Adequate Mathematical Models of the Cumulative Distribution
Function of Order Statistics to Construct Accurate Tolerance Limits
and Confidence Intervals of the Shortest Length or Equal Tails
NICHOLAS NECHVAL1*, GUNDARS BERZINS1, KONSTANTIN NECHVAL2
1BVEF Research Institute, University of Latvia, Riga LV-1586, LATVIA
2Transport and Telecommunication Institute, Riga LV-1019, LATVIA
*Corresponding Author
Abstract: - The technique used here emphasizes pivotal quantities and ancillary statistics relevant for
obtaining tolerance limits (or confidence intervals) for anticipated outcomes of applied stochastic models
under parametric uncertainty and is applicable whenever the statistical problem is invariant under a group of
transformations that acts transitively on the parameter space. It does not require the construction of any
tables and is applicable whether the experimental data are complete or Type II censored. The exact
tolerance limits on order statistics associated with sampling from underlying distributions can be found
easily and quickly making tables, simulation, Monte-Carlo estimated percentiles, special computer
programs, and approximation unnecessary. The proposed technique is based on a probability transformation
and pivotal quantity averaging. It is conceptually simple and easy to use. The discussion is restricted to one-
sided tolerance limits. Finally, we give practical numerical examples, where the proposed analytical
methodology is illustrated in terms of the exponential distribution. Applications to other log-location-scale
distributions could follow directly.
Key-Words: - anticipated outcomes, parametric uncertainty, unknown (nuisance) parameters, elimination,
pivotal quantities, ancillary statistics, new-sample prediction, within-sample prediction.
Received: July 2, 2022. Revised: January 4, 2023. Accepted: February 2, 2023. Published: March 2, 2023.
1 Introduction
Statistical prediction and optimization (under
parametric uncertainty) of future random quantities
(future outcomes, order statistics, etc.) based on the
past and current data is the most prevalent form of
statistical inference. Predictive inferences for future
random quantities are widely used in risk
management, finance, insurance, economics,
hydrology, material sciences, telecommunications,
and many other industries. Predictive inferences
(predictive distributions, prediction or tolerance
limits (or intervals), confidence limits (or intervals)
for future random quantities on the basis of the past
and present knowledge represent a fundamental
problem of statistics, arising in many contexts and
producing varied solutions. Statistical prediction is
the process by which values for unknown
observables (potential observations yet to be made
or past ones which are no longer available) are
inferred based on current observations and other
information at hand. The approach used here is a
special case of more general considerations
applicable whenever the statistical problem is
invariant under a group of transformations, which
acts transitively on the parameter space [1-12].
There are the following types of prediction
problems:
1.1 New-Sample Prediction Problem
In this case, the data from a past sample of size n are
used to make prediction on one or more future units
in a second sample of size m from the same process
or population. For example, based on previous
(possibly censored) life test data, one could be
interested in predicting the following: (1) time to
failure of a new item (m = 1); (2) time until the kth
failure in a future sample of m units, m ≥ k; (3)
number of failures by time
in a future sample of
m units. Formally we call the problems in this
category as two-sample problems.
1.2 Within-Sample Prediction Problem
In this case, the problem is to predict future events
in a sample or process based on the early-failure
data from that sample or process. For example, if n
units are followed until censoring time
c
and there
are r observed ordered failure times,
1... r
XX
,
one could be interested in predicting the following:
1) time of next failure; 2) time until l additional
failures, l ≤ n ‒ r; 3) number of additional failures in
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a future interval
,
c
. Formally we call the
problems in this category as one-sample problems.
2 Adequate Mathematical Models of
the Cumulative Distribution Function
of Order Statistics to Construct New-
Sample Tolerance Limits (One-Sided)
Theorem 1.Let us assume that there is a random
sample of t ordered observations Z1 … Zt from a
known distribution with a probability density
function (pdf)
( ),fz
cumulative ´distribution
function (cdf)
( ),Fz
where
is the parameter (in
general, vector), then the adequate mathematical
models of the cumulative distribution function (cdf)
of the rth order statistic Zr, r{1, 2, …, t}, to
construct one-sided
−content tolerance limits with
confidence level
,
are given (for a new sample) as
follows:
2.1 Adequate Mathematical Model 2.1
()
,1
0
( ) ( | ),
r
Fz
r t r r r
f u du P Z z t
(1)
where
,1
1 ( 1) 1
1
( ) (1 ) ,
,1
r t r
r t r
f u u u
r t r
0 1,u
(2)
is the probability density function (pdf) of the
beta distribution
( ( , - 1))Beta r t r
with shape
parameters r and t-r+1,
( | ) [ ( )] [1 ( )] .
tj t j
r r r l
jr
t
P Z z t F z F z
j
(3)
Proof. On the one hand, it follows from (1) that
()
,1
0
()
r
Fz
r t r
r
df u du
dz
()
1 ( 1) 1
0
1(1 )
,1
r
Fz
r t r
r
du u du
dy r t r
111
() 1 ( ) ( ).
,1
rtr
rrr
Fz F z f z
r t r
(4)
On the other hand, it follows from (1) that
1
()
( | ) ,1
r
r
rr
r
Fz
dP Z z t
dz r t r
11
1 ( ) ( ).
tr
rr
F z f z
(5)
Thus,
()
r
Fz
is the generalized pivotal quantity:
,1
( ) ~ ( )
r t r
r
F z u f u
1 ( 1) 1
1(1 ) , 0 1.
,1
r t r
u u u
r t r
(6)
This ends the proof.
2.2 Adequate Mathematical Model 2.2
1
1,
1 ( )
( ) ( | ),
r
t r r r r
Fz
f u du P Z z t
(7)
where
1,
( 1) 1 1
1
( ) (1 ) ,
1,
t r l
t r r
f u u u
t r r
0 1,u
(8)
is the probability density function (pdf) of the beta
distribution
( ( 1, ))Beta t r r
with shape
parameters t
r+1 and r,
Proof. It follows from (7) that
1
1,
1 ( )
()
r
t r r
rFz
df u du
dz
1( 1) 1 1
1 ( )
1(1 )
1,
r
t r r
rFz
du u du
dz t r r
11 1
11 ( ) ( ) ( ( ))
1,
tr r
r r r
F z F z f z
t r r
1
1( ) 1 ( ) ( ).
,1
tr
r
r r r
F z F z f z
r t r
(9)
It follows from (5) and (9) that
1
1,
1 ( )
( | ) ( ) .
r
r r t r r
rr
Fz
dd
P Z z t f u du
dz dz
(10)
Thus,
1 ( )
r
Fz
is the generalized pivotal quantity:
1,
1 ( ) ~ ( )
t r r
r
F z u f u
( 1) 1 1
1(1 ) , 0 1.
1,
m l l
u u u
t l l
(11)
This ends the proof.
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2.3 Adequate Mathematical Model 2.3
()
1
1 ( )
,1
0
( ) ( | ),
r
r
Fz
tr
r F z
r t r r r
u du P Z z t
(12)
where
,1 1
() ,1
r t r ur t r
1
1
1
, (0, ).
1
11
r
t
rur
tr u
tr
ru
tr
(13)
is the probability density function (pdf) of the F
distribution
( ( , 1))F r t r
with parameters r and
t−r+1, which are positive integers known as the
degrees of freedom for the numerator and the
degrees of freedom for the denominator.
Proof. It follows from (12) that
()
1
1 ( )
,1
0
( ) ( | ).
r
r
Fz
tr
r F z
r t r r r
rr
dd
u du P Z z t
dz dz
(14)
Thus,
()
1
1 ( )
r
r
Fz
tr
r F z
is the generalized pivotal
quantity:
,1
()
11
~ ( )
1 ( ) , 1
rl m l
r
Fz
tr uu
r F z r t r
1
1
1
, (0, ).
1
11
l
m
rur
tr u
tr
ru
tr
(15)
This ends the proof.
2.4 Adequate Mathematical Model 2.4
1,
1 ( )
1 ( )
( ) ( | ),
r
r
t r r r r
Fz
r
t r F z
u du P Z z t
(16)
where
1,
1
1
( ) ,
1, 1
1
tr
t r r tr
tr
tr u
r
r
ut r r tr u
r
(0, ),u
(17)
is the probability density function (pdf) of the F
distribution
( ( 1, )F t r r
with parameters t
r
+1 and r, which are positive integers known as
the degrees of freedom for the numerator and
the degrees of freedom for the denominator.
Proof. It follows from (16) that
1,
1 ( )
1 ( )
( ) ( | ).
r
r
t r r r r
Fz
rr
r
t r F z
dd
u du P Z z t
dz dz
(18)
Thus,
1 ( )
1 ( )
r
r
Fz
r
t r F z
is the generalized pivotal
quantity:
1,
1 ( ) 1
~ ( )
1 ( ) 1,
rt r r
r
Fz
ruu
t r F z t r r
1
11
, (0, ).
1
1
tr
t
tr utr
ru
r
tr u
r
(19)
This ends the proof.
3 Adequate Mathematical Models of
the Cumulative Distribution Function
(Conditional) of Order Statistics to
Construct Within-Sample Tolerance
Limits (One-Sided)
Theorem 2. Let us assume that there is a random
sample of t ordered observations Z1 … Zt from a
known distribution with a probability density
function (pdf)
()fz
, cumulative distribution
function (cdf)
( ),Fz
where
is the parameter (in
general, vector), then the adequate mathematical
models of the conditional cumulative distribution
function (cdf) of the rth order statistic Zr (1
k < r
t) given Zk=zk are determined (for the same
sample) as follows:
3.1 Adequate Mathematical Model 3.1
()
1()
,1
0
( ) ( | ; )
r
k
Fz
Fz
r k t r r r k k
f u du P Z z Z z t
( ) ( )
1( ) ( )
j t k j
tk rr
j r k kk
F z F z
tk
jF z F z
(20)
where
( ) 1 ( ),F z F z
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1 ( 1) 1
,1 (1 )
( ) , 0 1,
,1
r k t r
r k t r uu
f u du u
r k t r
(21)
is the probability density function (pdf) of the beta
distribution
( ( , 1))Beta r k t r
with shape
parameters r−k and t−r+1.
Proof. On the one hand, it follows from (20) that
()
1()
,1
0
()
r
k
Fz
Fz
r k t r
r
df u du
dz
()
1() 1 ( 1) 1
0
(1 )
,1
r
k
Fz
Fz r k t r
r
d u u du
dz r k t r
1
( ) ( ) ( )
1( ) ( ) ( ) .
( , 1)
r k t r
r r r
k k k
F z F z f z
F z F z F z
r k t r
(22)
On the other hand, it follows from (20) that
( | ; )
r r k k
r
dP Z z Z z t
dz
1
( ) ( ) ( )
1( ) ( ) ( ) .
( , 1)
r k t r
r r r
k k k
F z F z f z
F z F z F z
r k t r
(23)
Thus,
1 ( ) ( )
rk
F z F z
is the generalized pivotal
quantity:
1 ( 1) 1
,1
() (1 )
1 ~ ( ) ,
( ) , 1
r k t r
rr k t r
k
Fz uu
u f u
F z r k t r
0 1.u
(24)
This ends the proof.
3.2 Adequate Mathematical Model 3.2
1
1,
()
()
( ) , ( | ; )
r
k
t r r k r r k k
Fz
Fz
f u du P Z z Z z t
( ) ( )
1( ) ( )
j t k j
tk rr
j r k kk
F z F z
tk
jF z F z
(25)
where
11 1
1, (1 )
( ) , 0 1,
1,
ml lk
m l l k uu
f u u
m l l k
(26)
is the probability density function (pdf) of the beta
distribution
( ( 1, ))Beta t r r k
with shape
parameters t−r+1 and r−k.
Proof. On the one hand, it follows from (25) that
1
1,
()
()
()
r
k
t r r k
rFz
Fz
df u du
dz
111 1
()
()
(1 )
1,
r
k
tr rk
rFz
Fz
d u u du
dz t r r k
1 1 1
( ) ( ) ( )
1( ) ( ) ( ) .
( , 1)
r k t r
r r r
k k k
F z F z f z
F z F z F z
r k t r
(27)
On the other hand, it follows from (25) that
( | ; )
r r k k
r
dP Z z Z z t
dz
1
( ) ( ) ( )
1( ) ( ) ( ) .
( , 1)
r k t r
r r r
k k k
F z F z f z
F z F z F z
r k t r
(28)
Thus,
( ) ( )
rk
F z F z
is the generalized pivotal
quantity:
11 1
1,
() (1 )
~ ( ) ,
( ) 1,
tr rk
rt r r k
k
Fz uu
u f u
F z t r r k
0 1.u
(29)
This ends the proof.
3.3 Adequate Mathematical Model 3.3
( ) ( )
11( ) ( )
,1
0
()
rr
kk
F z F z
tr
r k F z F z
r k t r u du
( | ; )
r r k k
P Z z Z z t
( ) ( )
1,
( ) ( )
j t k j
tk rr
j r k kk
F z F z
tk
jF z F z
(30)
where
1
,1 1
1
1
( ) ,
( , 1) 11
rk
r k t r tk
rk
rk u
tr
tr
ur k t r rk
u
tr
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(0, ),u
(31)
is the probability density function (pdf) of the F
distribution
( ( , 1))F r k t r
with parameters r−k
and t−r+1, which are positive integers known as the
degrees of freedom for the numerator and the
degrees of freedom for the denominator.
Proof. It follows from (30) that
( ) ( )
11( ) ( )
,1
0
( ) .
rr
kk
F z F z
tr
r k F z F y
r k t r
r
du du
dz
( | ; )
r r k k
r
dP Z z Z z t
dz
(32)
Thus,
11 ( ) ( ) 1 ( ) ( )
r k r k
tr F z F z F z F z
rk
is
the generalized pivotal quantity:
,1
( ) ( )
11 ~ ( )
( ) ( )
rr
r k t r
kk
F z F z
tr uu
F z F z
rk
1
1
11,
( , 1) 1
11
rk
tk
rk
urk
tr
r k t r t r
rk
u
tr
(0, ).u
(33)
This ends the proof.
3.4 Adequate Mathematical Model 3.4
1, ,
( ) ( )
1
1 ( ) ( )
( ) ,
rr
kk
t r r k
F z F z
rk
t r F z F z
u du
( | ; )
r r k k
P Z z Z z t
( ) ( )
1( ) ( )
j t k j
tk rr
j r k kk
F z F z
tk
jF z F z
(34)
where
1, 1
() ( 1, )
t r r k ut r r k
1
1
11, (0, ).
1
1
tr
mk
tr utr
rk u
rk
tr u
rk
(35)
is the probability density function (pdf) of the F
distribution
( ( 1, ))F t r r k
with parameters t
r+1 and r
k, which are positive integers known as
the degrees of freedom for the numerator and the
degrees of freedom for the denominator.
Proof. It follows from (30) that
1, ,
( ) ( )
1
1 ( ) ( )
()
rr
kk
t r r k
rF z F z
rk
t r F z F z
du du
dz
( | ; )
r r k k
r
dP Z z Z z t
dz
(36)
Thus,
( ) ( ) 1 ( ) ( )
1r k r k
rk F z F z F z F z
tr
is the
generalized pivotal quantity:
( ) ( )
1
( ) ( )
1
rr
kk
F z F z
rk
F z F z
tr
1
1,
~ ( ) ( 1, )
t r r k
u u t r r k
1
1
11, (0, ).
1
1
tr
tr
tr utr
rk u
rk
tr u
rk
(37)
This ends the proof.
4 Two-Parameter Exponential
Distribution
Let Z = (Z1 ... Zr) be the first r ordered
observations (order statistics) in a sample of size t
from the two-parameter exponential distribution
with the probability density function (pdf)
1
( ) exp , 0, z 0,
z
fz
(38)
and the cumulative distribution function (cdf)
( ) 1 exp , ( ) 1 ( ),
z
F z F z F z
(39)
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where
( , ),
is the shift parameter and
is
the scale parameter. It is assumed that these
parameters are unknown. In Type II censoring,
which is of primary interest here, the number of
survivors is fixed and Zk is a random variable. In
this case, the likelihood function is given by
1
( , ) ( ) ( )
rtr
ir
i
L f z F z
1
1exp ( )( )
r
ir
ri
z t r z
11
111
1exp ( )( )
ri
rir
z z z
t r z z z
11
11
1exp ( ) ( )( )
r
ir
ri
z z t r z z
1
()
1exp tz
1
1
()
11
exp exp ,
r
r
s t s
(40)
where
11
11
1
,
( ) ( )( )
r
r i r
i
SZ
S Z Z t r Z Z
S
(41)
is the complete sufficient statistic for μ. The
probability density function of S=(S1, Sr) is given by
1
( , )
r
f s s
1
1
211
21
00
()
11
exp exp
()
11
exp exp
r
r
r
rr
r
rr
r
s t s
s s t s
t
ds ds
st
1
1
2
()
11
exp exp
( 1) 1
r
r
r
r
s t s
r
st
2
1
1exp
( 1)
rr
r
r
s
s
r
11
()
exp ,
r
ts
tf s f s
(42)
where
1
11
()
exp , ,
ts
t
f s s
(43)
2
1
1exp , 0.
( 1)
rr
r r r
r
s
f s s s
r
(44)
1
1S
V
(45)
is the pivotal quantity, the probability density
function of which is given by
1 1 1 1
( ) exp , 0,f v t tv v
(46)
r
r
S
V
(47)
is the pivotal quantity, the probability density
function of which is given by
2
1exp , 0.
( 1)
r
r r r r r
f v v v v
r
(48)
4.1 Constructing One-Sided γ-Content
Tolerance Limit with a Confidence Level
β (where Model 2.1 is used)
Theorem 3. Let Z1…Zr be the first r ordered
observations from the preliminary sample of size t
from a two-parameter exponential distribution
defined by the probability density function (37).
Then the lower one-sided γ-content tolerance limit
with a confidence level β, Lk = Lk (S) (on the kth
order statistic Yk from a set of n future ordered
observations Y1…Yn also from the distribution
(37)), which satisfies
Pr ( | ) ,
kk
E P Y L n
(49)
is given by Lk = Lk (S)
1
1
1
11
1
1
1
11
ln(1 )
1 , if ,
ln
1
ln(1 )
1 , if ,
ln
1
tr
r
tr
r
S
St
t
S
St
t
(50)
where
11
, 1 ,1
( , - 1), .1 quantile
k n k
q Beta k n k
(51)
Proof. It follows from (1), (39) and (49) that
Pr ( | )
kk
P Y L n
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()
,1
0
Pr ( ) 1
k
FL
k n k
f u du
, 1;1
Pr 1 exp kk n k
Lq
, 1;1
Pr exp 1
kk n k
Lq
11, 1;1
Pr ln 1
krk n k
r
LS
SS q
S
1
1, 1;1
Pr ln 1
krk n k
r
LS
SS
q
S
1
ln
1 1 1 1 1
0
Pr ln ( ) ,
Lr
k
k
V
Lr
V V f v dv
(52)
where
1
.
k
k
L
r
LS
S
(53)
It follows from (49) and (52) that
1
ln
1 1 1
0
Pr ( | ) ( )
Lr
kV
kk
E P Y L n E f v dv
1
ln
11
0
exp
Lr
kV
E t tv dv
1
1 exp ln
k
Lr
E t V
1
1 exp exp ln
k
t
Lr
E t V
1
1 exp k
tLr
E t V
1
0
1 exp ( )
k
tL r r r r
t v f v dv
2
1
0
1
1 exp exp
( 1)
k
tr
L r r r r
t v v v dv
r
1
1
1.
1k
t
r
L
t
(54)
It follows from (54) that
1
1
1
111.
1
k
tr
k
L
r
LS
St
(55)
It follows from (55) that
1
1
1
11.
1
tr
r
kS
LS t
(56)
It follows from (56) that
1
1
1
10
1
tr
1
ln(1 )
if ln
t
(57)
or
1
1
1
10
1
tr
1
ln(1 )
if .
ln
t
(58)
Then (50) follows from (56), (57) and (58). This
ends the proof.
Corollary 3.1. Let Z1…Zr be the first r
ordered observations from the preliminary sample of
size t from a two-parameter exponential distribution
defined by the probability density function (38).
Then the upper one-sided γ-content tolerance limit
with a confidence level β, Uk Uk (S) (on the kth
order statistic Yk from a set of n future ordered
observations Y1…Yn also from the distribution
(38), which satisfies
Pr ( | ) ,
kk
E P Y U n
(59)
is given by
1
1
1
1
1
1
ln
1 , if ,
ln
ln
1 , if ,
ln
tr
r
k
tr
r
S
St
t
U
S
St
t
(60)
where
, 1 ,
1 ( , - 1), . quantile
k n k
q Beta k n k
(61)
4.2 Numerical Practical Example
Let us assume that k =1, r = m = n =15, γ = β = 0.95,
11
1
11
9,
( ) .
( )( ) 192.2508
rt i
rir
SZ
ZZ
St r Z Z
S
(62)
Then the lower one-sided γ-content tolerance limit
with a confidence level β, Lk=1 Lk=1(S) can be
obtained from (50). Since
1, 1 ,1
ln(1 )
ln(1 )
15 876,
ln ln 1 k n k
tq
(63)
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where the quantile of
( , - 1) 1, Beta k n k
is
given by
, 1 ,1 0.003414,
k n k
q
(64)
it follows from (50) and (64) that
1
1
1
11
( ) 1 9 3 6.
1
tr
r
S
LS
t
S
(65)
Statistical inference. From (65) it follows that
there is a 95% certainty that failures will not occur
in the proportion γ=0.95 or more of a set of n
selected items before the end of the lower one-sided
γ-content tolerance limit L1(S) = 6 monthly
intervals.
5 Adequate Mathematical Models of
the Cumulative Distribution Function
of Order Statistics to Construct Equal
Tails or Shortest Length Confidence
Intervals
Let Z = (Z1 ... Zr) be the first r ordered
observations (order statistics) in a sample of size t
from the exponential distribution with the
probability density function
1
( ) exp( / ), 0, z 0,f z z
(66)
and the cumulative probability distribution function
( ) 1 exp( / ),F z z
(67)
where μ is the scale parameter. It is assumed that the
parameter
is unknown. In Type II censoring,
which is of primary interest here, the number of
survivors is fixed and Zr is a random variable. It is
known that
1
()
r
r j r
j
S Z t r Z
(68)
is the complete sufficient statistic for μ. The
probability density function of Sr is given by
1
1
( ) exp , 0.
()
rr
r r r
r
s
f s s s
r
(69)
rr
VS
(70)
is the pivotal quantity, the probability density
function of which is given by
1
1
( ) exp( ), 0.
()
r
r r r r
f v v v v
r
( ( ,1)).Gamma r
(71)
Consider the above example, where t units, whose
lifetimes are distributed according to the same
exponential distribution (66), are put on test
simultaneously, and where all units are observed
until failure. In this case, Z1 ≤ ... ≤ Zr are the first r
ordered observations (the Type II censored sample
and the parameter μ is unknown).
1) What is the 100(1
)% shortest-length
confidence interval for μ based on Zr ? Answer 1:
5.1 Application of Mathematical Model 2.1
It follows from (6) that
()
r
Fz
is the generalized
pivotal quantity:
,1
1 ( 1) 1
(1 )
( ) ~ ( ) ,
,1
r t r
r t r
ruu
F z u f u r t r
0 1;u
( ( , - 1)).Beta r t r
(72)
Using (72), it can be obtained a 100(1
)%
confidence interval for μ from
1 2 1 2
Pr ( ) Pr 1 exp r
rz
u F z u u u
21
Pr 1 exp 1
r
z
uu
12
11
Pr ln ln
11
r
z
uu
21
Pr ln 1/ 1 ln 1/ 1
rr
zz
uu
1
(73)
by suitably choosing the decision variables
1
u
and
2
u
. Hence, the statistical confidence interval for
is given by
21
,.
ln 1/ 1 ln 1/ 1
rr
zz
uu
(74)
The length of the statistical confidence interval for
is given by
12
12
( , | ) .
ln 1/ 1 ln 1/ 1
rr
rzz
L u u z uu
(75)
In order to find the shortest length confidence
interval
12
( , | )
r
L u u z
, we should find a pair of
decision variables
1
u
and
2
u
such that
12
( , | )
r
L u u z
is minimum.
It follows from (73) and (74) that
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2 2 1
, 1 , 1 , 1
100
( ) ( ) ( )
r t r r t r r t r
u u u
u
f u du f u du f u du
1 1 ,pp
(76)
where p
(0 )p
is a decision variable,
2
,1
0
( ) 1
r t r
u
f u du p
(77)
and
1
,1
0
( ) .
k m k
u
f u du p
(78)
Then
2
u
represents the
1p
- quantile, which
is given by
2 1 ;( , 1) ,
p r t r
uq
(79)
1
u
represents the
p
- quantile, which is given by
1 ;( , 1).
p r t r
uq
(80)
The shortest length confidence interval for μ can
be found as follows:
Minimize
2
212
12
( , | ) ln 1/ 1 ln 1/ 1
rr
rzz
L u u z uu
2
;( , 1)
1 ;( , 1)
ln 1/ 1
ln 1/ 1
r
p r t r
r
p r t r
z
q
z
q
(81)
subject to
0,p
(82)
Numerical Solutions. The optimal numerical
solution minimizing L(u1, u2| zr) can be obtained
using the computer software "Solver". If, for
example, t=10, r = 4,
= 0.05, then the optimal
numerical solution is given by
1 ;( , 1)
0.048394 0.148512,, p r t r
p u q
1 , 1)2 ;(
0.779435
p r t r
u q
(83)
with the 100(1
)% shortest-length confidence
interval
12
( , | ) 1026.313 109.1584 917.1544.
r
L u u z
(84)
The 100(1
)% equal tails confidence interval is
given by
12
( , | ; / 2)
r
L u u z p
1273.159 156.1236 1117.036.
(85)
with
21 0.121552, 0.652453.0.025, pu u
(86)
Relative efficiency. The relative efficiency of
L(u1,u2 | zr ; p =
/2) as compared with L(u1,u2 | zr) is
given by
2112
rel.eff. ( | ; (, / 2 , | ), )
Lr r
Luz Lu zu up
1
21
2
()
=( | )
,|
, ; / 2
r
r
u
u
L
z
u
p
z
Lu
917.1544 0.821061
1117. .
036
(87)
2) What is the 100(1
)% shortest-length
confidence interval for μ based on Sr ? Answer 2:
5.2 Application of Gamma (r,1)
It follows from (71) that
/
r
Su
represents the
pivotal quantity:
,1
~ ( )
rr
Su f u
1
1exp , 0,
()
r
u u u
r
( ( ,1)).Gamma r
(88)
Using (88), it can be obtained a 100(1
)%
confidence interval for μ from
12
21
Pr Pr 1
r r r
S S S
uu uu
(89)
by suitably choosing the decision variables
1
u
and
2
u
. Hence, the statistical confidence interval for
is given by
21
,.
rr
s u s u
(90)
The length of the statistical confidence interval for
is given by
1 2 1 2
( , | ) .
r r r
L u u s s u s u
(91)
In order to find the shortest length confidence
interval
12
( , | )
r
L u u s
, we should find a pair of
decision variables
1
u
and
2
u
such that
12
( , | )
r
L u u s
is minimum.
It follows from (88) and (89) that
2 2 1
,1 ,1 ,1
100
( ) ( ) ( )
r r r
u u u
u
f u du f u du f u du
1 1 ,pp
(92)
where p
(0 )p
is a decision variable,
2
,1
0
( ) 1
r
u
f u du p
(93)
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and
1
,1
0
( ) .
r
u
f u du p
(94)
Then
2
u
represents the
1p
- quantile, which
is given by
2 1 ;( ,1) ,
pr
uq
(95)
1
u
represents the
p
- quantile, which is given by
1 ;( ,1).
pr
uq
(96)
The shortest length confidence interval for μ can
be found as follows:
Minimize
2
212
12
( , | ) rr
rss
L u u s uu
2
;( ,1) 1 ;( ,1)
rr
p r p r
ss
qq
(97)
subject to
0,p
(98)
The optimal numerical solution minimizing L(u1,
u2 | sr) can be obtained using the standard computer
software "Solver" of Excel 2016. If, for example,
t=10, r = 4,
= 0.05, then the optimal numerical
solution is given by
1 ;( ,1)
, 1.350.048393 ,1362
pr
p u q
,1)2 1 ;(
12.45735
pr
u q
(99)
with the 100(1
)% shortest-length confidence
interval
12
( , | ) 1035.992 112.3884 923.6087.
r
L u u s
(100)
The 100(1
)% equal tails confidence interval is
given by
12
( , | ; / 2)
r
L u u s p
1284.562 159.6848 1124.878
(101)
with
1 2
0.025, 1.0898 ., 8.76727365p uu
(102)
Relative efficiency. The relative efficiency of
L(u1,u2 | sr ; p =
/2) as compared with L(u1,u2 | sr) is
given by
2112
rel.eff. ( | ; (, / 2 , | ), )
Lr r
Lus Lu su up
1 2
1 2 8
()
=,| 923.6087
, ; / 2( | ) 1124.87
r
r
u
u s p
L u s
Lu
.0.821075
(103)
Inference. Two completely different versions of
constructing confidence intervals of the shortest
length and equal tails gave practically the same final
results. This confirms the validity of the analytical
conclusions and computational algorithms presented
in this paper.
6 New Mathematical Approach to
Constructing Statistical Estimates of
the Probability Density and
Cumulative Distribution Function
Let Z = (Z1 ... Zr) be the first r ordered
observations (order statistics) in a sample of size t
from the two-parameter exponential distribution
with the probability density function (pdf) (38) and
the cumulative distribution function (cdf) (39),
where the parametric vector μ is equal to
( , );
the
shift parameter
and the scale parameter
are
unknown.
6.1 Example of Constructing Statistical
Estimates for the Two-Parameter
Exponential Distribution
Let us suppose that Z is a future observation from
the same distribution (39), independent of Z = (Z1
... Zr). Then a statistical estimate of (39) can be
determined as follows.
Step 1. Invariant embedding of S1 in (39) to
isolate the unknown parameter
from the problem
through V1 (45),
11
( ) 1 exp z s s
Fz
111
1 exp exp , ,
zs v z s
(104)
Step 2. Averaging (104) over the probability
distribution of the pivotal quantity V1 to eliminate
unknown parameter
from the problem. It follows
from (104) and (46) that the pivot-based estimate of
the cumulative distribution function (39) (obtained
through the pivot-based method) is given by
1, 1 1 1
0
( ) ( ) ( )
s
F z F z f v dv
11 1 1
0
1 exp exp exp
zs v t tv dv
111
0
1 exp exp [ 1)
zs t v t dv
11
1 exp , , .
1
zs tzs
t
(105)
Since
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1, 1
() 1exp
1
s
dF z zs
t
dz t
(106)
and
1
1
1
1
1exp 1
1exp ,
1exp
1
s
zs
tzs
t
zs
tdz
t
(107)
It follows from (107) that the probability density
function (pdf) of Z is given by
1
1
,1
1
( ) exp , ,
szs
f z z s
(108)
with the cumulative distribution function
1
1
,( ) 1 exp .
szs
Fz
(109)
Step 3. Invariant embedding of Sr in (109) to
isolate the unknown parameter
from the problem
through Vr (47),
1
1
,( ) 1 exp
szs
Fz
1
1 exp r
r
z s s
s
11
1 exp , .
r
r
zs
v z s
s
(110)
Step 4. Averaging (111) over the probability
distribution of the pivotal quantity Vr to eliminate
unknown parameter
from the problem. It follows
from (110) and (48) that the pivot-based estimate of
the cumulative distribution function (39) (obtained
through the pivot-based method) is given by
1
1
,
00
( ) 1 exp
s r r r r
r
zs
F z f v dv v
s
2
1
exp
( 1)
r
r r r
v v dv
r
( 1)
1
1 1 ( ).
r
r
zs Fz
s
s
(111)
where
( 1)
1
( ) 1 ( ) 1 .
r
r
zs
F z F z s
ss
(112)
The pivot-based estimate of the probability density
function (38) is given by
11
() 1
( ) 1 , .
r
rr
dF z zs
r
f z z s
dz s s
s
s
(113)
It follows from (111) that the cumulative
distribution function of the ancillary statistic
1
r
ZS
WS
(114)
is given by
1
1
( ) 1 .
1r
Fw w
(115)
The probability density function of the ancillary
statistic (114) is given by
( ) 1
( ) , 0.
(1 )r
dF w r
f w w
dw w
(116)
Constructing Confidence Interval for Z. Using
(114) and (115), it can be obtained a 100(1
)%
confidence interval for Z from
1
1 2 1 2
Pr Pr
r
ZS
w W w w w
S
1 1 2 1
Pr 1 .
rr
w S S Z w S S
(117)
by suitably choosing the decision variables
1
w
and
2
w
. Hence, the statistical confidence interval for Z
is given by
1 1 2 1
,.
rr
w s s w s s
(118)
The length of the statistical confidence interval for Z
is given by
1 2 2 1 2 1
( , | ) .
r r r r
L w w s w s w s w w s
(119)
In order to find the shortest length confidence
interval
12
( , | )
r
L w w s
, we should find a pair of
decision variables
1
w
and
2
w
such that
12
( , | )
r
L w w s
is minimum.
It follows from (116) and (117) that
2 2 1
100
( ) ( ) ( )
w w w
w
f w dw f w dw f w dw
21
( ) ( ) 1 1 ,F w F w p p
(120)
where p
(0 )p
is a decision variable,
2
2
0
( ) ( ) 1
w
f w dw F w p
(121)
and
1
1
0
( ) ( ) .
w
f w dw F w p
(122)
Then
2
u
represents the
1p
- quantile, which
is given by
1/( 1)
21 11,
r
p
wq p
(123)
1
w
represents the
p
- quantile, which is given by
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1/( 1)
111.
1
r
p
wq p
(124)
The shortest length confidence interval for Z can
be found as follows:
Minimize
2
21 2 2 1 1
( , | )
r r p
L w w s w w s q
2
1/( 1) 1/( 1)
2
11
.
1
rr
r
s
pp
(125)
subject to
0,p
(126)
The optimal numerical solution minimizing L(w1,
w2 | sr) can be obtained using the standard computer
software "Solver" of Excel 2016. If, for example, r
= 4,
= 0.05, then the optimal numerical solution is
given by
0p
(127)
with the 100(1
)% shortest-length confidence
interval
12
( , | ) 1.114743 .
rr
L w w s s
(128)
The 100(1
)% equal tails confidence interval is
given by
12
( , | ; / 2) 1.508517
rr
L w w s p s
(129)
with
5.0.02p
(130)
Relative efficiency. The relative efficiency of
1 2 ;, / 2( | )
r
spL w w
as compared with L(w1,w2| sr)
is given by
2112
rel.eff. ( | ; (, / 2 , | ), )
Lr r
Lws Lw sw wp
1
21
2
( ) 1.114743
=1
,|
., ; /( | ) 5 72 0851 r
r
r
r
Lws
w w sL p
ws
s
0.738966.
(131)
7 Conclusion
The new intelligent computational methods
proposed in this paper are conceptually simple,
efficient, and useful for constructing accurate
statistical tolerance limits and shortest-length or
equal-tailed confidence intervals under the
parametric uncertainty of applied stochastic models.
The methods listed above are based on adequate
mathematical models of the cumulative distribution
function of order statistics and constructive use of
the principle of invariance in mathematical
statistics. We have illustrated proposed intelligent
computational methods for the exponential
distribution. Applications to other log-location-scale
distributions can follow directly.
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Gundars Berzins, Konstantin Nechval
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165
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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