Exponential Selection Problems Under Progressive
Type II Censoring
By
Lii-Yuh Leu
Department of Industrial Engineering and management,
St. John’s University, Tamsui, Taipei, Taiwan,25135,
R.O.C.,E-mail address: leu@mail.sju.edu.
Abstract
This paper considers the selection problems of exponential
distributions under progressive type II censoring mechanism. The
selection problems of the largest location and scale parameters are
proposed. The problems concerning with good populations and bad
populations are considered. Simultaneous confidence intervals, that can
be derived with the proposed selection procedure , are discussed.
Simultaneous upper confidence bounds for all distances of the parameters
from the largest one are proposed.
Keywords: subset selection, exponential distribution, good
population, bad population, multiple comparison with best.
Let
 i , i  1,..., k
parameter
i
be k exponential populations with location
and scale parameter
i
and density
function:
1
 e x p (x (i  i) x/  ) ,i  i
f ( x :i ,i )i
0 , o t h e r w i s e .

We denote the distribution by
Let
Xij , j  1,..., n be
 i , i  1,..., k ,under
,
0
E( i ,i ) .
i.i.d. samples from population
progressive type II censoring scheme
( R1 ,..., Rm ) ,we
will let
Rm )
Yij  X ij( :Rm1 :,...,
, j  1,...m, i  1,..., k .
n
We have
Z1  n(Yi1  i ) / i , Z 2  (n  R1  1)(Yi 2  Yi1 ) / i ,..., Z m  (n  R1  R2  ...  Rm 1  m  1)(Yim  Yi ( m 1) ) / i
are i.i.d.
E(0,1)
We will consider the cases for known or unknown
parameters.
1.  known, unknown
If

that
are known and
  0 .For

are unknown, we may assume
the selection problem, let
the ordered value of
 ' s.
[1]  [2]  ...  [ k ]
Our goal is to select t (1  t  k 1)
best populations associated with the t largest scale
parameters
[i ] , i  k  t  1,..., k .
1.1 Indifference zone formulation
Let
( )  {  (1 ,..., k ) | [ k t ] / [ k t 1]   }, 0    1
fixed be the
preference zone. Under progressive type II censoring
scheme
( R1 ,..., Rm ) ,  i
ˆi 
is estimated by
1 m
 ( R j 1Y)
m j 1
We have
2mˆi / i ~ 22 .m
.
i j
be
Let
Yi  mˆi ,then Yi ~ Gamma(m,i ) ,
the proposed selection
procedure is
R: select t populations corresponding to
Y [ k ] , Y [ k 1] ,..., and Y [ k t 1] .
Let
Y( i )
be the unknown observed sample corresponding
to the parameter
[i ] ,
then
P (CS | R)  P (max(Y(1) ,..., Y( k t ) )  min(Y( k t 1) ,..., Y( k ) ))
The least favorable configuration in
( )
is given by
[1]  ...  [ k t ] ,[ k t 1]  ...  [ k ] and [ k t ]  [ k t 1] .
Thus

inf P (CS | R)  (k  t )  Fmk t 1 (u )(1  Fm ( u ))t dFm (u )
  ( )
0
Where
Fm (u ) 
For t=1, it is
u
1
x m1e x dx .

0
 ( m)


0
0
(k  1) Fmk 2 (u)(1  Fm ( u))dFm (u)   Fmk 1 (u /  )dFm (u) .
1.2 Subset selection formulation
The proposed selection is
R*:select
i
iff
Yi  cY[ k t 1] , 0  c  1.
If S is the selected subset, then
P (CS | R*)  P ( [i ]  S , i  k  t  1,..., k | R*)
 P (Y(i )  cY[ k t 1] , i  k  t  1,..., k | R*)
 P (c max(Y(1) ,..., Y( k t ) )  min(Y( k t 1) ,..., Y( k ) ))
Hence
i nP
f C( S

0
 u(m ) ( F1 t cm u(. d) F
)


0
0
u( )
(k  1) Fmk 2 (u)(1  Fm (cu))dFm (u)   Fmk 1 (u / c)dFm (u) .
For t=1, it is
2.
 unknown, known
If

that

|R * ) k ( t mk  )t1F
are unknown and
  1.For

are known, we may assume
[1]  [2]  ...  [ k ]
the selection problem, let
the ordered value of
 ' s.
be
Our goal is to select t (1  t  k 1)
best populations associated with the t largest location
parameters
[i ] , i  k  t  1,..., k .
n(Yi1  i ) ~ E(0,1) .
Let
The estimation of
i is Yi1
and
Ti  Yi1 , i  1,..., k.
2.1 Indifference zone formulation
Let
( )  {  ( 1 ,..., k ) | [ k t ]  [ k t 1]   },   0
fixed be the
preference zone. The proposed selection procedure is
R: select t populations corresponding to
T [ k ] , T [ k 1] ,..., and T [ k t 1]
Then
inf P (CS | R)  P(max( Z1 ,..., Z k t )  n  min( Z k t 1 ,..., Z k ))
 ( )
 (1  e  n ) k t  (k  t )e nt I (e n ; t  1, k  t )
I (en ; t  1, k  t )
where
is an incomplete beta function.
(1  e n )k 1  (k  1)ent I (en ; 2, k  1) .
For t=1, it is
2.2 Subset selection formulation
The proposed selection is
R*:select
i
iff
Ti  T[ k t 1]  c, c  0.
and
inf P (CS | R*)  P(max( Z1 ,..., Z k t )  nc  min( Z k t 1 ,..., Z k ))

 (1  e  nc ) k t  (k  t )e ntc I (e nc ; t  1, k  t )
(1  e nc )k 1  (k  1)entc I (enc ; 2, k  1) .
For t=1, it is
2.3 Good Populations, Bad Populations, and Selection
Problems.
Let
 i ~ E ( i ,1)
scheme
and under progressive type II censoring
( R1 ,..., Rm ) ,
1/( k 1)
q=-log(1- ( P*)
Theorem 2.1:
Proof:
let
Ti  Yi1 , i  1,..., k.
) and S={i| T  T
i
[k ]
G  {i | i  [ k ]   } ,
If
   q / n }.
Then
P (G  S )  P *
n(Ti  i ), i  1,..., k
are i.i.d. E(0,1), we have
P(max(n(Ti  i ))  min(n(Ti  i ))  q)  P *
If
i  [ k ]  
, then
T[ k ]  ( k )  Ti  i  q / n ,
so
T[ k ]  Ti    T[ k ]  Ti  i  [ k ]  T[ k ]  Ti  i  ( k )  q / n .
Next, we let
G  {i | i  [ k ]  1}, and B  {i | i  [ k ]   2 }, 1   2
be
the set of good populations and bad populations. We
want
P(G  S  Bc )  P * .
( 2  1 )n  2q,
Let
For this problem we take n so that
and take t so that
S  {i | Ti  T[ k ]  t / n} ,
n 2  q  t  q  n1.
then
Theorem 2.2: P(G  S  B )  P *
c
Proof:
i  [ k ]  1 , and max(n(Ti  i ))  min(n(Ti  i ))  q ,
T[ k ]  Ti  1  q / n  t / n,
and hence
then
iS .
On the other hand,
i  [ k ]   2 , and max( n(Ti  i ))  min( n(Ti  i ))  q
then
Ti  i  T( k )  [ k ]  q / n, so T( k )  [ k ]  Ti  i  q / n
Hence
T[ k ]  T( k )  Ti  [ k ]  i  q / n  Ti  [ k ]  i  t / n   2  Ti  t / n, and i  S
2.4 Simultaneous Confidence Intervals
Let
 i  [ k ]  i , i  1,..., k ,
our goal is to find two sided
confidence intervals for
i ' s
with confidence coefficient
at least P*. We have
Theorem 2.3: P(G  S , (T[ k ]  Ti  q / n)   i  (max(T j  Ti )  q / n)  , i  1,..., k )  P *
j i
Proof:
max(n(Ti  i ))  min(n(Ti  i ))  q, and i  G, then G  S ,
Ti  i  T( k )  [ k ]  q / n, then  i  T( k )  Ti  q / n  (max(T j  Ti )  q / n)  , i  1,..., k
j i
T[ k ]  ( k )  Ti  i  q / n, so i  ( k )  i  T[ k ]  Ti  q / n  (T[ k ]  Ti  q / n) , i  1,..., k
Further, we will consider the pairwise comparison of the
ranked parameters
[i ]  [ j ] , i  j.
We will use the following
lemmas:
Lemma 1:
min(T[ j ]  ( j ) )  T[i ]  [i ]  max(T[ j ]  ( j ) ) .
1 j i
i  j k
Proof:
min(T[ j ]  ( j ) )  min(T[i ]  ( j ) )  T[i ]  max [ j ]  T[i ]  [i ]
1 j i
1 j i
1 j i
Similarly,
max(T[ j ]  ( j ) )  max(T[i ]  ( j ) )  T[i ]  min ( j )  T[i ]  [i ]
i  j k
i  j k
i  j k
Lemma 2:
min(T[ i ]  ( i ) )  min(T[ i ]  [ i ] ), max(T[ i ]  [ i ] )  max(T[ i ]  ( i ) )
1i  k
1i  k
1i  k
1i  k
Proof:
Let min(T[i ]  [i ] )  T[l ]  [l ] , then min(T[ i ]  ( i ) )  T[ l ]  [ l ] , by Lemma 1.
1i  k
1i l
Hence
min(T[i ]  (i ) )  min(T[i ]  (i ) )  min(T[i ]  [i ] ).
1i  k
1i l
1i  k
Let max(T[i ]  [i ] )  T[l ]  [l ] , then T[l ]  [l ]  max(T[i ]  (i ) ), by Lemma 1.
1i  k
l i  k
Hence
max(T[i ]  (i ) )  max(T[i ]  (i ) )  T[l ]  [l ]  max(T[ i ]  [ i ] )
1i  k
l i  k
1i  k
Theorem 2.4:
P(T[i ]  T[ j ]  q / n  [ i ]  [ j ]  T[ i]  T[ j]  q / n , i , j  1,..., k )  P * .
Proof:
If max(Ti  i )  min(Ti  i )  q / n,
then, by Lemma 2
max(T[ j ]  [ j ] )  max(T[ j ]  ( j ) )  min(T[ i ]  ( i ) )  q / n  min(T[ i ]  [ i ] )  q / n,
1 j  k
1i  k
1 j  k
1i  k
Hence
T[ j ]  [ j ]  T[ i ]  [ i ]  q / n, i  j , and [ i ]  [ j ]  T[ i ]  T[ j ]  q / n, i  j.
Similarly,
If max(Ti  i )  min(Ti  i )  q / n,
then, by Lemma 2
max(T[ j ]  [ j ] )  max(T[ j ]  ( j ) )  min(T[ i ]  ( i ) )  q / n  min(T[ i ]  [ i ] )  q / n,
1 j  k
1i  k
1 j  k
1i  k
Hence
T[i ]  [i ]  q / n  T[ j ]  [ j ] , i  j , and T[ i ]  T[ j ]  q / n  [ i ]  [ j ] , i  j.
3.
 unknown, unknown,  is the parameter of interest
Under progressive type II censoring scheme
will let
( R1 ,..., Rm ) ,we
Rm )
Yij  X ij( :Rm1 :,...,
, j  1,...m, i  1,..., k .
n
We have
Z1  n(Yi1  i ) / i , Z 2  (n  R1  1)(Yi 2  Yi1 ) / i ,..., Z m  (n  R1  R2  ...  Rm 1  m  1)(Yim  Yi ( m 1) ) / i
are i.i.d.
E(0,1) .
Hence
m
U i   Z j ~ Gamma(m  1,i )
j 2
3.1 Indifference zone formulation
Let
( )  {  (1 ,..., k ) | [ k t ] / [ k t 1]   }, 0    1
fixed be the
preference zone.
As in section 1, the proposed selection procedure is
R: select t populations corresponding to
U [ k ] , U [ k 1] ,..., and U [ k t 1] .
Then

inf P (CS | R)  (k  t )  Fmk1t 1 (u )(1  Fm  (1 u ))t dFm  (u1 )
  ( )
0
3.2 Subset selection formulation
The proposed selection is
i
R*:select
iff
U i  cU[ k t 1] , 0  c  1.
Then
i nP
f C( S

4.

|R * ) k ( t mk1)t1F
 u(m 1) ( F1
0
t
cm u1(
d) F
)
u( )
 unknown, unknown,  is the parameter of interest
We will assume that
i   , i  1,..., k .
II censoring scheme
( R1 ,..., Rm ) ,
Under progressive type
let
Rm )
Yij  X ij( :Rm1 :,...,
, j  1,...m, i  1,..., k .
n
We have
Z1  n(Yi1  i ) /  , Z 2  (n  R1  1)(Yi 2  Yi1 ) /  ,..., Z m  (n  R1  R2  ...  Rm 1  m  1)(Yim  Yi ( m 1) ) / 
are i.i.d.
and
Ti  Yi1

E(0,1) .
are
and
Yi1
ˆ
The maximum likelihood estimators of
and
k
m
ˆ   ( R j  1)(Yij  Yi1 ) / k (m  1) .
i 1 j  2
are independent.
4.1 Indifference zone formulation
Let
( )  {  ( 1 ,..., k ) | [ k t ]  [ k t 1]   },   0
fixed be the
i
preference zone. The proposed selection procedure is
R: select t populations corresponding to
T [ k ] , T [ k 1] ,..., and T [ k t 1]
Then
inf P (CS | R)  P(max( Z1 ,..., Z k t )  n  min( Z k t 1 ,..., Z k ))
 ( )
 (1  e  n ) k t  (k  t )e nt I (e n ; t  1, k  t )
4.2 Subset selection formulation
The proposed selection is
R*:select
i
iff
Ti  T[ k t 1]  cˆ / n, c  0.
and
inf P (CS | R*)  P(max( Z1 ,..., Z k t )  cˆ /   min( Z k t 1 ,..., Z k ))

  P(max( Z1 ,..., Z k t )  cx  min( Z k t 1 ,..., Z k ))dG ( x)
  ((1  e cx )k t  (k  t )ecxt I (ecx ; t  1, k  t ))dG ( x)
where
G(x) is the cdf of
ˆ /  ,
which is
Gamma(k(m-1),k(m-1)).
4.3 Good Populations, Bad Populations, and Selection
Problems.
Theorem 4.1: If
k 2
 k  1
v
G  {i | i  [ k ]   } , 1   (1)i 
 (1  q '(i  1) / v)  P*, v  k (m  1),
i 0
 i 1 
and
S={i| T  T
i
[k ]
   q 'ˆ / n }.
Then
P (G  S )  P *
Proof:
If
i  [ k ]  
, and
max(n(Ti  i ) / ˆ)  min(n(Ti  i ) / ˆ)  q ' ,then
T[ k ]  ( k )  Ti  i  q 'ˆ / n ,
so
T[ k ]  Ti    T[ k ]  Ti  i  [ k ]  T[ k ]  Ti  i  ( k )  q 'ˆ / n .
P(max(n(Ti  i ) / ˆ)  min(n(Ti  i ) / ˆ)  q ') 
P(max(n(T   ) /  )  min(n(T   ) /  )  q 'ˆ /  )
i
i
i
i
  P(max(n(Ti  i ) /  )  min(n(Ti  i ) /  )  q ' x)dG ( x)
  (1  e q ' x ) k 1 dG ( x)
v v 1  vx
k 1 k  1
k 1 k  1




e
i  iq ' x v x
i
v
  
(

1)
e
dx




(1) (1  iq '/ v)
i 
i 
 (v )
i 0 
i 0 
k 2
 k  1
v
 1   (1)i 
 (1  q '(i  1) / v)  P *
i 0
 i 1 
Next, we let
G  {i | i  [ k ]  1}, and B  {i | i  [ k ]   2 }, 1   2
be
the set of good populations and bad populations. We
want
P(G  S  Bc )  P * .
( 2  1 )n / ˆ  2q ',
and take t so that
Theorem 4.2: Let
Proof:
For this problem we take n so that
n 2 / ˆ  q '  t  q ' n1 / ˆ.
S  {i | Ti  T[ k ]  tˆ / n} ,
then
P(G  S  Bc )  P *
i  [ k ]  1 , and max(n(Ti  i ) / ˆ)  min(n(Ti  i ) / ˆ)  q ' ,
T[ k ]  Ti  1  q 'ˆ / n  tˆ / n,
and hence
iS .
On the other hand,
i  [ k ]   2 , and max(n(Ti  i ) / ˆ)  min(n(Ti  i ) / ˆ)  q '
then
then
Ti  i  T( k )  [ k ]  q 'ˆ / n, so T( k )  [ k ]  Ti  i  q 'ˆ / n
Hence
T[ k ]  T( k )  Ti  [ k ]  i  q 'ˆ / n  Ti  [ k ]  i  tˆ / n   2  Ti  tˆ / n, and i  S
4.4 Simultaneous Confidence Intervals
 i  [ k ]  i , i  1,..., k ,
Let
our goal is to find two sided
confidence intervals for
i ' s
with confidence coefficient
at least P*. We have
Theorem 4.3: P(G  S , (T[ k ]  Ti  q 'ˆ / n)    i  (max(T j  Ti )  q 'ˆ / n)  , i  1,..., k )  P *
j i
Proof:
max(n(Ti  i ) / ˆ)  min(n(Ti  i ) / ˆ)  q ', and i  G, then G  S ,
T    T    q 'ˆ / n, then   T  T  q 'ˆ / n  (max(T  T )  q 'ˆ / n)  , i  1,..., k
i
i
(k )
[k ]
i
(k )
i
j i
j
i
T[ k ]  ( k )  Ti  i  q 'ˆ / n, so i  ( k )  i  T[ k ]  Ti  q 'ˆ / n  (T[ k ]  Ti  q 'ˆ / n)  , i  1,..., k
Further, we will consider the pairwise comparison of the
ranked parameters
[i ]  [ j ] , i  j.
We will use the following
lemmas:
Lemma 1:
min(T[ j ]  ( j ) )  T[i ]  [i ]  max(T[ j ]  ( j ) ) .
1 j i
i  j k
Proof:
min(T[ j ]  ( j ) )  min(T[i ]  ( j ) )  T[i ]  max [ j ]  T[i ]  [i ]
1 j i
Similarly,
1 j i
1 j i
max(T[ j ]  ( j ) )  max(T[i ]  ( j ) )  T[i ]  min ( j )  T[i ]  [i ]
i  j k
i  j k
i  j k
Lemma 2:
min(T[ i ]  ( i ) )  min(T[ i ]  [ i ] ), max(T[ i ]  [ i ] )  max(T[ i ]  ( i ) )
1i  k
1i  k
1i  k
1i  k
Proof:
Let min(T[i ]  [i ] )  T[l ]  [l ] , then min(T[ i ]  ( i ) )  T[ l ]  [ l ] , by Lemma 1.
1i  k
1i l
Hence
min(T[i ]  (i ) )  min(T[i ]  (i ) )  min(T[i ]  [i ] ).
1i  k
1i l
1i  k
Let max(T[i ]  [i ] )  T[l ]  [l ] , then T[l ]  [l ]  max(T[i ]  (i ) ), by Lemma 1.
1i  k
l i  k
Hence
max(T[i ]  (i ) )  max(T[i ]  (i ) )  T[l ]  [l ]  max(T[ i ]  [ i ] )
1i  k
l i  k
1i  k
Theorem 4.4:
P(T[i ]  T[ j ]  q 'ˆ / n  [ i]  [ j]  T[ i]  T[ j]  q 'ˆ / n, i, j  1,..., k )  P * .
Proof:
If max(Ti  i )  min(Ti  i )  q 'ˆ / n,
then, by Lemma 2
max(T[ j ]  [ j ] )  max(T[ j ]  ( j ) )  min(T[i ]  (i ) )  q 'ˆ / n  min(T[i ]  [i ] )  q 'ˆ / n,
1 j  k
1 j  k
1i  k
1i  k
Hence
T[ j ]  [ j ]  T[i ]  [i ]  q 'ˆ / n, i  j , and [i ]  [ j ]  T[i ]  T[ j ]  q 'ˆ / n, i  j.
Similarly,
If max(Ti  i )  min(Ti  i )  q 'ˆ / n,
then, by Lemma 2
max(T[ j ]  [ j ] )  max(T[ j ]  ( j ) )  min(T[i ]  (i ) )  q 'ˆ / n  min(T[i ]  [i ] )  q 'ˆ / n,
1 j  k
1 j  k
1i  k
1i  k
Hence
T[ i ]  [ i ]  q 'ˆ / n  T[ j ]  [ j ] , i  j , and T[ i ]  T[ j ]  q 'ˆ / n  [i ]  [ j ] , i  j.