Abstract—Prediction intervals for future order statistics are widely used for reliability problems and other related problems. The determination of these intervals has been extensively investigated. But the optimality property of these intervals has not been fully explored. In this paper we discuss this problem for extreme value distributions. Introducing a risk function to compare prediction intervals, the interval which minimizes it among the class of invariant prediction intervals is obtained. The technique used here for optimization of prediction intervals based on censored data emphasizes pivotal quantities relevant for obtaining ancillary statistics and factors. It allows one to solve the optimization problems in a simple way.
Index Terms — Extreme value distribution, future order statistic, prediction interval, risk function, optimization
I. INTRODUCTION
REDICTION of an unobserved random variable is a fundamental problem in statistics. Patel [1] provides an extensive survey of literature on this topic. In the areas of reliability and life-testing, this problem translates to obtaining prediction intervals for life distributions such as the Exponential and the Weibull. One of the earlier works on prediction for the Weibull distribution is by Mann and Saunders [2]. They considered prediction intervals for the smallest of a set of future observations, based on a small (two or three) preliminary sample of past observations. An expression for the warranty period (time before the failure of the first ordered observation from a set of future observations or a lot) was derived as a function of the ordered past observations. Mann [3] extended the results for lot sizes n = 10 (5) 25 and sample sizes m = 2 (1) n − 3 for a specified assurance level of 0.95. This method requires numerical integration. In addition, the tables provided are limited to sample sizes less than 25 and are given only for the assurance level of 0.95. Antle and Rademaker [4] Manuscript received March 06, 2012. This work 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.
Nicholas A. Nechval is with the Statistics Department, EVF Research Institute, University of Latvia, Riga LV-1050, Latvia (e-mail: [email protected]).
Maris Purgailis is with the Cybernetics Department, University of Latvia, Riga LV-1050, Latvia (e-mail: [email protected]).
Konstantin N. Nechval is with the Applied Mathematics Department, Transport and Telecommunication Instutute, Riga LV-1019, Latvia (e-mail: [email protected]).
Inta Bruna is with the Cybernetics Department, University of Latvia, Riga LV-1050, Latvia (e-mail: [email protected]).
provided a method of obtaining a prediction bound for the largest observation from a future sample of the Type I extreme value distribution, based on the maximum likelihood estimates of the parameters. They used Monte Carlo simulations to obtain the prediction intervals. Using the well-known relationship between the Weibull distribution and the Type I extreme value distribution one can use their method to construct an upper prediction limit for the largest among a set of future Weibull observations. However this method is valid only for complete samples and limited to constructing an upper prediction limit for the largest among a set of future observations. Lawless [5] proposed a method for constructing prediction intervals for the smallest ordered observation among a set of k future observations based on a Type II censored sample of past observations. These results are based on the conditional distribution of the maximum likelihood estimates given a set of ancillary statistics. This procedure is exact, but it requires numerical integration, for each new sample obtained, to determine the prediction bounds. Mee and Kushary [6] provided a simulation based procedure for constructing prediction intervals for Weibull populations for Type II censored case. This procedure is based on maximum likelihood estimation and requires an iterative process to determine the percentile points.
To develop appropriate probabilistic models and assess the risks caused by stochastic events, business analysts and engineers frequently use the extreme value distributions (EVD). In this paper we assume that the parent EVD are the Gumbel distribution,
−
− =
>x xσµ
X } exp exp
Pr{ , −∞ < x < ∞, (1)
where µ is the location parameter, and σ is the scale
parameter (σ > 0), and the Weibull distribution,
− = >
δ
β y y
Y } exp
Pr{ , y≥ 0, (2)
where both distribution parameters (δ− shape, β− scale) are
positive.
Let Y be a random variable with the Weibull distribution (2), and define X = lnY. Then X becomes a random variable with the Gumbel distribution (1) where µ=lnβ and σ= δ−1.
Therefore it is enough to consider only the Gumbel distribution, because the results for the Weibull distribution are easily obtained from those for the Gumbel distribution.
Optimal Prediction Intervals for Future Order
Statistics from Extreme Value Distributions
Nicholas A. Nechval,
Member, IAENG
, Maris Purgailis, Konstantin N. Nechval, and Inta Bruna
II. WITHIN −SAMPLE PREDICTION
A. Mathematical Preliminaries
Theorem 1. Let X1 ≤ ... ≤ Xk be the first k ordered
observations (order statistics) in a sample of size m from a continuous distribution with some probability density function fθ(x) and distribution function Fθ(x), where θ is a parameter (in general, vector). Then the joint probability density function of X1≤ ... ≤Xk and the lth order statistics Xl
(1 ≤k < l ≤m) is given by
), | ( ) ..., , ( ) , ..., ,
(x1 xk xl f x1 xk f xl xk
fθ = θ θ (3)
where
) ,..., (x1 xk
fθ ( )[1 ( )] ,
)! ( ! 1
∏
= − − − = k i k m ki F x
x f k m m θ θ (4) 1 ) ( 1 ) ( ) ( )! ( )! 1 ( )! ( ) | ( − − − − − − − − = k l k k l k l x F x F x F l m k l k m x x f θ θ θ θ ) ( 1 ) ( ) ( 1 ) ( ) ( 1 k l l m k k l x F x f x F x F x F θ θ θ θ θ − − − − × −
∑
− − = − − − − − − − = 1 0 ) 1 ( 1 )! ( )! 1 ( )!( l k
j j j k l l m k l k m ) ( 1 ) ( ) ( 1 ) ( 1 k l j l m k l x F x f x F x F θ θ θ θ − − − × + −
∑
− = − − − − − −= m l
j j j l m l m k l k m 0 ) 1 ( )! ( )! 1 ( )! (
1 ( )
) ( ) ( 1 ) ( ) ( 1 k l j k l k k l x F x f x F x F x F θ θ θ θ θ − − − × + − − (5)
represents the conditional probability density function of Xl
given Xk=xk.
Proof. The joint density of X1≤ ... ≤Xk and Xlis given by
) , ..., , (x1 xk xl
fθ
∏
= − − − = k i i x f l m k l m 1 ) ( )! ( )! 1 ( ! θ l m l l k l k
l F x f x F x
x
F − − − − −
×[ ( ) ( )] 1 ( )[1 ( )]
θ θ
θ θ
= fθ(x1 ,...,xk)fθ(xl|xk). (6)
It follows from (4) and (6) that
( ,..., ) ( | ), ) , ..., , ( ) ..., , | ( 1 1
1 l k
k l k k
l f x x
x x f x x x f x x x f θ θ θ
θ = =
(7) i.e., the conditional distribution of Xl, 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 (5). This ends the
proof.
Theorem 2. Let X1 ≤ ... ≤ Xk be the first k ordered
observations from a sample of size m, which follow the Gumbel distribution (1) with the density
( ) 1exp exp exp (−∞< <∞), − − −
= x x x
x f σµ σµ σ θ (8) where θ = (µ,σ). Then the joint probability density function of the pivotal quantities
1=µ −σµ,
)
S 2 ,
σ σ) =
V (9)
conditional on fixed
) ..., , ( 1 ) ( k k z z =
z , (10)
where , ..., , 1
, i k
X
Zi i =
− =
σ) µ
)
(11) are ancillary statistics, any k−2 of which form a functionally independent set,µ) and σ)are the maximum likelihood estimates for µ and σ based on the first k ordered observations (X1≤ ... ≤Xk) from a sample of size m from the
Gumbel distribution (1), which can be found from solution of
ln ( ) ,
1 / / − + =
∑
= k e k m e k i xxi σ k σ
σ
µ) ) ) ) (12)
and − + =
∑
= σ σσ) ) /)
1
/ ( ) k
i x
k k
i x
ie m k x e
x , 1 ) ( 1 1 1 / /
∑
∑
= − = − − + × k i i k i x x x k e k mei σ) k σ) (13)
is given by
=
∑
= − • k i i k k k z v v v s f 1 2 2 2 ) ( ) ( 21, | ) ( ) exp
( z ϑ z
− + − ×
∑
= ) exp( ) ( ) exp( exp 2 1 2 11 e zv m k z v
e k k i i s ks ), | ( ) |
(v2 (k) f s1 v2 (k)
f z ,z
= s1∈(−∞, ∞), v2∈(0, ∞), (14)
where Γ = ∞
∫
∑
= − •0 2 1
2 2 )
( ) ( ) exp
( k i i k k z v v k z ϑ 1 2
1 2 2
) exp( ) ( ) exp( − − = − +
×
∑
zv m k z v dvk k
i
k
i (15)
=
∑
= − k i i k kk v v z
v f 1 2 2 2 ) ( ) (
2| ) ( ) exp
( z ϑ z
exp( ) ( )exp( ) ,
1 2 2
k k
i
k
iv m k z v
z − = − +
×
∑
v2∈(0,∞), (16) = ∞
∫
∑
= − 0 1 2 2 2 )( ) exp
( k i i k k z v v z ϑ
exp( ) ( )exp( ) ,
1
2
1 2 2
− − = − +
×
∑
zv m k z v dvk k
i
r
i (17)
k k i k i k v z k m v z k v s f − + Γ =
∑
=1 2 2
) ( 2
1| , ) (1) exp( ) ( )exp( )
( z , ) exp( ) ( ) exp( exp 2 1 2 1 1 − + − ×
∑
= v z k m v z e e r k i i s kss1∈(−∞, ∞). (18)
Proof. The joint density of X1≤ ... ≤Xk is given by
) , | ..., ,
(x1 xk µσ
f
∏
= − − − − = k i i i x x k m m 1 exp exp 1 )! ( ! σ µ σ µ σ . exp ) ( exp − − − × σ µ k x km (19)
Using the invariant embedding technique [7-14], we then find in a straightforward manner, that the probability element of the joint density of S1, V2, conditional on fixed
z(k)= (z1 ,...,zk), is
2 1 ) ( 2
1, | )
(s v dsdv
f zk
1 1 2 2 2 )
( ) exp
( k ks
i i k k e z v
v
=
∑
= − • z ϑ , ) exp( ) ( ) exp(exp 2 1 2
1 2
1 zv m k z v dsdv
e k k i i s − + − ×
∑
=s1∈(−∞, ∞), v2∈(0, ∞). (20) This ends the proof.
Theorem 3. Let X1 ≤ ... ≤ Xk be the first k ordered
observations (order statistics) in a sample of size m from the Gumbel distribution (1). Then the joint probability density function of the pivotal quantities
,
1 XlσXk
V = − 2 ,
σ µ − = Xk
S (21)
where Xl(1 ≤ k < l ≤ m)is the lth order statistic from the
same sample, is given by
f−(v1,s2)= f(v1|s2)f(s2), (22)
where 1 1 ) 1 ( 1 ) 1 , ( 1 ) | ( 1 0 2
1 − − + +
− − + − − Β =
∑
−−= j m l j
k l l m k l s v f k l j j , ) 1
( 2 1
1
2( 1)
) 1
(ml j e e s v
e e j l m
e s v − + +
× − −+ + − 0 < v
1 < ∞, (23)
, ) 1 ( 1 ) 1 , ( 1 )
( 1 ( 1) 2 2
0
2 m k j e s
k
j
je e
j k k m k s
f − − + + s
− =
∑
− − + − Β =−∞ < s2 < ∞. (24) Proof.The joint density function of the order statistics Xk,
Xl (1≤ k < l ≤ m) is given by
). ( ) | ( ) ,
(xk xl f xl xk f xk
fθ+ = θ+ θ+ (25)
It will be noted that
∑
− − = + − − − − − − − = 1 0 ) 1 ( 1 )! ( )! 1 ( )! ( ) |( l k
j j l k l j k l l m k l k m dx x x fθ l k l j l m k l dx x F x f x F x F ) ( 1 ) ( ) ( 1 ) ( 1 θ θ θ θ − − − × + − 1 1 ) 1 ( 1 ) 1 , ( 1 1
0 − + +
− − − + − − Β =
∑
− −= j m l j
k l l m k l k l j j 1 ) 1 ( ) 1
( 2 1 (m l j 1)e2e1dv
e ml j es ev − + + s v
× − −+ + −
≡ f(v1|s2)dv1, 0 < v1 < ∞, (26)
)! ( )! 1 ( ! ) ( k m k m dx x
f k k
− − = + θ k k k m k k
k F x f x dx
x
F( )] [1 ( )] ( )
[ 1 θ − − − × 2 ) 1 ( 1 0 2 2 ) 1 ( 1 ) 1 , (
1 e e ds
j k k m k s e j k m k j
j − −+ + s
− =
∑
− − + − Β = , ) (s2 ds2 f≡ −∞ < s2 < ∞. (27)
This ends the proof.
Corollary 3.1. The probability density function of the pivotal quantity V1 is given by
( 1)
∫
( 1, 2) 2 ∞∫
( 1| 2) ( 2) 2.∞ − ∞ ∞ − −
− v = f v s ds = f v s f s ds
f (28)
V1= Xlσ−Xk, σσ
)
=
2
V (29)
is given by
f(v1,v2)= f−(v1)f(v2|z(k)). (30)
B. Piecewise-Linear Loss Function
We shall consider the interval prediction problem for the lth order statistic Xl, k<l≤m, in the same sample of size m for
the situation where the first k observations X1 < X2 < ⋅⋅⋅< Xk,
1≤k<m, have been observed. Suppose that we assert that an interval d=(d1,d2) contains Xl. If, as is usually the case, the
purpose of this interval statement is to convey useful information we incur penalties if d1 lies above Xl or if d2
falls below Xl. Suppose that these penalties are c1(d1− Xl)
and c2(Xl−d2), losses proportional to the amounts by which
Xl escapes the interval. Since c1 and c2 may be different the
possibility of differential losses associated with the interval overshooting and undershooting the true µ is allowed. In addition to these losses there will be a cost attaching to the length of interval used. For example, it will be more difficult and more expensive to design or plan when the interval
d=(d1,d2) is wide. Suppose that the cost associated with the interval is proportional to its length, say c(d2−d1). In the specification of the loss function, σ is clearly a ‘nuisance parameter’ and no alteration to the basic decision problem is caused by multiplying all loss factors by 1/σ. Thus we are led to investigate the piecewise-linear loss function
> −
+ −
≤ ≤ −
< −
+ −
=
). (
) (
) (
), (
) (
), (
) ( ) (
) , (
2 2
2 1 2
2 1
1 2
1 1
2 1
1
d X d
X c d d c
d X d d
d c
d X d
d c X d c
r
l l
l l l
σ σ
σ
σ σ
d
θθθθ (31)
The decision problem specified by the informative experiment density function (1) and the loss function (31) is invariant under the group of transformations, which takes µ (the location parameter) and σ (the scale) into cµ + b and cσ, respectively, where b lies in the range of µ, c > 0. This group acts transitively on the parameter space. Thus, the problem is to find the best invariant interval predictor of Xl,
), , ( min
arg d
d
d∈DR θθθθ
∗= (32)
where D is a set of invariant interval predictors of Xl,
R(θθθθ,d)=Eθθθθ{r(θθθθ,d)} is a risk function.
C. Transformation of the Loss Function
It follows from (31) that the invariant loss function, r(θθθθ,d), can be transformed as follows:
), , ( ) ,
(θθθθd r V ηηηη
r =&& (33)
where
> −
+ −
≤ ≤ −
< +
+ − =
), (
)
( ) (
), (
)
(
), (
)
-( ) (
) , (
2 2 1 2
1 2 2 2 1 2
2 2 1 2 1 2
1 2
2 1 1 2
1 2 2 1 1 1
V V V
c V V c
V V V V
c
V V V
c V V c r
η η
η η
η η
η η
η η
η η
ηηηη
V
& &
(34)
V=(V1,V2), V1=(Xl−Xk)/σ , V2=σ)/σ;
ηηηη=(η1,η2), η1=(d1−Xk)/σ), η2=(d2−Xk)/σ). (35)
D. Risk Function
It follows from (34) that the risk associated with d and θθθθ
can be expressed as
{
( , )} {
( , )}
),
(θθθθd Eθθθθ r θθθθd Er V ηηηη
R = = &&
∫ ∫
∞
+ − =
0 0
2 1 2 1 2 1 1 1
2 1
) , ( ) (
v
dv dv v v f v v c
η
η
∫ ∫
∞ ∞
− +
0
2 1 2 1 2 2 1 2
2 2
) , ( ) (
v
dv dv v v f v v c
η
η
( ) ( , ) ,
0 0
2 1 2 1 2 1
2
∫ ∫
∞ ∞
−
+cη η v f v v dvdv (36)
which is constant on orbits when an invariant predictor (decision rule) d is used, where f(v1,v2) is defined by (30).
E. Risk Minimization and Invariant Prediction Rules The following theorem gives the central result in this section.
Theorem 4. Suppose that (U1,U2) is a random vector having density function
( , ) ( , ) ( 1 , 2 0),
1
0 0
2 1 2 1 2 2 1
2 >
∞ ∞ −
∫∫
u f u u dudu u uu u f
u (37)
where f is defined by f(v1,v2), and let Q be the probability distribution function of U1/U2.
(i) If c/c1+c/c2<1 then the optimal invariant linear-loss interval predictor of Xl based on X is d*=(Xk+η1Sk, Xk+η2Sk),
where
Q(η1)=c/c1, Q(η2)=1−c/c2. (38) (ii) If c/c1+c/c2≥1 then the optimal invariant linear-loss interval predictor of Xl based on X degenerates into a point
predictor Xk+η•Sk, where
). /( )
( c2 c1 c2
Qη• = + (39)
Proof. From (36)
{
}
1 ) , (
η ∂
∂E r&&V ηηηη
∫ ∫
∫ ∫
∞ ∞∞
− =
0 0
2 1 2 1 2 0 0
2 1 2 1 2
1 ( , ) ( , )
2 1
dv dv v v f v c dv dv v v f v c
v
η
( , ) [ 1 ( 1) ],
0 0
2 1 2 1
2f v v dvdv cQ c
v −
=∞ ∞
∫ ∫
η (40)and
{
( , )}
( , ) [ (1 ( )) ,]0
2 2
0
2 1 2 1 2
2
∫ ∫
∞ ∞
+ −
− =
∂ ∂
c Q
c dv dv v v f v r
E η
η ηηηη
V
& &
where
=
∫
η
η
0
, ) ( )
( qw dw
Q (42)
, ) , (
) , ( )
(
0 0
2 1 2 1 2 0
2 2 2 2 2
∫ ∫
∫
∞ ∞ ∞
=
dv dv v v f v
dv v wv f v
w
q (43)
. / 2
1 V
V
W = (44)
Now ∂E{r&&(V,ηηηη)}/∂η1 = ∂E{r&&(V,ηηηη)}/∂η2 = 0if and only
if (38) hold. Thus, E{r&&(V,ηηηη)} provided (38) has a solution
with η1<η2 and this is so if 1−c/c2>c/c1. It is easily
confirmed that this ηηηη=(η1,η2) gives the minimum value of
E{r&&(V,ηηηη)}. Thus (i) is established.
If c/c1+c/c2≥1 then the minimum of E{r&&(v,η)} in the
region η2≥η1 occurs where η1=η2=η•, η• being determined
by setting
∂E{r&&(V,(η•,η•))}/∂η•=0 (45)
and this reduces to
, 0 )] ( 1 [ )
( 2
1Qη• −c −Qη• =
c (46)
which establishes (ii).
Corollary 4.1. The minimum risk of the optimal invariant predictor of Xl based on X is given by
{
( , )}
{
( , )}
) ,
(θθθθd Eθθθθ r θθθθd E r V ηηηη
R ∗ = ∗ = &&
∫ ∫
∫ ∫
∞ ∞∞
+ −
=
0
2 1 2 1 1 2 0 0
2 1 2 1 1 1
2 2 2
1
) , ( )
, (
v v
dv dv v v f v c dv dv v v f v c
η η
(47) for case (i) with ηηηη=(η1,η2) as given by (38) and for case (ii)
with η1=η2=η• as given by (39).
Proof. These results are immediate from (36) when use is made of ∂E{r&&(V,ηηηη)}/∂η1 = ∂E{r&&(V,ηηηη)}/∂η2 = 0 in case (i)
and ∂E{r&&(V,(η•,η•))}/∂η•=0 in case (ii).
The underlying reason why c/c1+c/c2 acts as a separator of interval and point prediction is that for c/c1+c/c2≥1 every interval predictor is inadmissible, there existing some point predictor with uniformly smaller risk.
Theorem 5. Suppose that µ=0 and
V=(V1,V2), V1=(Xl−Xk)/σ , V2=Xk/σ;
), , ( 1o 2o o = η η
ηηηη η1o=(d1−Xk)/Xk, η2o =(d2−Xk)/Xk.
(48) Let us assume that (U1,U2) is a random vector having density function
( , ) ( , ) ( 1 , 2 0),
1
0 0
2 1 2 1 0 2 2 1 0
2 >
∞ ∞ −
∫ ∫
u f u u dudu u uu u f
u (49)
where f0 is defined by f0(v1,v2), and let Q0 be the probability distribution function of u1/u2.
(i) If c/c1+c/c2<1 then the optimal invariant linear-loss interval predictor of Xl based on Xk is d*=((1+η1o)Xk,
(1+ o
2
η )Xk), where
. / 1 ) ( , / )
( 1 1 0 2 2
0 c c Q c c
Q ηo = ηo = − (50)
(ii) If c/c1+c/c2≥1 then the optimal invariant linear-loss interval predictor of Xl based on Xk degenerates into a point
predictor (1+ o)
•
η Xk, where
). /( )
( 2 1 2
0 c c c
Q η•o = + (51)
Proof. For the proof we refer to Theorem 1.
Corollary 5.1. The minimum risk of the optimal invariant predictor of Xl Based on Xk is given by
{
( , )} {
( , )}
),
(θθθθ θθθθ && ηηηηo
θθθθ d V
d E r Er
R ∗ = ∗ =
∫ ∫
∫ ∫
∞ ∞∞
+ −
=
0
2 1 2 1 0 1 2
0 0
2 1 2 1 0 1 1
2 2 2
1
) , ( )
, (
v v
dv dv v v f v c dv dv v v f v c
o o
η η
(52) for case (i) with ηηηηo =(η1o,η2o)as given by (50) and for case
(ii) with o o o
•
=
=η η
η1 2 as given by (51).
Proof. For the proof we refer to Corollary 4.1. III. EQUIVALENT CONFIDENCE COEFFICIENT
For case (i) when we obtain an interval predictor for Xl we
may regard the interval as a confidence interval in the conventional sense and evaluate its confidence coefficient. The general result is contained in the following theorem.
Theorem 6. Suppose that V=(V1,V2) is a random vector
having density function f(v1,v2) (v1,v2>0) where f is defined by (30) and let H be the distribution function of W=V1/V2, i.e., the probability density function of W is given by
( ) ( 2, 2) 2. 0
2f wv v dv
v w
h
∫
∞
= (53)
Then the confidence coefficient based on X and associated
with the optimum prediction interval d*=(d1,d2), where d1=Xk+η1σ), d2=Xk+η2σ), is
} , | :
{
Pr d∗ d1<Xl <d2 µσ
)]. / ( [ )] / 1 (
[Q 1 c c2 H Q 1 c c1
H − − − −
Proof. The confidence coefficient for d∗ corresponding to
(µ,σ) is given by
} , | :
) ,
Pr{(Xk σ) Xk+η1σ)<Xl <Xk+η2σ) µσ
} /
: ) ,
Pr{( 1 2 η1< 1 2 <η2
= v v v v
)]. / ( [ )] / 1 ( [ ) ( )
( 2 H 1 H Q 1 c c2 H Q 1 c c1
H − = − − − −
= η η (55)
This is independent of (µ,σ).
Theorem 7. Suppose that V=(V1,V2) is a random vector
having density function f0(v1,v2) (v1 real, v2>0), where f0 is defined by
) 1 , ( ) , (
1 )
, (
+ − Β − Β =
l m l k l k x x fθ k l
1
1[ ( ) ( )]
)] (
[ − − −−
× l k
k l
k
k F x F x
x
Fθ θ θ
) ( ) ( )] ( 1
[ k l
l m
l f x f x
x
Fθ − θ θ
−
× , (56)
where µ=0, and let H0 be the distribution function of W=V1/V2, i.e., the probability density function of W is given by
( ) 0( 2, 2) 2. 0
2
0 w v f wv v dv
h
∫
∞
= (57)
Then the confidence coefficient based on Xk and associated
with the optimum prediction interval d*=(d1,d2), where d1=(1+η1o)Xk, d2=(1+η2o)Xk, is
} , | :
{
Pr d∗ d1<Xl <d2 µσ
)]. / ( [ )] / 1 (
[ 1 1
0 0 2 1
0
0 Q c c H Q c c
H − − − −
= (58)
Proof. For the proof we refer to Theorem 6.
The way in which (54) (or (58)) varies with c, c1 and c2, and the fact that c1 and c2 are the factors of proportionality associated with losses from overshooting and undershooting relative to loss involved in increasing the length of interval, provides an interesting interpretation of confidence interval prediction.
IV. CONCLUSION
In many statistical decision problems it is reasonable co confine attention to rules that are invariant with respect to a certain group of transformations. If a given decision problem admits a sufficient statistic, it is well known that the class of invariant rules based on the sufficient statistic is essentially complete in the class of all invariant rules under some assumptions. This result may be used to show that if there exists a minimax invariant rule among invariant rules based on sufficient statistic, it is minimax among all invariant rules. In this paper, we consider statistical prediction problems which are invariant with respect to a certain group
of transformations and construct the optimal invariant interval predictors. The method used is that of the invariant embedding of sample statistics in a loss function in order to form pivotal quantities which allow one to eliminate unknown parameters from the problem. This method 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.
ACKNOWLEDGMENT
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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