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On the nonexistence of certain optimal confidence sets for the rectangular problem
ELSEVIER
Statistics
& Probability
Letters
21 (1994) 263-269
On the nonexistence of certain optimal confidence for the rectangular problem Shoutir Department
Kishore
Chatterjee *, Gaurangadeb
sets
Chattopadhyay’
of Statistics, Calcutta University, 35, Ballygunge Circular Road, Calcutta 700 019. India Received
September
1992; revised August
1993
Abstract It has been shown that there does not exist any uniformly most accurate unbiased or uniformly most accurate invariant confidence set for 0 at level 1 - a where the sample is drawn from a rectangular (0, 6 + 1) distribution. So Pratt’s (1961) statement in this connection is seen to be fallacious. Keywords:
Confidence set; Unbiased; Invariant;
Most accurate; Rectangular distribution
1. Introduction distributed according to the rectangular distribution over (0, 0 + 1). Let X1, X1, . . . , X, be independently Denote X = (X,, . . , X,), Xc,,) = max {X,, X1, . . . , X,} and Xcl) = min {X,, X1, . . . , X,}. Pratt (1961) stated for the rectangular (0, 8 + 1) problem, the uniformly most accurate unbiased (or invariant) (UMAU or UMAI) confidence interval for 8 at confidence level 1 - c( is (see also Kiefer, 1977, Ex. l(a), Lehmann, 1986, Ch. 10, Problem 28) S*(x) =
{e: 0 = max
where d is the solution 2d”=cc
(xcl)
+
d, xc,)) - 1 < 8 < min(x(,,, x(,) - d) = s},
(1.1)
of the equation
ifa<2l-“,
2d”-(2d-l)“=cc
(1.2a) ifcc>2l-“.
(1.2b)
But, as a point of fact, there does not exist any uniformly most accurate unbiased or invariant confidence set for 0 for this problem. In Sections 2 and 4 of this note we shall justify this statement. A locally best confidence set is incidentally suggested in Section 3.
* Corresponding author. ‘Work supported by a Research 8%EMR-1).
Fellowship
from
the Council
0167-7152/94/$7.00 @) 1994 Elsevier Science B.V. All rights reserved SSDI 0167-7152(93)E0202-5
of Scientific
and
Industrial
Research
(Sanction
No. 9/28(199)/
264
S.K. Chatterjee.
2. Nonexistence of a UMAU
G. Chattopadhyay
/ Statistics
& Probability
Letters 21 (1994) 263-269
confidence set
We disprove the claim of Pratt (1961) mentioned earlier by using the relationship between confidence estimation and hypothesis testing. Let, for any real 9, [0, 0 + 11” denote the Cartesian product [e, 8 + 1-j x [e, 8 + i-j x . . . n factors. Also let X = (5: x (,,) - xcl) < 1, x E R”}. Clearly whatever Q we can disregard points x$X as such points would be impossible. Result 2.1. Any test C#I’(ZC)ofthefirm
4Ok) =
1
if x ~3
-
[eo,eo +
1-y or 5 E qe,),
(2.1)
0, otherwise,
where B(f3,) c [e,, B0 + 11” is any set such that P,,{x
E mm
(2.2)
= 4
is unbiased size CIfor testing H,: f3 = 0,-, against H,: 8 # f& Proof. Size of the test 4’ = -%, C$Oml
= P~,cx~,,> e. + 11+ ho cx,,, < eel + h, cx E we,)1 = PO,[& E qe,)l = LY(by
(2.2)).
So to prove the result it is sufficient Ed [40(x)1
to show that for the test 4’ the power is
3 ~1v e f eo.
(2.3)
Now
~e~~w~ =
p,cx,,,> e.+ 11+ p,cx,,,< eoi+ pe[xE wo)i.
(2.4)
For convenience, we denote, for any real 0 [e, 0 + 11” = AO. If AO, n A0 = 8, power of test would be 1 and hence the test would be trivially unbiased against 8. If ABOn A0 # 8, consider the following two cases: Case 1: P,Q
E B(e,)]
E~~~~(x)I=
= 0. From
(2.4) we have
pecx,,,> e. + 11+ ~~cx,~, < eoi
= 1 - Pe[&E
(2.5)
AB,nAO].
But
POCX E Aen
&,I = PeaCXE Ae
n Ae,l
< 1 - c1 [since
B(8,) c x - (A@,,n A@) and PO, {& E B(&)}
Using (2.6), from (2.5) we get (2.3). Case 2: Pe[& E B(e,)] > 0. Let B(B,) = &(0,) 4%)
n CAe, n Ael = 8.
u B2(e0), where
&(0,)
= a].
(2.6)
n B2(e0) = 8, MOO) = Aeo n Ae,
S.K. Chatterjee,
G. Chattopadhyayl
Statistics
& Probability
Letters
21 (1994) 263-269
265
Hence,
Q?C~“(X)I= 1 - pocx E‘4Jn &,I + PeCXE &v&)1 = 1 - Pe,CX6 At3n AJ,l + Pe,CXE&(&)I 3 1 - (1 - poocx E~lv%3)1 + Pe,CXE&(Wl) = Pe,{& E B(8,)) = a. From the above we can conclude Now, define
44(x)=
(2.7)
that 4” is unbiased
against
1
if max (x(r) + d, xc,,)) - 1 2 B0 or min(xCI,,
0
otherwise,
all 0 # QO. xc,) - d) < 00,
where d < 1 is such that 4&/4
(2.8)
WI = a.
Here +“, is the test corresponding Note that 4”, can be rewritten
4: (xl =
1
if(i)
0
otherwise.
to the confidence as
x~%-[C8,,,&,+1]”
interval
(1.1) suggested
or (ii) xo,BfIO-d+l
by Pratt (1961).
and/or
xt,)
(2.9)
Note that if d < f (Fig. l(a)), J&l%:
(X)1 = P,o[Xc,, 2 80 - d + 11+ PO,,[IX,,, < 4, + d] = 2d”.
Again if d > f (Fig. l(b)), &&#C
ml = P,JX(I, 2 e. - d + 11+ PO0 cx,,, d e. + 4 - PO, [e,, - d + 1 < Xo, < x,,, -C f$, + d] if d > f = 2d” - (2d - 1)“.
Therefore Define
(2.8) implies
4%) = &,[4;(&)]
d is determined
by (1.2) as in Pratt
(1961).
[e,, e. + 11” or e. + 1 - CP d xtl) < x(,) G e. + 1,
1
if x $
0
otherwise. = P,,[X,,,
2 &, + 1 - alln] = CI (Fig. 2).
By Result 2.1 both C/I”,and 4: are unbiased
size-a against
HI: 8 # QO.
Result 2.2. C#I~ is at least as powerful as c#J”,against all 8 > do and strictly more powerful than 4: for some
8 > eo.
Proof. From well-known results for the rectangular distribution (Lehmann, 1986, p. 115), it follows that 4 y is UMP for testing H,,: 8 = 8,, against HI: 8 > &,. So we have only to show that EO[~T($)] From
> &[c$O*(X)]
for some
e > eo.
Fig. 2 it is clear that for 8 3 f3,, + 1 - a””
J%C~~ml = 1.
266
S.K. Chatterjee,
I
C.
Chattopadhyay / Statistics & Probability Letters 21 (1994) 263-269
(
(eo-dtl,OO-d+l)
Fig.
1. (a) 4: with d < f, n = 2; (b) c#I”,with d > & n = 2.
(q@o) Fig. 2. +y,n=2.
Again from Figs. l(a) and (b) we have for &, < 0 < &, - d + 1
G+#Jm1 < 1. Note that from (1.2a) for u < 2l-“d” < a and from (1.2b) for c( > 2r-“, have d” < ct. Thus d < ~8” always. Thus for 6 E [l&, - c?~ + 1, B0 - d + 1) we would have
Kmm1
’ w#JO*ml.
Hence the result. From the above against all 8 # 8,,. sets it follows that Again we know
4: (xl =
since 4 -C d < 1, (2d - 1)” < d” we
0 we get that 4”, is not uniformly most powerful among unbiased size-a tests for Ho: 8 = 8, Hence from the well-known correspondence between UMPU tests and UMAU confidence the confidence interval suggested by Pratt (1961) is not uniformly most accurate unbiased. that (Lehmann, 1986, p. 115) c$~(x) given by
1
if X$ [e,, e. + 11” or e. < xcl) < xc,) < e. + d/n,
0
otherwise,
S.K. Chatterjee. G. ChattopadhyaylStatistics
& Probability Letters 21 (1994) 263-269
261
is the UMP size-a test for testing He: 9 = BO against HI. * 8 < &,. By Result 2.1 it is unbiased against HI: 0 # EJO.Note that q!~yand 42 are entirely different. Hence there does not exist any UMPU size-a test for testing HO: 0 = &, against HI: 0 # &. So there does not exist any UMAU confidence set with confidence coefficient 1 - CL.
3. Locally most accurate confidence set We first set up a locally best nonrandomized
test for H,: 6’ = OOagainst
HI: /3 # O,,.
Result 3.1. The test @L given by
4L (xl =
1
if SEA?-[e,,e,+
0
otherwise,
11” or e0+3-~cr1’n
is (upto equivalence) the unique locally best size-a nonrandomized test for testing HO: 8 = tIO against H,: i9/8O. Proof. Define E = (5: 60 + ; - &zx~‘”<
Xcl) -C Xc,) -C 60 + f + &din}.
Since PB,[x E E] = a, by Result 2.1 4L is an unbiased Note that when
(3.1) size-x test. (see Fig. 3.)
e,-%+~ccl’ndedeo+3-fa1’n
(3.2)
at 8 the power of &, is ~~~~~ (x)]
= C(+ P,cx,,,
Let ~(_LC)be any nonrandomized
< e,] + P,cx,,, level-a
> eO +
II.
test. Let { _LC: r#~(x)= l} = C u D, where
CnD=@, C = 4,
with PO,[& E C] < CI,
D c (5: xc,,) > e. + 1
or
xcl) < eo, x E 5~).
Fig. 3. r#~~, n = 2.
(3.3) C and D are such that
268
S.K. Chatterjee,
G. Chattopadhyay
/ Statistics
& Probability
Letters 21 (1994) 263-269
We suppose
&J(X)is not equivalent to +L(x) in the sense that not both the relations C = E and D = (5: xc,) > 00 + 1 or x(1) < do, 5 E a} hold (for two sets E, and Ez in a Euclidean space we write E, s E2 if the Lebesgue measure of the symmetric difference EIAEZ is zero). At 0 the power of 4 is
&C4(Bl= PeCXE Cl + PeC&E01.
(3.4)
Per& E
Cl d P&p E Cl d a,
(3.5)
P~CX E
01 d ~dx,,, < 44 + hex,,, > e. + ii.
(3.6)
Now,
From (3.3)-(3.6) E,&,(X)
we have for all e E [e, - f + f&n,
b E&(X)
e. + f - &din].
If PO[X E C] = c1we must have equality at both places in (3.5). But P,J$ holds for all 8 satisfying (3.2) we must have CC e~[e~-(1/2)+(1/2)~““,
n
(3.7) E C] = Pe,[& E C] iff C c Ae. If this
A, = E, eo+(lpm(l/2)d~~]
where E is defined by (3.1). As PO,{& E E} = CI,we get that Pe,[f7L~C]=a=Pe[g~C] iff C-E. Further
foralleE[eo-f+fa”‘,e,+~-la”“]
.‘ we would have equality
D = tx: xc,j > e. + i
or
in (3.6) for all 0 iff
xtlj < eo, x E cf}.
c@,
Since &(x) and 4(x) are not equivalent (3.7) holds with strict inequality for at least one 0E [&, - $ + f e0 + f ill”‘). This holds for all level-a nonrandomized tests 4 not equivalent to $L. So we can conclude that 4L gives up to equivalence the unique locally best size-a nonrandomized test for testing, H,: 8 = 8,, against H,: Q # f&.
f
Definition 2.1. A family of (1 - a) level confidence confidence sets at level (1 - a) if Pe{B E S(s)}
2 1 - C(
sets {S(x)} is said to be a locally most accurate
ve,
family of
(3.8)
Pe{@ E S(X)} d Pe{ 8’ E S(X)},
(3.9)
for all 8’ E (0 - E, 8 + E) for some E > 0 for all 8, for any (1 - a)-level family of confidence sets {S’(x)}. Since 4L is unique locally best size-cl test for testing H,: 0 = 8,, against Hl: 0 # &, we get strict inequality in f a”“) and hence we can say that the corresponding confidence set (3.7)forsome@E(e-++3c(““,e++S,(x) = (0 : xc,) - 1 < 8 < xcl) is unique
locally most accurate
4. Nonexistence
and
[e > xcl) - 4 + idin
or 0 < xc,) - 4 - f~+]}
(3.10)
at level (1 - a) in the above case.
of a UMAI confidence set
It can be shown that S,(x) given by (3.10) and S*(x) (th e confidence interval (1.1) suggested by Pratt (1961)) are equivariant under the group of transformation generated by g where for a single observation x, gx = -x
S.K. Chatterjee. G. Chattopadhyay / Statistics & Probability Leiters 21 (1994) 263-269
269
and the induced transform of 8 is a0 = -8 - 1. Now, since S,(x) is unique locally most accurate at level (1 - E) and clearly S*(x) is not equivalent to S&), the latter is not UMAI. Now, when for E given by (3.1), P0[f7L E E] = 0 i.e E,I[~I,(&)] do - 1 < 0 < I!&,- f - :a”“, Pe{X E % - [e,, e. + l]“}. But from Figs. l(a) and (b) it is clear that P,[X,,, 2 &, - d + 1 and/or &, d d + Q,] > 0. Hence E. [4”, (&)I = P,[X,,,
z e. - d + 1 and/or
+ P,{X > P,{X
Xc,,) G d + e,]
E 3 - [e,, e. + i]n}
E .fz - [e,, e. + I]“)
=E,[&,(X)]
for
alleE(e,-l,e,-~-3cr”~).
(4.1)
So, from (3.1) we have pO{e’ E S* (x)1 < pep
E sL(x))
for all 8 and for all 8’ E (0 + 3 + : c?“, 8 + 1) i.e. for every 8, S*(x) strictly beats S,(x) for some 8’. But SL(x) is unique locally best confidence set of level (1 - a) for 8. Since SL(x) and S*(x) are both equivariant under g it follows that there does not exist any UMAI level (1 - E) confidence set for 8. Note. Since X(r) given R = Xc”) - Xcl) = r follows rect (0, 8 + 1 - Y)by the same line of argument it can be shown that there does not exist any uniformly most accurate unbiased (or invariant) conditional confidence set for f3(see Lehmann, 1986, Ch. 10, Problem 28 (iii)). Here also a conditionally locally best confidence set can be suggested.
References Kiefer, J. (1977), Conditional confidence statements and confidence estimators (with discussion), J. Amer. Statist. Assoc. 72, 789-827. Lehmann, E.L. (1986), Testing Statistical Hypotheses (Wiley, New York, 2nd ed.). Pratt, J.W. (1961), Review of Testing Statistical Hypotheses by E.L. Lehmann, J. Amer. Statist. Assoc. 56, 163-167.
Statistics
& Probability
Letters
21 (1994) 263-269
On the nonexistence of certain optimal confidence for the rectangular problem Shoutir Department
Kishore
Chatterjee *, Gaurangadeb
sets
Chattopadhyay’
of Statistics, Calcutta University, 35, Ballygunge Circular Road, Calcutta 700 019. India Received
September
1992; revised August
1993
Abstract It has been shown that there does not exist any uniformly most accurate unbiased or uniformly most accurate invariant confidence set for 0 at level 1 - a where the sample is drawn from a rectangular (0, 6 + 1) distribution. So Pratt’s (1961) statement in this connection is seen to be fallacious. Keywords:
Confidence set; Unbiased; Invariant;
Most accurate; Rectangular distribution
1. Introduction distributed according to the rectangular distribution over (0, 0 + 1). Let X1, X1, . . . , X, be independently Denote X = (X,, . . , X,), Xc,,) = max {X,, X1, . . . , X,} and Xcl) = min {X,, X1, . . . , X,}. Pratt (1961) stated for the rectangular (0, 8 + 1) problem, the uniformly most accurate unbiased (or invariant) (UMAU or UMAI) confidence interval for 8 at confidence level 1 - c( is (see also Kiefer, 1977, Ex. l(a), Lehmann, 1986, Ch. 10, Problem 28) S*(x) =
{e: 0 = max
where d is the solution 2d”=cc
(xcl)
+
d, xc,)) - 1 < 8 < min(x(,,, x(,) - d) = s},
(1.1)
of the equation
ifa<2l-“,
2d”-(2d-l)“=cc
(1.2a) ifcc>2l-“.
(1.2b)
But, as a point of fact, there does not exist any uniformly most accurate unbiased or invariant confidence set for 0 for this problem. In Sections 2 and 4 of this note we shall justify this statement. A locally best confidence set is incidentally suggested in Section 3.
* Corresponding author. ‘Work supported by a Research 8%EMR-1).
Fellowship
from
the Council
0167-7152/94/$7.00 @) 1994 Elsevier Science B.V. All rights reserved SSDI 0167-7152(93)E0202-5
of Scientific
and
Industrial
Research
(Sanction
No. 9/28(199)/
264
S.K. Chatterjee.
2. Nonexistence of a UMAU
G. Chattopadhyay
/ Statistics
& Probability
Letters 21 (1994) 263-269
confidence set
We disprove the claim of Pratt (1961) mentioned earlier by using the relationship between confidence estimation and hypothesis testing. Let, for any real 9, [0, 0 + 11” denote the Cartesian product [e, 8 + 1-j x [e, 8 + i-j x . . . n factors. Also let X = (5: x (,,) - xcl) < 1, x E R”}. Clearly whatever Q we can disregard points x$X as such points would be impossible. Result 2.1. Any test C#I’(ZC)ofthefirm
4Ok) =
1
if x ~3
-
[eo,eo +
1-y or 5 E qe,),
(2.1)
0, otherwise,
where B(f3,) c [e,, B0 + 11” is any set such that P,,{x
E mm
(2.2)
= 4
is unbiased size CIfor testing H,: f3 = 0,-, against H,: 8 # f& Proof. Size of the test 4’ = -%, C$Oml
= P~,cx~,,> e. + 11+ ho cx,,, < eel + h, cx E we,)1 = PO,[& E qe,)l = LY(by
(2.2)).
So to prove the result it is sufficient Ed [40(x)1
to show that for the test 4’ the power is
3 ~1v e f eo.
(2.3)
Now
~e~~w~ =
p,cx,,,> e.+ 11+ p,cx,,,< eoi+ pe[xE wo)i.
(2.4)
For convenience, we denote, for any real 0 [e, 0 + 11” = AO. If AO, n A0 = 8, power of test would be 1 and hence the test would be trivially unbiased against 8. If ABOn A0 # 8, consider the following two cases: Case 1: P,Q
E B(e,)]
E~~~~(x)I=
= 0. From
(2.4) we have
pecx,,,> e. + 11+ ~~cx,~, < eoi
= 1 - Pe[&E
(2.5)
AB,nAO].
But
POCX E Aen
&,I = PeaCXE Ae
n Ae,l
< 1 - c1 [since
B(8,) c x - (A@,,n A@) and PO, {& E B(&)}
Using (2.6), from (2.5) we get (2.3). Case 2: Pe[& E B(e,)] > 0. Let B(B,) = &(0,) 4%)
n CAe, n Ael = 8.
u B2(e0), where
&(0,)
= a].
(2.6)
n B2(e0) = 8, MOO) = Aeo n Ae,
S.K. Chatterjee,
G. Chattopadhyayl
Statistics
& Probability
Letters
21 (1994) 263-269
265
Hence,
Q?C~“(X)I= 1 - pocx E‘4Jn &,I + PeCXE &v&)1 = 1 - Pe,CX6 At3n AJ,l + Pe,CXE&(&)I 3 1 - (1 - poocx E~lv%3)1 + Pe,CXE&(Wl) = Pe,{& E B(8,)) = a. From the above we can conclude Now, define
44(x)=
(2.7)
that 4” is unbiased
against
1
if max (x(r) + d, xc,,)) - 1 2 B0 or min(xCI,,
0
otherwise,
all 0 # QO. xc,) - d) < 00,
where d < 1 is such that 4&/4
(2.8)
WI = a.
Here +“, is the test corresponding Note that 4”, can be rewritten
4: (xl =
1
if(i)
0
otherwise.
to the confidence as
x~%-[C8,,,&,+1]”
interval
(1.1) suggested
or (ii) xo,BfIO-d+l
by Pratt (1961).
and/or
xt,)
(2.9)
Note that if d < f (Fig. l(a)), J&l%:
(X)1 = P,o[Xc,, 2 80 - d + 11+ PO,,[IX,,, < 4, + d] = 2d”.
Again if d > f (Fig. l(b)), &&#C
ml = P,JX(I, 2 e. - d + 11+ PO0 cx,,, d e. + 4 - PO, [e,, - d + 1 < Xo, < x,,, -C f$, + d] if d > f = 2d” - (2d - 1)“.
Therefore Define
(2.8) implies
4%) = &,[4;(&)]
d is determined
by (1.2) as in Pratt
(1961).
[e,, e. + 11” or e. + 1 - CP d xtl) < x(,) G e. + 1,
1
if x $
0
otherwise. = P,,[X,,,
2 &, + 1 - alln] = CI (Fig. 2).
By Result 2.1 both C/I”,and 4: are unbiased
size-a against
HI: 8 # QO.
Result 2.2. C#I~ is at least as powerful as c#J”,against all 8 > do and strictly more powerful than 4: for some
8 > eo.
Proof. From well-known results for the rectangular distribution (Lehmann, 1986, p. 115), it follows that 4 y is UMP for testing H,,: 8 = 8,, against HI: 8 > &,. So we have only to show that EO[~T($)] From
> &[c$O*(X)]
for some
e > eo.
Fig. 2 it is clear that for 8 3 f3,, + 1 - a””
J%C~~ml = 1.
266
S.K. Chatterjee,
I
C.
Chattopadhyay / Statistics & Probability Letters 21 (1994) 263-269
(
(eo-dtl,OO-d+l)
Fig.
1. (a) 4: with d < f, n = 2; (b) c#I”,with d > & n = 2.
(q@o) Fig. 2. +y,n=2.
Again from Figs. l(a) and (b) we have for &, < 0 < &, - d + 1
G+#Jm1 < 1. Note that from (1.2a) for u < 2l-“d” < a and from (1.2b) for c( > 2r-“, have d” < ct. Thus d < ~8” always. Thus for 6 E [l&, - c?~ + 1, B0 - d + 1) we would have
Kmm1
’ w#JO*ml.
Hence the result. From the above against all 8 # 8,,. sets it follows that Again we know
4: (xl =
since 4 -C d < 1, (2d - 1)” < d” we
0 we get that 4”, is not uniformly most powerful among unbiased size-a tests for Ho: 8 = 8, Hence from the well-known correspondence between UMPU tests and UMAU confidence the confidence interval suggested by Pratt (1961) is not uniformly most accurate unbiased. that (Lehmann, 1986, p. 115) c$~(x) given by
1
if X$ [e,, e. + 11” or e. < xcl) < xc,) < e. + d/n,
0
otherwise,
S.K. Chatterjee. G. ChattopadhyaylStatistics
& Probability Letters 21 (1994) 263-269
261
is the UMP size-a test for testing He: 9 = BO against HI. * 8 < &,. By Result 2.1 it is unbiased against HI: 0 # EJO.Note that q!~yand 42 are entirely different. Hence there does not exist any UMPU size-a test for testing HO: 0 = &, against HI: 0 # &. So there does not exist any UMAU confidence set with confidence coefficient 1 - CL.
3. Locally most accurate confidence set We first set up a locally best nonrandomized
test for H,: 6’ = OOagainst
HI: /3 # O,,.
Result 3.1. The test @L given by
4L (xl =
1
if SEA?-[e,,e,+
0
otherwise,
11” or e0+3-~cr1’n
is (upto equivalence) the unique locally best size-a nonrandomized test for testing HO: 8 = tIO against H,: i9/8O. Proof. Define E = (5: 60 + ; - &zx~‘”<
Xcl) -C Xc,) -C 60 + f + &din}.
Since PB,[x E E] = a, by Result 2.1 4L is an unbiased Note that when
(3.1) size-x test. (see Fig. 3.)
e,-%+~ccl’ndedeo+3-fa1’n
(3.2)
at 8 the power of &, is ~~~~~ (x)]
= C(+ P,cx,,,
Let ~(_LC)be any nonrandomized
< e,] + P,cx,,, level-a
> eO +
II.
test. Let { _LC: r#~(x)= l} = C u D, where
CnD=@, C = 4,
with PO,[& E C] < CI,
D c (5: xc,,) > e. + 1
or
xcl) < eo, x E 5~).
Fig. 3. r#~~, n = 2.
(3.3) C and D are such that
268
S.K. Chatterjee,
G. Chattopadhyay
/ Statistics
& Probability
Letters 21 (1994) 263-269
We suppose
&J(X)is not equivalent to +L(x) in the sense that not both the relations C = E and D = (5: xc,) > 00 + 1 or x(1) < do, 5 E a} hold (for two sets E, and Ez in a Euclidean space we write E, s E2 if the Lebesgue measure of the symmetric difference EIAEZ is zero). At 0 the power of 4 is
&C4(Bl= PeCXE Cl + PeC&E01.
(3.4)
Per& E
Cl d P&p E Cl d a,
(3.5)
P~CX E
01 d ~dx,,, < 44 + hex,,, > e. + ii.
(3.6)
Now,
From (3.3)-(3.6) E,&,(X)
we have for all e E [e, - f + f&n,
b E&(X)
e. + f - &din].
If PO[X E C] = c1we must have equality at both places in (3.5). But P,J$ holds for all 8 satisfying (3.2) we must have CC e~[e~-(1/2)+(1/2)~““,
n
(3.7) E C] = Pe,[& E C] iff C c Ae. If this
A, = E, eo+(lpm(l/2)d~~]
where E is defined by (3.1). As PO,{& E E} = CI,we get that Pe,[f7L~C]=a=Pe[g~C] iff C-E. Further
foralleE[eo-f+fa”‘,e,+~-la”“]
.‘ we would have equality
D = tx: xc,j > e. + i
or
in (3.6) for all 0 iff
xtlj < eo, x E cf}.
c@,
Since &(x) and 4(x) are not equivalent (3.7) holds with strict inequality for at least one 0E [&, - $ + f e0 + f ill”‘). This holds for all level-a nonrandomized tests 4 not equivalent to $L. So we can conclude that 4L gives up to equivalence the unique locally best size-a nonrandomized test for testing, H,: 8 = 8,, against H,: Q # f&.
f
Definition 2.1. A family of (1 - a) level confidence confidence sets at level (1 - a) if Pe{B E S(s)}
2 1 - C(
sets {S(x)} is said to be a locally most accurate
ve,
family of
(3.8)
Pe{@ E S(X)} d Pe{ 8’ E S(X)},
(3.9)
for all 8’ E (0 - E, 8 + E) for some E > 0 for all 8, for any (1 - a)-level family of confidence sets {S’(x)}. Since 4L is unique locally best size-cl test for testing H,: 0 = 8,, against Hl: 0 # &, we get strict inequality in f a”“) and hence we can say that the corresponding confidence set (3.7)forsome@E(e-++3c(““,e++S,(x) = (0 : xc,) - 1 < 8 < xcl) is unique
locally most accurate
4. Nonexistence
and
[e > xcl) - 4 + idin
or 0 < xc,) - 4 - f~+]}
(3.10)
at level (1 - a) in the above case.
of a UMAI confidence set
It can be shown that S,(x) given by (3.10) and S*(x) (th e confidence interval (1.1) suggested by Pratt (1961)) are equivariant under the group of transformation generated by g where for a single observation x, gx = -x
S.K. Chatterjee. G. Chattopadhyay / Statistics & Probability Leiters 21 (1994) 263-269
269
and the induced transform of 8 is a0 = -8 - 1. Now, since S,(x) is unique locally most accurate at level (1 - E) and clearly S*(x) is not equivalent to S&), the latter is not UMAI. Now, when for E given by (3.1), P0[f7L E E] = 0 i.e E,I[~I,(&)] do - 1 < 0 < I!&,- f - :a”“, Pe{X E % - [e,, e. + l]“}. But from Figs. l(a) and (b) it is clear that P,[X,,, 2 &, - d + 1 and/or &, d d + Q,] > 0. Hence E. [4”, (&)I = P,[X,,,
z e. - d + 1 and/or
+ P,{X > P,{X
Xc,,) G d + e,]
E 3 - [e,, e. + i]n}
E .fz - [e,, e. + I]“)
=E,[&,(X)]
for
alleE(e,-l,e,-~-3cr”~).
(4.1)
So, from (3.1) we have pO{e’ E S* (x)1 < pep
E sL(x))
for all 8 and for all 8’ E (0 + 3 + : c?“, 8 + 1) i.e. for every 8, S*(x) strictly beats S,(x) for some 8’. But SL(x) is unique locally best confidence set of level (1 - a) for 8. Since SL(x) and S*(x) are both equivariant under g it follows that there does not exist any UMAI level (1 - E) confidence set for 8. Note. Since X(r) given R = Xc”) - Xcl) = r follows rect (0, 8 + 1 - Y)by the same line of argument it can be shown that there does not exist any uniformly most accurate unbiased (or invariant) conditional confidence set for f3(see Lehmann, 1986, Ch. 10, Problem 28 (iii)). Here also a conditionally locally best confidence set can be suggested.
References Kiefer, J. (1977), Conditional confidence statements and confidence estimators (with discussion), J. Amer. Statist. Assoc. 72, 789-827. Lehmann, E.L. (1986), Testing Statistical Hypotheses (Wiley, New York, 2nd ed.). Pratt, J.W. (1961), Review of Testing Statistical Hypotheses by E.L. Lehmann, J. Amer. Statist. Assoc. 56, 163-167.