applied statistics.
Let be having and
, if it contains the joint of as the
following form
1
1
n
j
j
k
1
1
1
[ ( )] ( ) [ ( )] ( )
i
nmk
i i n n
i
F x f x F x f x
(7)
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where
and
),1()( mink
i
From (7) the of the
th
r
is
11
1
: , , ( ) ( )[ ( )] [ ( )]
( 1)!
rr
r
r n m k m
C
f x f x F x g F x
r
,
(8)
The joint of the
th
r
and
th
s
GOS
is
1
, : , , ( , ) [ ( )] ( )
( 1)!( 1)!
m
s
r s n m k
C
f x y F x f x
r s r
1
11
( )[ ( ( )) ( ( ))] [ ( )] ( )
s
r s r
m m m
g F x h F y h F x F y f y
, (9)
and are needed
for (9). Further, we note that
1
1
s
si
i
C
,
1,)1log(
1,)1(
1
1
)(
1
mx
mx
m
xh m
m
and
)0()()( mmm hxhxg
,
)1,0[x
.
Ordinary order statistics , sequential ,
progressively Type -II censoring , and record
values are main examples of the model. For
more details [6-7].
The doubly truncated distribution of develops
from when a sample is from non-truncated
distribution. Many authors have developed the
moment properties of for doubly truncated
distribution. Detailed information can be noticed in,
[8-14]and among others.
Reducing the number of direct computations is the
main characteristic of recurrence relations. The
characterization outcomes play an essential part to
finds out the probability distributions. This article
addresses the moments of doubly truncated
using , which are unseen in the literature.
The remainder of the manuscript is as follows:
Section 2 contains the recurrence relations for single
moments and numerical computations for mean and
variance for several values of parameters. Product
moments are elaborated in Section 3.
Characterization result from doubly truncated
based on is in Section 4. Section 5 ends
with conclusion.
2 Single Moments
Here use,
Theorem 2.1. For reported in (4) and ,
(10)
where
1
11
r
ii
i
,
1
1
),1(
r
i
mkn
i
and
Proof: Applying (6) in (8), we have
.
Next, one can write the above expression as
(11)
where
.
Integrating by parts taking for integration, we
obtain
.
Similarly
Inserting the terms of
and
in (10) and solving, the Theorem 2.1 is proved.
Some corollaries and remarks based on single
moments of , when sample from doubly
truncated is described as follows.
2.1 Corollary
(i) For , Theorem 2.1 reduces
to single moments of , as
(ii) Single moments of record can be given
from Theorem 2.1.
(iii) Setting and (for non-
truncated case) in Theorem 2.1,
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as reported by similar result in [15] for
(iv) As stated in Corollary (iii) and setting
in (10), the corresponding result
is same as obtained by [16] for .
Remark 2.1
(i) Doubly truncated exponential distribution
can be obtained at in Theorem 2.1,
(ii) Setting , in Theorem 2.1, we get
doubly truncated Weibull distribution as
discussed by [10].
(iii) Setting in Theorem 2.1, it gives
doubly truncated linear exponential
distribution, as established by [12].
(iv) Setting
and in Theorem 2.1,
it yields, doubly truncated Rayleigh
distribution.
For arbitrarily selected values of and sample
sizes
,...20,10n
, Table 1-2, represents the
numerical computations of first four moments and
variances of from .
Table 1. Moments of for .
Table 2. Variances of for
n
r
10
1
0.0538
0.0572
0.0381
0.0434
0.0402
0.0419
0.0365
0.0386
20
1
0.0381
0.0434
0.0269
0.0329
0.0319
0.0344
0.0289
0.0316
2
5.0862
4.6901
3.5965
3.5544
2.9269
2.8471
2.6592
2.5416
,
,
10
1
0.1381
0.0648
0.0333
0.0184
0.0992
0.0250
0.0074
0.0025
20
1
0.1097
0.0408
0.0167
0.0073
0.0701
0.0125
0.0026
0.0006
2
2.0835
0.7760
0.3167
0.1389
1.3312
0.2375
0.0499
0.0119
30
1
0.0958
0.0312
0.0111
0.0043
0.0572
0.0083
0.0014
0.0003
2
2.7781
0.9038
0.3222
0.1235
1.6599
0.2417
0.0415
0.0081
3
19.4466
6.3269
2.2556
0.8643
11.6127
1.6917
0.29032
0.0564
40
1
0.0870
0.0257
0.0083
0.0029
0.0495
0.0062
0.0009
0.0002
2
3.3944
1.0034
0.3250
0.1131
1.9321
0.2438
0.0362
0.0061
3
32.2469
9.5322
3.0875
1.0749
18.3551
2.3156
0.3442
0.0579
4
132.5707
39.1879
12.6931
4.4190
75.4601
9.5198
1.4149
0.2380
50
1
0.0808
0.0222
0.0067
0.0022
0.0443
0.0050
0.0007
1e-04
2
3.960
1.0864
0.3267
0.1056
2.1712
0.2450
0.0326
0.0049
3
47.5088
13.0369
3.92
1.2669
26.0551
2.9400
0.3908
0.0588
4
248.1016
68.0816
20.4711
6.6160
136.0654
15.3533
2.0410
0.3071
5
713.2921
195.7347
58.8544
19.0211
391.188
44.1408
5.8678
0.8828
,
,
10
1
0.0931
0.0455
0.0232
0.0122
0.0901
0.0350
0.0144
0.0062
20
1
0.0810
0.0345
0.0153
0.0070
0.0758
0.0248
0.0086
0.0031
2
1.5384
0.6555
0.2911
0.1338
1.4396
0.4706
0.1632
0.0594
30
1
0.0746
0.0293
0.0120
0.0051
0.0685
0.0202
0.0063
0.0021
2
2.1652
0.8507
0.3483
0.1477
1.9854
0.5865
0.1838
0.0604
3
15.1565
5.955
2.4383
1.0334
13.8983
4.1057
1.2865
0.4229
40
1
0.0705
0.0261
0.0101
0.0040
0.0637
0.0175
0.0051
0.0016
2
2.7490
1.0197
0.3942
0.1577
2.4848
0.6831
0.1992
0.0609
3
26.1159
9.6872
3.7447
1.4983
23.6058
6.4895
1.8923
0.5789
4
107.3652
39.8252
15.3947
6.1597
97.0462
26.6792
7.7793
2.3799
50
1
0.0674
0.0239
0.0088
0.0033
0.0603
0.0157
0.0043
0.0013
2
3.3032
1.1718
0.4332
0.1658
2.9526
0.76775
0.2117
0.0612
3
39.6380
14.0612
5.1982
1.9891
35.4308
9.2119
2.5403
0.7350
4
206.9984
73.4309
27.1463
10.3877
185.0275
48.1064
13.2662
3.8383
5
595.1203
211.1139
78.0455
29.8646
595.1203
211.1139
78.0455
29.8646
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30
1
0.0311
0.0369
0.0220
0.0279
0.0279
0.0306
0.0253
0.0282
2
9.0573
8.5955
6.4045
6.5142
5.4497
5.1827
4.9513
4.7738
3
378.1142
347.2498
267.3671
263.1662
213.0991
196.2422
193.6136
180.7573
40
1
0.0269
0.0329
0.0190
0.0250
0.0253
0.0282
0.0230
0.0260
2
13.7149
13.3101
9.6979
10.0871
8.5490
8.1934
7.7673
7.3853
3
1101.492
1043.385
778.872
790.7382
654.1655
612.1221
594.3487
563.8213
4
18889.27
17950.38
13356.74
13603.85
11291.1
10600.43
9763.988
8274.752
50
1
0.0241
0.0306
0.0170
0.0228
0.0235
0.0228
0.0214
0.0244
2
18.9706
19.8566
13.4142
14.2102
12.1759
11.7492
11.0626
10.8430
3
2492.259
2416.392
1762.294
1831.283
1539.214
1457.132
1398.469
1342.153
4
68566.56
66607.95
48483.87
50479.38
42512.04
40325.07
38624.75
37143.15
5
565673.6
416202.2
399991.6
416281.5
350430.3
332261.5
318387
306043.6
3 Product Moments
Here use,
Theorem 3.1. For outlined in (4) and
(12)
where
1
11
s
ii
i
.
Proof: Using (8), we have
(13)
where
(14)
Next, using (5) in (13), we get
(15)
where
and
.
Integrating by parts taking for integration,
we get
Similarly
.
Following the same steps as reported in Theorem
2.1. Hence the Theorem 3.1 is complete.
Note: Product moments is reduced to single
moments at The corollaries and remarks as
listed in Section 2 are the same for the product
moments.
4 Characterization
The characterization of doubly is
addressed in this section.
Theorem 4.1: Let follows doubly , then
the necessary and sufficient condition for is
listed below
(16)
where is defined in Section 2.
Proof: Using (10), it is easy to determine the
necessary part.
For sufficiency part: On rearranging the terms in
(16), and after some simplification, it gives
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,
(17)
where
(18)
and
Now, integrating RHS in (16) by parts. Utilizing
the values of and from (18), we
get
. (19)
Next, generalization of the Müntz-Szász theorem
[17] apply to (19), we get
5 Conclusion
Moment's properties of from doubly
are investigated. For selected values, means and
variances for order statistics are enumerated.
Characterization of doubly via is
given.
Acknowledgement:
The researcher wishes to extend his sincere
gratitude to the Deanship of Scientific Research at
the Islamic University of Madinah for the support
provided to the Post-Publishing Program 1.
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