Advances in Systems Science and Applications (2012) Vol.12 No.4 373-387

Predictive Inferences for a Future Number of Failures
Coming from Underlying Models under Parametric

Uncertainty
Konstantin N. Nechval1, Nicholas A. Nechval2, Maris Purgailis2,
Uldis Rozevskis2, Vladimir F. Strelchonok3 and Max Moldovan4

1Applied Mathematics Department, Transport and Telecommunication Institute
Lomonosov Street 1, LV-1019, Riga, Latvia

2Statistics Department, EVF Research Institute, University of Latvia
Raina Blvd 19, LV-1050, Riga, Latvia

3Informatics Department, Baltic International Academy
Lomonosov Street 4, LV-1019, Riga, Latvia

4Australian Institute of Health Innovation, University of New South Wales,
Level 1 AGSM Building, Sydney NSW 2052, Australia

Abstract

In this paper, we present an accurate procedure to obtain prediction limits for
the number of failures that will be observed in a future inspection of a sample
of units, based only on the results of the first in-service inspection of the same
sample. The failure-time of such units is modeled with a two-parameter Weibull
distribution indexed by scale and shape parameters β and δ, respectively. It will
be noted that in the literature only the case is considered when the scale param-
eter β is unknown, but the shape parameter δ is known. As a rule, in practice
the Weibull shape parameter δ is not known. Instead it is estimated subjectively
or from relevant data. Thus its value is uncertain. This δ uncertainty may con-
tribute greater uncertainty to the construction of prediction limits for a future
number of failures. In this paper, we consider the case when both parameters
β and δ,are unknown. In literature, for this situation, usually a Bayesian ap-
proach is used. Bayesian methods are not considered here. We note, however,
that although subjective Bayesian prediction has a clear personal probability in-
terpretation, it is not generally clear how this should be applied to non-personal
prediction or decisions. Objective Bayesian methods, on the other hand, do not
have clear probability interpretations in finite samples. The technique proposed
here for constructing prediction limits emphasizes pivotal quantities relevant for
obtaining ancillary statistics. and represents a special case of the method of in-
variant embedding of sample statistics into a performance index. Two versions
of prediction limits for a future number of failures are given.
Keywords Weibull distribution, parametric uncertainty, future number of fail-
ures, prediction limits



374 Konstantin N. Nechval:Predictive Inferences for a Future Number of Failures Coming from...

1 Introduction

This paper extends the results of Nelson [1]. Nelson’s prediction limits were mo-
tivated by the following application. Nuclear power plants contain large heat
exchangers that transfer energy from the reactor to steam turbines. Such ex-
changers typically have 10,000 to 20,000 stainless steel tubes that conduct the
flow of steam. Due to stress and corrosion, the tubes develop cracks over time.
Cracks are detected during planned inspections. The cracked tubes are subse-
quently plugged to remove them from service. To develop efficient inspection and
plugging strategies, plant management can use a prediction of the added number
of tubes that will need plugging by a specified future time.

Nelson presents simple prediction limits for the number of failures that will be
observed in a future inspection of a sample of units. The past data consist of
the cumulative number of failures in a previous inspection of the same sample
of units. Life of such units is modeled with a Weibull distribution with a given
shape parameter value.

Prediction of an unobserved random variable is a fundamental problem in s-
tatistics. Hahn and Nelson [2], Patel [3], and Hahn and Meeker [4] provided
surveys of methods for statistical prediction for a variety of situations on this
topic. In the areas of reliability and life-testing, this problem translates to ob-
taining prediction intervals for lifetime distributions. Nordman and Meeker [5]
compared probability ratio, simplified probability ratio and likelihood ratio meth-
ods proposed by Nelson [1], assuming known the Weibull shape parameter δ.

In this paper, we use a frequentist procedure, which is called ‘within-sample
prediction of future order statistics’, when the time-to-failure follows the two-
parameter Weibull distribution indexed by scale and shape parameters β and δ.
We consider the case when both parameters β and δ are unknown. The technique
proposed here for constructing prediction limits emphasizes pivotal quantities rel-
evant for obtaining ancillary statistics and represent a special case of the method
of invariant embedding of sample statistics into a performance index applicable
whenever the statistical problem is invariant under a group of transformations,
which acts transitively on the parameter space (Nechval et al. [6-7]).

Conceptually, it is useful to distinguish between “new-sample” prediction,
“within-sample” prediction, and “new-within-sample” prediction. Some math-
ematical preliminaries for the within-sample prediction are given below.

2 Mathematical Preliminaries for Within-Sample Prediction

Theorem 1 Let X1 ≤ . . . ≤ Xk be the first k ordered observations (order statis-
tics) in a sample of size m from a continuous distribution with some probability
density functio fθ(x) and distribution function Fθ(x), where θ is a parameter (in
general, vector). Then the joint probability density function of X1 ≤ . . . ≤ Xk



Advances in Systems Science and Applications (2012) Vol.12 No.4 375

and the lth order statistics Xl(1 < k < l < m) is given by

gθ(x1, . . . , xk, xl) = gθ(x1, . . . , xk)gθ(xl|xk), (1)

where

gθ(x1, . . . , xk) =
m!

(m− k)!
Πk

i=1fθ(xi)[1− Fθ(xk)]
m−k, (2)

gθ(xl|xk)

=
(m− k)!

(l − k − 1)!(m− l)!
[
Fθ(xl)− Fθ(xk)

1− Fθ(xk)
]l−k−1[1− Fθ(xl)− Fθ(xk)

1− Fθ(xk)
]m−l fθ(xl)

1− Fθ(xk)

=
(m− k)!

(l − k − 1)!(m− l)!

l−k−1X
j=0

�
l − k − 1

j

�
(−1)j [

1− Fθ(xl)

1− Fθ(xk)
]m−l+j fθ(xl)

1− Fθ(xk)

=
(m− k)!

(l − k − 1)!(m− l)!

m−lX
j=0

�
m− l
j

�
(−1)j [

Fθ(xl)− Fθ(xk)

1− Fθ(xk)
]l−k−1+j fθ(xl)

1− Fθ(xk)

(3)

represents the conditional probability density function of Xl given Xk = xk.

Proof.The joint density of X1 ≤ . . . ≤ Xk and Xl is given by

gθ(x1, . . . , xk, xl) =
(m)!

(l − k − 1)!(m− l)!

kY
i=1

fθ(xi)[Fθ(xl)− Fθ(xk)]
l−k−1fθ(xl)

[1− Fθ(xl)]
m−l = gθ(x1, . . . , xk)gθ(xl|xk). (4)

It follows from (4) that

gθ(xl|x1, . . . , xk) =
gθ(x1, . . . , xk, xl)

gθ(x1, . . . , xk)
= gθ(xl|xk), (5)

i.e., the conditional distribution of Xl l, given Xi = xi for all i = 1, . . . , k , is the
same as the conditional distribution of Xl , given only Xk = xk,which is given by
(3). This ends the proof.
Corollary 1.1.The conditional probability distribution function of Xl given
Xk = xk is

Pθ(Xl ≤ xl|Xk = xk)

=1− (m− k)!

(l − k − 1)!(m− l)!

l−k−1X
j=0

�
l − k − 1

j

�
(−1)j

m− l + 1 + j

h 1− Fθ(xl)

1− Fθ(xk)

im−l+1+j

=
(m− k)!

(l − k − 1)!(m− l)!

m−lX
j=0

�
m− l
j

�
(−1)j

l − k + j

hFθ(xl)− Fθ(xk)

1− Fθ(xk)

il−k+j
.

(6)



376 Konstantin N. Nechval:Predictive Inferences for a Future Number of Failures Coming from...

Corollary 1.2. Let X1 ≤ . . . ≤ Xk be the first k order statistics in a sample of
size m from the two-parameter Weibull distribution with the probability density
function

fθ(x) =
δ

β
(
x

β
)δ−1 exp[−(

x

β
)δ] (x > 0), (7)

where θ = (β, σ),β > 0 and σ > 0 are the scale and shape parameters, respective-
ly. Then the conditional probability distribution function of Xl given Xk = xk
is

Pθ{Xl ≤ xl|Xk = xk} = 1− (m− k)!

(l − k − 1)!(m− l)!

l−k−1X
j=0

�
l − k − 1

j

�

(−1)j

m− l + 1 + j

h
exp

�
− xδl − xδk

βδ

�im−l+1+j
.

(8)

Theorem 2 If in (8) the scale parameter is unknown, then the predictive prob-
ability distribution function of Xl based on (xk, δ) is given by

Pδ

n�Xl

Xk

�δ
≤
� xl
xk

�δ
} = 1− m!

(l − k − 1)!(m− l)!
×
�

l − k − 1
j

�
(−1)j

m− l + 1 + j

�
Πk−1

s=0

h� xl
xk

�δ
− 1)(m− l + 1 + j) + (m− k + 1 + s)

i�−1

.

(9)

Proof.We reduce (8) to

Pθ

n�Xl

Xk

�δ
≤
�Xl

Xk

�δ
|
�Xk

β

�δ
=
�Xk

β

�δo

=1− (m− k)!

(l − k − 1)!(m− l)!

l−k−1X
j=0

�
l − k − 1

j

�
(−1)j

m− l + 1 + jh
exp(−ω[νδ − 1])

im−l+1+j

=Pδ{V δ ≤ νδ|W = ω},

(10)

where V = Xl/Xk is the ancillary statistic whose distribution does not depend
on the parameter β. Since Xk does not depend on V , W = (Xk/β)

δ is the pivotal
quantity, whose distribution is known and does not depend on the parameters β
and δ, we eliminate the parameter from the problem as

Pδ{Xl ≤ xl} =
Z ∞

0
Pθ{Xl ≤ xl|Xk = xk}gθ(xk)dxk, (11)

where

gθ(xk) =
m!

(k − 1)!(m− k)!
F k−1
θ (xk)

h
1− Fθ(xk)

im−k
fθ(xk), xk ∈ (0,∞), (12)



Advances in Systems Science and Applications (2012) Vol.12 No.4 377

represents the probability density function of the kth order statisticXk k. Indeed,
it follows from (12) that

gθ(xk)dxk =
m!

(k − 1)!(m− k)!

h
1− exp

�
−
�xk
β

�δ�ik−1
exp

�
−
�xk
β

�δ(m−k)
�

exp
�
−
�x
β

�δ�
d
�x
β

�δ
=

m!

(k − 1)!(m− k)!
[1− e−ω]k−1e−ω(m−k+1)dω = g(ω)dω.

(13)

It follows from (10) and (13) that

Pδ{V δ ≤ νδ} =
Z ∞

0
Pδ{V δ ≤ νδ|W = ω}g(ω)dω

=1− (m)!

(l − k − 1)!(m− l)!

l−k−1X
j=0

�
l − k − 1

j

�
(−1)j

m− l + 1 + j

�
Πk−1

s=0

�
(νδ

− 1)(m− l + 1 + j) + (m− k + 1 + s)
��−1

.

(14)

Now (9) follows from (14). This ends the proof.
Corollary 2.1.If the parameter δ = 1 , i.e. we deal with the exponential distri-
bution, then the predictive probability distribution function of Xl based on xk is
given by

P
n�Xl

Xk

�
≤
� xl
xk

�o
= 1− m!

(l − k − 1)!(m− l)!
×

l−k−1X
j=0

�
l − k − 1

j

�

(−1)j

m− l + 1 + j

�
Πk−1

s=0

h� xl
xk

− 1
�
(m− l + 1 + j) + (m− k + 1 + s)

i�−1
.

(15)

Theorem 3 Let X1 ≤ . . . ,≤ Xk be the first k ordered observations from a sam-
ple of size m from the two-parameter Weibull distribution (7). Then the joint
probability density function of the pivotal quantities

W2 =
δóδ , W3 =

� óβ
β

�δ̂
, (16)

conditional on fixed zk = (zi, . . . , zk),where Zi = (Xi/óβ)δ̂, i = 1, . . . , k,are
ancillary statistics, any k − 2 of which form a functionally independent set,óβ
and óδ are the estimators of β and δ,based on the first k ordered observations
(X1 ≤ . . . ≤ Xk) from a sample of size m from the two-parameter Weibull dis-
tribution (7), such that W2 and W3 are the pivotal quantities (in particular, the



378 Konstantin N. Nechval:Predictive Inferences for a Future Number of Failures Coming from...

maximum likelihood estimators of β and δ,

óβ =

�h kX
i=1

xδ̂i + (m− k)xδ̂k

i
/k

�1/δ̂

(17)

and

óδ =

�� kX
i=1

xδ̂i lnxi+(m−k)xδ̂k lnxk
�� kX

i=1

xδ̂i +(m−k)xδ̂k

�−1
− 1

k

kX
i=1

lnxi

�−1

(18)

respectively, lead to the pivotal quantities W2 and W3 )is given by

f(ω2, ω3|z(k))

=ϑ•(z(k))ωk−1
2

kY
i=1

zω2
i ωkω2−1

3 exp
�
− ωω2

3

� kX
i=1

zω2
i + (m− k)zω2

k

��

=ϑ•(z(k))ωk−2
2

kY
i=1

zω2
i ω

ω2(k−1)
3 exp

�
− ωω2

3

� kX
i=1

zω2
i + (m− k)zω2

k

��
ω2ω

ω2−1
3

=f(ω2|z(k))f(ω3|ω2, z
(k)), ω3 ∈ (0,∞),

(19)

where

ϑ•(z(k)) =
h Z ∞

0
Γ(k)ωk−2

2

kY
i=1

zω2
i

� kX
i=1

zω2
i + (m− k)zω2

k

�−k
dω2

i−1
(20)

is the normalizing constant,

f(ω2|z(k)) = ϑ(z(k))ωk−2
2

kY
i=1

zω2
i

� kX
i=1

zω2
i + (m− k)zω2

k

�−k
, ω2 ∈ (0,∞), (21)

ϑ(z(k)) =
h Z ∞

0
ωk−2
2

kY
i=1

zω2
i

� kX
i=1

zω2
i + (m− k)zω2

k

�−k
dω2

i−1
, (22)

f(ω3, ω2|z(k)) =

hPk
i=1 z

ω2
i + (m− k)zω2

k

ik
Γ(k)

ω
w2(k−1)
3

× exp
�
− ωw2

3

h
(

kX
i=1

zω2
i + (m− k)zω2

k

i�
ω2ω

ω2−1
3 , ω3 ∈ (0,∞). (23)

Proof.The joint density X1 ≤ . . . ≤ Xk is given by

fθ(x1, . . . , xk) =
m!

(m− k)!

kY
i=1

δ

β
(
xi
β
)δ−1 exp(−(

xi
β
)δ) exp(−(m− k)((

xk
β
)δ). (24)



Advances in Systems Science and Applications (2012) Vol.12 No.4 379

Using óβ and óδ (the maximum likelihood estimators of β and δ obtained from solu-
tion of (17) and (18)) and the invariant embedding technique [8-14], we transform
(24) as follows:

fθ(x1, . . . , xk)dóβdóδ
=

m!

(m− k)!

kY
i=1

x−1
i δk

kY
i=1

�xi
β

�δ
exp

�
−

kX
i=1

�xi
β

�δ − (m− k)
�xk
β

�δ�
dóβdóδ

=− m!

(m− k)!
óβóδk kY

i=1

x−1
i

�δóδ
�k−2

kY
i=1

�xióβ
�δ̂( δ

δ̂
)� óβ
β

�δ̂( δ
δ̂
)(k−1) × exp

�
−
� óβ
β

�δ̂( δ
δ̂
)

� kX
i=1

�xióβ
�δ̂( δ

δ̂
)
+ (m− k)

�xkóβ
�δ̂( δ

δ̂
)���óδ( δóδ )

β

� óβ
β

�δ̂( δ
δ̂
)−1

dóβ��− δóδ2dóδ
�

=− m!

(m− k)!
óβóδk kY

i=1

x−1
i ωk−2

2

kY
i=1

zω2
i ω

ω2(r−1)
3 exp

�
− ωω2

3

h kX
i=1

zω2
i +

(m− k)zω2
k

i�
d(ωω2

3 )dω2

=− m!

(m− k)!
óβóδk kY

i=1

x−1
i ωk−2

2

kY
i=1

zω2
i ω

ω2(k−1)
3 exp

�
− ωω2

3

h kX
i=1

zω2
i

+ (m− k)zω2
k

i�
ω2ω

ω2−1
3 dω2dω3.

(25)

Normalizing (25), we obtain (19). This ends the proof.
It will be noted that more general case of distributions indexed by location

and scale parameters has been considered in [15].

Theorem 4 If in (8) both parameters β and δ are unknown, then the predictive
probability distribution function of Xl based on (xk, óδ) and conditional on fixed
z(k) is given by

P
n�Xl

Xk

�δ̂ ≤ � xl
xk

�δ̂|z(k)o

=1− m!

(l − k − 1)!(m− l)!
×
Z ∞

0

l−k−1X
j=0

�
l − k − 1

j

�
(−1)j

m− l + 1 + j� k−1Y
s=0

h��� xl
xk

�δ̂�ω2 − 1
�
(m− l + 1 + j) + (m− k + 1 + s)

i�−1

f
�
ω2|z(k)

�
dω2.

(26)



380 Konstantin N. Nechval:Predictive Inferences for a Future Number of Failures Coming from...

Proof. We reduce (9) to

Pδ

n�Xl

Xk

�δ̂( δ
δ̂
) ≤

� xl
xk

�δ̂( δ
δ̂
)
o
= 1− m!

(l − k − 1)!(m− l)!
×

l−k−1X
j=0

�
l − k − 1

j

�

(−1)j

m− l + 1 + j

� k−1Y
s=0

h�� xl
xk

�δ̂( δ
δ̂
) − 1

�
(m− l + 1 + j) + (m− k + 1 + s)

i�−1

=1− m!

(l − k − 1)!(m− l)!
×

l−k−1X
j=0

�
l − k − 1

j

�
(−1)j

m− l + 1 + j

� k−1Y
s=0

h
(νω2

2 − 1)(m− l + 1 + j) + (m− k + 1 + s)
i�−1

=P
¦
V W2
2 ≤ νω2

2

©
,

(27)

where V2 = (Xl/Xk)
δ̂ is the ancillary statistic whose distribution does not depend

on the parameters β and δ. Since the pivotal quantity W2, whose distribution is
given by (21), does not depend on V2, it follows from (21) and (27) that

P
¦
V2 ≤ ν2|z(k)

©
=
Z ∞

0
P
¦
V W2
2 ≤ νω2

2

©
f
�
ω2|z(k)

�
dω2, (28)

where the unknown parameters β and δ are eliminated from the problem. Now
(26) follows from (28). This ends the proof.

3 Prediction Limits for a Future Number of Failures

Consider the situation in which m units start service at time 0 and are observed
until a time tc when the available Weibull failure data are to be analyzed. Failure
times are recorded for the k units that fail in the interval [0, tc]. Then the data
consist of the k smallest-order statistics X1 ≤ . . . ≤ Xk ≤ tc and the information
that the other m?k units will have failed after tc.With time (or Type I) censored
data, tc is prescribed and k is random. With failure (or Type II) censored data,
k is prescribed and tc = Xk is random.

The problem of interest is to use the information obtained up to tc to construct
the Weibull within-sample prediction limits (lower and upper) for the number of
units that will fail in the time interval [tc, tω].For example, this tω could be the
end of a warranty period.

Consider the situation when tc = Xk. Under conditions of Theorem 4, the
lower prediction limit for the number of units that will fail in the time interval
[tc, tω] is given by

Llower = lmax − k, (29)



Advances in Systems Science and Applications (2012) Vol.12 No.4 381

where

lmax = max
k<l≤m

argP
�¦

Xl > tω|zk
©
≤ α

�
(30)

The upper prediction limit for the number of units that will fail in the time
interval [tc, tω] is given by

Lupper = lmin − k − 1, (31)

where

lmin = min
k<l≤m

argP
�¦

Xl > tω|zk
©
≥ 1− α

�
(32)

In the above case, where both parameters β and δ are unknown, the prediction
limits (lower and upper) for the number of units that will fail in the time interval
[tc, tω] are based on (xk, óδ) and conditional on fixed z(k). If l, which satisfies
(30), does not exist then lmax = k and the lower prediction limit for the number
of units that will fail in the time interval [tc, tω] is given by

Llower = lmax − k = 0. (33)

If l, which satisfies (32), does not exist then lmin = m+1 and upper prediction
limit for the number of units that will fail in the time interval [tc, tω] is given by

Lupper = lmin − k − 1 = m− k, (34)

4 Second Version of Prediction Limits for a Future Number of Failures

In this section, we wish to show how to obtain the second version of prediction
limits for a future number of failures. The methodology is based on the following
results.

Theorem 5 Let X1 ≤ . . . ≤ Xk be the first k ordered observations from a sam-
ple of size m from the two-parameter Weibull distribution (7). Then the joint
probability density function of the pivotal quantities

W1 =
� óβ
β

�δ
, W3 =

δóδ , (35)

conditional on fixed z(k) = (zi, . . . , zk), where Zi = (Xi/óβ)δ̂, i = 1, . . . , k are
ancillary statistics, any k−2 of which form a functionally independent set,óβ andóδ are, for instance, the maximum likelihood estimators for β and δ based on the
first k ordered observations (X1 ≤ . . . ≤ Xk) from a sample of size m from the



382 Konstantin N. Nechval:Predictive Inferences for a Future Number of Failures Coming from...

two-parameter Weibull distribution (7), which can be found from solution of (17)
and (18), is given by

f(ω1, ω2|z(k)) = ϑ•(z(k))ωk−2
2

kY
i=1

zω2
i ωk−1

1 exp(−ω1[
kX

i=1

zω2
i + (m− k)zω2

k ])

= f(ω2|z(k))f(ω1|ω2, z
(k)), ω1 ∈ (0,∞), ω2 ∈ (0,∞),

(36)

where

ϑ•(z(k)) =
� Z ∞

0
Γ(k)ωk−2

2

kY
i=1

zω2
i

� kX
i=1

zω2
i + (m− k)zω2

k

�−k
dω2

�−1
(37)

is the normalizing constant, f(ω2|z(k)) is given by (21),

f(ω1, ω2|z(k)) =

�Pk
i=1 z

ω2
i + (m− k)zω2

k

�k
Γ(k)

ωk−1
1 exp

�
− ω1

� kX
i=1

zω2
i

+ (m− k)zω2
k )

��
ω1 ∈ (0,∞),

(38)

Proof. The joint density of X1 ≤ . . . ≤ Xk is given by

fθ(x1, . . . , xk) =
m!

(m− k)!

kY
i=1

δ

β
(
xi
β
)δ−1 exp(−(

xi
β
)δ) exp(−(m− k)(

xk
β
)δ). (39)

Using the invariant embedding technique [8-14], we transform (39) to

fθ(x1, . . . , xk)dóβdóδ
=

m!

(m− k)!

kY
i=1

x−1
i δk

kY
i=1

�xi
β

�δ
exp

�
−

kX
i=1

�xi
β

�δ − (m− k)
�xk
β

�δ�
dóβdóδ

=− m!

(m− k)!
óβóδk kY

i=1

x−1
i

�δóδ
�k−2

kY
i=1

�xióδ
�δ̂( δ

δ̂
)� óβ
β

�δ(k−1) × exp
�
−
� óβ
β

�δ
h kX
i=1

(
xióδ )

δ̂( δ
δ̂
)
+ (m− k)

�xkóδ
�δ̂( δ

δ̂
)
i�� δ

β

� óβ
β

�δ(k−1)
dóβ�(− δóδ2 )dóδ2

=− m!

(m− k)!
óβóδk kY

i=1

x−1
i ωk−2

2

kY
i=1

x−1
i zω2

i ωk−1
1 exp

�
− ω1

h kX
i=1

zω2
i +

(m− k)zω2
k

i�
dω1ω2.

(40)

Normalizing (40), we obtain (36). This ends the proof.
Corollary 5.1. If the parameter δ is known then

W1 ∼ f(ω1) =
kk

Γ(k)
ωk−1
1 exp(−ω1k), ω1 ∈ (0,∞). (41)



Advances in Systems Science and Applications (2012) Vol.12 No.4 383

Theorem 6 If in (8) the scale parameter β is unknown, then the predictive prob-
ability distribution function of Xl based on (óβ, δ) and conditional on fixed xk is
given by

Pδ{Xl ≤ xl|Xk = xk} =1− (m− k)!

(l − k − 1)!(m− l)!

l−k−1X
j=0

�
l − k − 1

j

�

(−1)j

m− l + 1 + j

h
1− (m− l + 1 + j)

xδl − xδk
k óβδ

�−k

(42)

Proof.We reduce (8) to

Pθ{Xl ≤ xl|Xk = xk}

=1− (m− k)!

(l − k − 1)!(m− l)!

l−k−1X
j=0

�
l − k − 1

j

�
(−1)j

m− l + 1 + j

h
exp

�
−
� óβ
β

�δ xδl − xδkóβδ

�im−l+1+j

=1− (m− k)!

(l − k − 1)!(m− l)!

l−k−1X
j=0

�
l − k − 1

j

�
(−1)j

m− l + 1 + j

h
exp

�
− ω1

xδl − xδkóβδ

�im−l+1+j
.

(43)

Now, we eliminate the unknown parameter β from the problem and find (42) as

Pδ{Xl ≤ xl|Xk = xk} =
Z ∞

0
Pθ{Xl ≤ xl|Xk = xk}f(ω1)dω1. (44)

This ends the proof.
Corollary 6.1.If the parameter δ = 1, i.e. we deal with the exponential dis-
tribution, then the predictive probability distribution function of Xl based on óβ
and conditional on fixed xk is given by xk

P{Xl ≤ xl|Xk = xk} = 1− 1

B(l − k,m− l + 1)

l−k−1X
j=0

�
l − k − 1

j

�
(−1)j

m− l + 1 + j

h
1 + (m− l + 1 + j)

xl − xk

k óβ
i−k

, (45)

where

k óβ =
kX

i=1

xi + (m− k)xk. (46)

Theorem 7 If in (8) both parameters β and δ are unknown, then the predictive
probability distribution function of Xl based on (wideparenβ,wideparenδ) and



384 Konstantin N. Nechval:Predictive Inferences for a Future Number of Failures Coming from...

conditional on fixed xk and z(k) is given by

Pθ{Xl ≤ xl|Xk = xk; z
(k)} = 1− (m− k)!

(l − k − 1)!(m− l)!
×
Z ∞

0

l−k−1X
j=0

�
l − k − 1

j

�

(−1)j

m− l + 1 + j

h
1 + (m− l + 1 + j)

��xlóβ
�δ̂ω2 −

�xkóβ
�δ̂ω2

�� kX
i=1

zω2
i + (m− k)

zω2
i

�−1i−k
× f(ω2|z(k))dω2.

(47)

Proof. We reduce (8) to

Pθ{Xl ≤ xl|Xk

=xk} = 1− (m− k)!

(l − k − 1)!(m− l)!

l−k−1X
j=0

�
l − k − 1

j

�

(−1)j

m− l + 1 + j

�
exp

�
−
� óβ
β

�δh�xlóβ
�δ̂( δ

δ̄
) −

�xkóβ
�δ̂( δ

δ̄
)
i��m−l+1+j

=1− (m− k)!

(l − k − 1)!(m− l)!

l−k−1X
j=0

�
l − k − 1

j

�
(−1)j

m− l + 1 + j

exp
h�

− ω1

��xlóβ
�δ̂ω2 −

�xkóβ
�δ̂ω2

��im−l+1+j
.

(48)

Now, we eliminate the unknown parameters β and δ from the problem and find
(47) as

Pθ{Xl ≤ xl|Xk = xk; z
(k)}

=
Z ∞

0

Z ∞

0
Pθ{Xl ≤ xl|Xk = xk}f(ω1, ω2|z(k))dω1dω2

=
Z ∞

0

Z ∞

0
Pθ{Xl ≤ xl|Xk = xk}f(ω1|ω2, z

(k))f(ω2|z(k))dω1dω2.

(49)

This ends the proof.
Under conditions of Theorem 7, the lower prediction limit for the number of

units that will fail in the time interval [tc, tω] is given by

Llower = lmax − k, (50)

where

lmax = max
k<l≤m

arg
�
P{Xl > tω|Xk = xk; z

(k))} ≤ α
�
, (51)



Advances in Systems Science and Applications (2012) Vol.12 No.4 385

The upper prediction limit for the number of units that will fail in the time
interval [tc, tω] is given by

Lupper = lmin − k − 1, (52)

lmin = min
k<l≤m

arg
�
P{Xl > tω|Xk = xk; z

(k))} ≥ 1− α
�
, (53)

In the above case, when both parameters β and δ are unknown, the prediction
limits (lower and upper) for the number of units that will fail in the time interval
[tc, tω] are based on (óβ, óδ) and conditional on fixed xk,z

(k).
If l, which satisfies (51), does not exist then lmax = k and the lower prediction

limit for the number of units that will fail in the time interval [tc, tω] is given by
Llower = 0. If l, which satisfies (53), does not exist then lmin = m+ 1 and upper
prediction limit for the number of units that will fail in the time interval [tc, tω]
is given by Lupper = m− k.

5 Numerical Example

For the sake of simplicity, but without loss of generality, we consider (for illus-
tration) the special case of Theorem 2 where m = 40 items simultaneously tested
have life times, which follow the Weibull distribution with δ = 1 . In other words,
we deal with the exponential distribution. Two items have failed by the inspec-
tion at times, X1 = 45 and X2 = 100 hours. Let us assume that the situation
takes place when tc = Xk = 100 hours, where k = 2. Suppose, say, tω = 450
hours. Taking into account (15), we find the lower prediction limit for the number
of units that will fail in the time interval [tc, tω] as

Llower = lmax − k = 3− 2 = 1, (54)

lmax = max
k<l≤m

arg
�
P{Xl > tω} ≤ α

�
= 3, α = 0.05 (55)

P{Xl > tω} =
m!

(l − k − 1)!(m− l)!

l−k−1X
j=0

�
l − k − 1

j

�
(−1)j

m− l + 1 + j� k−1Y
s=0

h� tω
xk

− 1
�
(m− l + 1 + j) + (m− k + 1 + s)

i�−1

,

(56)

The upper prediction limit for the number of units that will fail in the time
interval [tc, tω] is given by

Lupper = lmin − k − 1 = 17− 2− 1 = 14, (57)

lmin = min
k<l≤m

arg
�
P{Xl > tω|Xk = xk; z

(k))} ≥ 1− α
�
= 17. (58)



386 Konstantin N. Nechval:Predictive Inferences for a Future Number of Failures Coming from...

It will be noted that when both parameters β and δ are unknown, the lower
and upper prediction limits for the number of units that will fail in the time inter-
val [tc, tω] can be found either from (29) and (31), which are based on (xk, óδ),or
from (50) and (52), which are based on (óβ, óδ).
Conclusion and Future Work

The methodology described here can be extended in several different directions
to handle various problems that arise in practice.

We have illustrated the prediction method for log-location-scale distributions
(such as the Weibull or exponential distributions). Application to other distribu-
tions could follow directly.

Acknowledgements
This research was supported in part by Grant No. 06.1936, Grant No. 07.2036,
Grant No. 09.1014, and Grant No. 09.1544 from the Latvian Council of Science
and the National Institute of Mathematics and Informatics of Latvia.

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Corresponding author
Konstantin N. Nechval can be contacted at:e-mail: konstan@tsi.lv