- Email: [email protected]
Improved confidence estimators for the usual one-sided confidence intervals for the ratio of two normal variances
Statistics & Probability Letters 59 (2002) 307–315
Improved con dence estimators for the usual one-sided con dence intervals for the ratio of two normal variances Hsiuying Wang Institute of Statistical Science, Academia Sinica, Taipei 115, Taiwan Received January 2001; received in revised form May 2002
Abstract The usual con dence intervals for the ratio of two normal variances are constructed from F-distribution. These con dence intervals are especially important in the variance component model. In this paper, I focus on con dence estimate and conditional performance of these con dence intervals. Under some conditions, con dence estimators better than the con dence coe1cient are provided in this article. Moreover, conditional properties of these intervals are also discussed. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Semirelevant betting procedures; Betting procedures; Con dence estimators
1. Introduction Consider a one-way classi cation model yij = + ui + ij ;
(1)
where i = 1; 2; : : : ; k, denotes the treatment, class or group and j = 1; 2; : : : ; m, denotes the sample observations. The mean of ith group is denoted by + ui and the overall mean of all the groups by . The ui are assumed to be random variables and have normal distributions with mean zero and variance u2 and uncorrelated with ij . The errors ij are assumed to have normal distributions with mean zero and variance 2 . The errors ij and ij (j = j ) are assumed to be uncorrelated. The variances u2 and 2 are accordingly called variance components. In this model, we are interested in estimating the ratio of the variance components. Let (y: − y) : 2 U1 = i2 i 2 u + =m E-mail address: [email protected] (H. Wang). c 2002 Elsevier Science B.V. All rights reserved. 0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 2 ) 0 0 1 9 8 - 0
308
H. Wang / Statistics & Probability Letters 59 (2002) 307–315
and
U2 =
i
j
(yij − y: i )2 2
:
The U1 and U2 have independent 2 -distribution with (k − 1) and k(m − 1) degrees of freedom, respectively. The ratio U1 =(k − 1) E = ; U2 =k(m − 1) mR + 1 : 2 =[ i j (yij − y: i )2 ], has a central F-distribution with (k −1) where R= u2 = 2 , and E =m i (y: i − y) and k(m − 1) degrees of freedom. Hence the usual (1 − ) con dence intervals for mR + 1 and R are (E=Fu ; E=Fl ) and ((1=m)(E=Fu − 1), (1=m)(E=Fl − 1)), where P(Fl 6 E=(mR + 1) 6 Fu ) = 1 − . In this paper, the conditional procedure introduced in the next paragraph will be used to examine the properties of these con dence intervals and several improved con dence estimators are provided for these con dence intervals. Assume that X is a random variable with distribution F(X |), where is an unknown parameter. For a con dence set C(X ) of , the con dence coe1cient 1 − , given by inf P [ ∈ C(X )] = 1 − , is the traditional way to report the con dence for the con dence set C(X ). If there exists a subset S of the sample space such that (i) P ( ∈ C(X ) | X ∈ S) ¿ 1 − + for all , or (ii) P ( ∈ C(X ) | X ∈ S) ¡ 1 − − for all , for some ¿ 0, then the con dence report should be considered to be greater than 1 − or less than 1 − with respect to the case (i) or case (ii), respectively, when observation X belongs to S. The subset S is called a relevant subset. The relevant subsets of Buehler (1959) was extended and formalized by Robinson (1979a) to the concept of relevant betting procedures. The notation C(X ); (X ) is used to denote the con dence interval C(X ) associated with a con dence report (X ), where (X ) states a degree of belief or level of con dence in some sense in the proposition that ∈ C(X ) after X = x has been observed. If there exists a function S(X ) (= 1; 0), whose absolute bound is 1, such that F=
E[(I ( ∈ C(X )) − (X ))S(X )] ¿ 0
for all
or E[(I ( ∈ C(X )) − (X ))S(X )] ¿ E|S(X )| for all ; where I (·) is the indicator function, then S(X ) is called a semirelevant or a relevant betting procedure, respectively. If 0 6 S(X ) 6 1, then S(X ) is called a positive semirelevant or a relevant betting procedure. On the other hand, if −1 6 S(X ) 6 0, then S(X ) is called a negative semirelevant or a relevant betting procedure. If (x) = 1 − and S(X ) is the indicator function of some set S, then the de nition of a relevant betting procedure reduces to the relevant set. Robinson (1979a) pointed out that the existence of a semirelevant betting procedure is a mild criticism of an interval estimator, and the existence of a relevant betting procedure is a criticism on a level which seems to be just serious enough to bother about. Other related literature can be found in Casella (1987) and Maatta and Casella (1987). In Section 2, some semirelevant procedures are provided for one-sided
H. Wang / Statistics & Probability Letters 59 (2002) 307–315
309
con dence intervals of the ratio of two normal variances and in Section 3, with some parameter space restriction, relevant procedures are provided for these con dence intervals. Under the same conditions in Section 3, improved con dence estimators are provided in Section 4. Moreover, the MSE comparisons of the improved estimators and con dence coe1cient are also present in Section 4. The results in Section 3 imply that the con dence coe1cient can be improved by other con dence reports under the criterion in Kiefer (1977). Kiefer (1977) points out that in place of the con dence coe1cient 1−, a better approach is to provide a data-dependent estimate of the value of the coverage function I ( 12 = 22 ∈ CXY ). Throughout this paper, we consider the squared error loss function 2 2 2 1 1 = E r(X; Y ) − I ∈ CXY ∈ CXY ; (2) L r(X; Y ); I 22 22 where r(X; Y ) is an estimator of I ( 12 = 22 ∈ CXY ). 2. Semirelevant betting procedure In this section, we show that there exist semirelevant betting procedures for the usual one-sided con dence intervals of the ratio of two variances of two independent normal distributions. Assume that Xi , i = 1; : : : ; m and Yj , j = 1; : : : ; n are two samples from normal distributions N (1 ; 12 ) and N (2 ; 22 ), respectively. The usual (1 − ) con dence intervals for 12 = 22 are : 2 2 12 (n − 1) i (Xi − X ) = 1 ¡ c2 ; CXY = (3) : c1 ¡ × : 2 2 (m − 1) 22 j (Yj − Y ) = 2 c2 satisfy P( 12 = 22 ∈ CXY )=1− if 1 and 2 are unknown. Let S12 = i (Xi −X: )2 =(m−1) where c1 and u ‘ and S22 = j (Yi − Y: )2 =(n − 1). In the following, the notations CXY and CXY are used to denote the one-sided (1 − ) con dence intervals of (3) with respect to c2 = 0 and c1 = 0. ‘ Theorem 1. For the one-sided (1 − ) con0dence interval CXY = { 12 = 22 : (S12 = 12 )=(S22 = 22 ) ¡ c2 }; there exists a negative semirelevant betting procedure : =S1 X: =S1 |6k −1 if | YX: =S 2 = V1 : Y =S2 0 otherwise
such that
2 X: =S1 1 ‘ ¿ 0 for all 1 ; 2 ; 12 and 22 ; E I ∈ CXY − (1 − ) V1 22 Y: =S2 where k is some positive constant. Proof. The left-hand side of (4) is 2 1 ‘ (1 − ) − I dF(x; : y; : s12 ; s22 ) ∈ CXY 22 |X: |=S1 = |Y: |=S2 6k
(4)
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H. Wang / Statistics & Probability Letters 59 (2002) 307–315
=
|X: |=S1 = |Y: |=S2 6k
dF(x; : y; : s12 ; s22 )
× (1 − ) −
|X: |=S1 = |Y: |=S2 6k |X: |=S1 = |Y: |=S2 6k
‘ I ( 12 = 22 ∈ CXY ) dF(x; : y; : s12 ; s22 )
‘ C [I ( 12 = 22 ∈ CXY ) + I ( 12 = 22 ∈ CXY )] dF(x; : y; : s12 ; s22 )
;
(5)
C ‘ where CXY is the complement set of CXY . Let 2 1 ‘ A= dF(x; : y; : s12 ; s22 ) I ∈ CXY 22 |X: |=S1 = |Y: |=S2 6k
and
2 1 C B= I dF(x; : y; : s12 ; s22 ): ∈ CXY 2 : : |X |=S1 = |Y |=S2 6k 2
Hence (5) is equal to (A + B) (1 − ) −
A A+B
:
Therefore; we only need to show 1 − ¿ A=(A + B) for all 1 ; 2 ; 12 and 22 . Let 2 1 ‘ A = dF(x; : y; : s12 ; s22 ) I ∈ C √ 2 2 XY 2 |X: |= |Y: |6k
and
B =
c2 1 = 2
2
|X: |= |Y: |6k
√
c2 12 = 22
I
12 C ∈ CXY 22
dF(x; : y; : s12 ; s22 ):
‘ = { 12 = 22 : (S12 = 12 )=(S22 = 22 ) 6 c2 }. Then A 6 A and B ¿ B . Moreover; since X: ; Y: ; S12 Note that CXY 2 and S2 are independent;
2 2 ‘ 2 2 √ = ∈ C ) dF(s ; s ) dF(x; : y) : I ( 1 2 XY 1 2 A |X: |= |Y: |¿k c2 12 = 22
= ‘ C A + B ) + I ( 12 = 22 ∈ CXY )] dF(s12 ; s22 ) |X: |=|Y: |¿k √c 2 = 2 dF(x; : y) : [I ( 12 = 22 ∈ CXY 2 1
= 1 − : Moreover; A B − AB AB − AB A A = ¿ − A + B A + B (A + B)(A + B ) (A + B)(A + B ) =
A(B − B ) ¿ 0: (A + B)(A + B )
Hence (4) ¿ 0 for all 1 ; 2 ; 12 and 22 and the proof is completed.
2
H. Wang / Statistics & Probability Letters 59 (2002) 307–315
311
By the similar argument, we also can show that there exists a positive semirelevant betting procedure : =S1 X: =S1 | ¿ k; 1 if | YX: =S = 2 V2 Y: =S2 0 otherwise ‘ . for a one-sided (1 − ) con dence interval CXY u Moreover, for the one-sided (1 − ) con dence interval CXY , by a similar argument as Theorem 1, : =S1 : | 6 k; 1 if | YX: =S ∗ X =S1 2 = V1 Y: =S2 0 otherwise
and
: −1 X =S 1 = V2∗ Y: =S2 0
:
=S1 | ¿ k; if | YX: =S 2
otherwise
can be shown to be positive and negative semirelevant betting procedures respectively. Besides the above results, the following theorem provides other positive semirelevant betting procedures for one-sided con dence intervals. ‘ Theorem 2. For the one-sided (1 − ) con0dence interval CXY ; 2 S2 1 if S12 ¡ k S1 2 T1 = S22 0 otherwise;
where k is some positive constant; is a positive semirelevant betting procedure. Proof. Let H = (S12 = 12 )=(S22 = 22 ); which is Fm−1; n−1 distribution. 2
2 S1 1 ‘ l= E I ∈ CXY − (1 − ) T1 [(h 6 c2 ) − (1 − )] dF(h): 22 S22 {h¡( 22 = 12 )k } The last expression can be written as {h¡( 22 = 12 )k } I (h 6 c2 ) dF(h)
− (1 − ) : dF(h) × {h¡( 22 = 12 )k } {h¡( 2 = 2 )k } dF(h) 2
1
(i) If ( 22 = 12 ) k ¿ c2 , (6) equals {h6c2 } dF(h) − (1 − ) ¿ 0: dF(h) × {h¡( 22 = 12 )k } {h¡( 2 = 2 )k } dF(h) The last inequality is due to
2
{ h 6c 2 }
1
dF(h) = 1 − and
{h¡( 22 = 12 )k }
dF(h) 6 1.
(6)
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H. Wang / Statistics & Probability Letters 59 (2002) 307–315
(ii) If ( 22 = 12 )k ¡ c2 , (6) equals dF(h) × ¿ 0: {h¡( 22 = 12 )k }
Combining cases 1 and 2, the proof is completed. 3. Relevant betting procedures The previous section provides semirelevant betting procedures against (1−) for one-sided (1−) con dence intervals if the parameter space is { 12 = 22 : 12 = 22 ¿ 0}. In this section, the parameter space is assumed to be bounded. Then the semirelevant betting procedures in Section 2 are shown to be relevant betting procedures in this section. Theorem 3. Assume that 0 ¡ b1 6 22 = 12 6 b2 ¡ ∞; then T1 (S12 =S22 ) is a positive betting procedure ‘ for (1 − ) con0dence interval CXY . Proof. We only need to show that (6) in the proof of Theorem 2 is strictly greater than 0. Note that in the proof of Theorem 2; (6) equals 0 only if 22 = 12 = 0 and 22 = 12 = ∞. Hence in this parameter space assumption; (6) is strictly greater than 0. Moreover, the above results can apply to the case of con dence intervals of variance. Assume that
2 ‘ 2 (m − 1)S1 ∗ CX = 1 : 6 c 12 is the usual (1−) con dence interval for 12 , where c∗ is the upper cutoL point of m2 −1 distribution. If 12 is bounded above, then according to a similar argument as in the case of the ratio of two variances, 1 if S12 ¡ k; 2 Z1 (S1 ) = 0 otherwise and
Z2 (S12 ) =
−1
if S12 ¿ k;
0
otherwise
are betting procedures for CX‘ . 4. Improved con dence estimators ‘ u In the previous section, it is shown that there exist betting procedures for CXY and CXY under some parameter space assumption. It reveals that there exists some subset of sample space such that the con dence report of those con dence intervals can be improved when the observation is
H. Wang / Statistics & Probability Letters 59 (2002) 307–315
313
in this subset. This means that there exist better con dence estimators for the coverage function of those con dence intervals. The relationship between relevant betting procedures and admissibility properties for interval estimation has been discussed in Proposition 8.2 in Robinson (1979b). Proposition (Robinson (1979a)). For an interval estimator C(X ); (X ) of a parameter , the absence of semirelevant betting procedures implies admissibility with respect to squared-error loss, and admissibility with respect to squared-error loss implies the absence of relevant betting procedures. In the proof of Theorem 3, E 12 = 22 T (S12 =S22 ) is bounded because the parameter space of 12 = 22 is bounded, thus, it is not necessary to specify the value of in the proof of existing relevant betting procedures. Applying the above proposition guarantees the existence of improved con dence estimators, but cannot specify them. In Theorem 4, the improved con dence estimators are provided under the loss function (2). Theorem 4. For 0 ¡ b1 6 22 = 12 6 b2 ¡ ∞; 2 S2 1 − + w if 21 6 k S1 S2 r = S22 1− otherwise; ‘ ) under the loss function (2); where is an estimator better than 1 − for estimating I ( 12 = 22 ∈ CXY w is a constant satis0ed ‘ I ( 12 = 22 ∈ CXY ) dF(x; : y; : s12 ; s22 ) S12 =S22 6k
2 0¡w¡ inf − (1 − ) : (7) dF(x; : y; : s12 ; s22 ) b1 6 22 = 12 6b2 S 2 =S 2 6k 1
Proof.
2
2 2 2 2 2 1 S1 1 ‘ ‘ −I ∈ CXY −E r ∈ CXY E 1−−I 22 S22 22 2 1 ‘ 2 2 2 1−−I dF(x; : y; : s1 ; s2 ) − w = − 2w ∈ CXY dF(x; : y; : s12 ; s22 ) 2 2 2 2 2 S1 =S2 6k S1 =S2 6k 2 2 1 ‘ 2 2 2 2 =−w 2 1−−I dF(x; : y; : s1 ; s2 ) + w ∈ CXY dF(x; : y; : s1 ; s2 ) : 22 S12 =S22 6k S12 =S22 6k
(8)
Eq. (8) is greater than zero for all 12 = 22 if w satis es (7). By the same argument that the right-hand side of (6) is not smaller than zero in the proof Theorem 2; we have the right-hand side of (7) is not smaller than zero. But note that in the proof of Theorem 3; we know that (6) equals zero only if 12 = 22 = 0 and 12 = 22 = ∞; thus; the right-hand side of (7) is strictly greater than zero due to the parameter space assumption. Hence there exists w ¿ 0 such that (7) holds for all 12 = 22 in this parameter space.
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H. Wang / Statistics & Probability Letters 59 (2002) 307–315 Table 1 Assume that 0:2 6 22 = 12 6 10. m, n and 1 − are 5, 7 and 0.95, respectively. Let w and k in r(S12 =S22 ) are 0:0368 and 0:7, respectively 22 12
S 2 2
MSE r
1
S22
MSE 1−; I
1
;I
2
22
1 2 2
‘ ∈CXY
‘ ∈CXY
1
3
5
7
9
0.983
0.962
0.957
0.967
0.994
Table 2 Assume that 0:01 6 22 = 12 6 35. m, n and 1 − are 25, 12 and 0.95. Let w and k in r(S12 =S22 ) are 0:0335 and 0:1, respectively 22 12
S 2 2
MSE r
1
S22
MSE 1−; I
1
;I
2
22
1
22
‘ ∈CXY
‘ ∈CXY
5
10
20
25
30
0.995
0.976
0.957
0.955
0.980
Table 3 Assume that 5 ¡ 22 = 12 6 20. m, n and 1 − are 61, 25 and 0.95, respectively. Let w and k in r(S12 =S22 ) are 0.0304 and 0.1, respectively 22 12
S 2 2
MSE r
1
S22
MSE 1−; I
1
;I
2
22
1 22
u ∈CXY
u ∈CXY
6
8
10
15
20
0.958
0.967
0.977
0.994
0.999
Table 4 Assume that 1 ¡ 12 ¡ 8, m = 7 and 1 − = 0:9. Let w and k in r ∗ (S12 ) are 0.097 and 2, respectively 12 ∗
‘ MSE(r (S12 ); I ( 12 ∈CXY )) ‘ )) MSE(1−; I ( 12 ∈CXY
2
3
5
7
0.940
0.968
0.989
0.996
u For estimating I ( 12 = 22 ∈ CXY ) and I ( 12 ∈ CX‘ ), 2 S2 1 − + w if S12 ¿ k; ∗ S1 2 r = S22 1− otherwise
H. Wang / Statistics & Probability Letters 59 (2002) 307–315
and
r(S12 )
=
1−+w
if S12 6 k;
1−
otherwise
315
can also be shown to be estimators better than 1− under the assumptions 0 ¡ b1 6 22 = 12 6 b2 ¡ ∞ and b1 6 12 6 b2 , respectively. In Tables 1– 4, the mean squared errors (MSE) of the new estimator and 1 − are compared. The ‘ MSE of the 1 − is (1 − ) for all 12 = 22 . The MSE of r(S12 =S22 ) for estimating I ( 12 = 22 ∈ CXY ), by straightforward calculation, is ‘ ))2 E(r(S12 =S22 ) − I ( 12 = 22 ∈ CXY
=(1 − + !)2 Fm−1; n−1 (k 22 = 12 ) + (1 − )2 (1 − Fm−1; n−1 (k 22 = 12 )) −2[(1 − )(Fm−1; n−1 (c2 ) − Fm−1; n−1 (k 22 = 12 ))I (k 22 = 12 6 c2 ) +(1 − + !)Fm−1; n−1 (min(k 22 = 12 ; c2 ))] + (1 − ); where Fm−1; n−1 (x) is the cumulative F-distribution with m−1 and n−1 degrees of freedom and c2 is the (1−)th quantile of F-distribution with m−1 and n−1 degrees of freedom. The MSE of r ∗ (S12 =S22 ) and r(S12 ) can also be computed from theoretical computation by straightforward calculation. Acknowledgements The author wishes to thank Professor C.F. JeL Wu for pointing out this valuable problem. The author would also like to thank referees for helpful comments and suggestions. References Buehler, R.J., 1959. Some validity criteria for statistical inferences. Ann. Math. Statist. 30, 845–863. Casella, G., 1987. Conditionally acceptable reventered set estimators. Ann. Statist. 15, 1363–1371. Kiefer, J., 1977. Conditional con dence statements and con dence estimators. J. Amer. Statist. Assoc. 72, 789–827. Maatta, J.M., Casella, G., 1987. Conditional properties of interval estimators of the normal variance. Ann. Statist. 4, 1372–1388. Robinson, G.K., 1979a. Conditional properties of statistical procedures. Ann. Statist. 7, 742–755. Robinson, G.K., 1979b. Conditional properties of satistical procedures for location and scale parameters. Ann. Statist. 7, 756–771.
Improved con dence estimators for the usual one-sided con dence intervals for the ratio of two normal variances Hsiuying Wang Institute of Statistical Science, Academia Sinica, Taipei 115, Taiwan Received January 2001; received in revised form May 2002
Abstract The usual con dence intervals for the ratio of two normal variances are constructed from F-distribution. These con dence intervals are especially important in the variance component model. In this paper, I focus on con dence estimate and conditional performance of these con dence intervals. Under some conditions, con dence estimators better than the con dence coe1cient are provided in this article. Moreover, conditional properties of these intervals are also discussed. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Semirelevant betting procedures; Betting procedures; Con dence estimators
1. Introduction Consider a one-way classi cation model yij = + ui + ij ;
(1)
where i = 1; 2; : : : ; k, denotes the treatment, class or group and j = 1; 2; : : : ; m, denotes the sample observations. The mean of ith group is denoted by + ui and the overall mean of all the groups by . The ui are assumed to be random variables and have normal distributions with mean zero and variance u2 and uncorrelated with ij . The errors ij are assumed to have normal distributions with mean zero and variance 2 . The errors ij and ij (j = j ) are assumed to be uncorrelated. The variances u2 and 2 are accordingly called variance components. In this model, we are interested in estimating the ratio of the variance components. Let (y: − y) : 2 U1 = i2 i 2 u + =m E-mail address: [email protected] (H. Wang). c 2002 Elsevier Science B.V. All rights reserved. 0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 2 ) 0 0 1 9 8 - 0
308
H. Wang / Statistics & Probability Letters 59 (2002) 307–315
and
U2 =
i
j
(yij − y: i )2 2
:
The U1 and U2 have independent 2 -distribution with (k − 1) and k(m − 1) degrees of freedom, respectively. The ratio U1 =(k − 1) E = ; U2 =k(m − 1) mR + 1 : 2 =[ i j (yij − y: i )2 ], has a central F-distribution with (k −1) where R= u2 = 2 , and E =m i (y: i − y) and k(m − 1) degrees of freedom. Hence the usual (1 − ) con dence intervals for mR + 1 and R are (E=Fu ; E=Fl ) and ((1=m)(E=Fu − 1), (1=m)(E=Fl − 1)), where P(Fl 6 E=(mR + 1) 6 Fu ) = 1 − . In this paper, the conditional procedure introduced in the next paragraph will be used to examine the properties of these con dence intervals and several improved con dence estimators are provided for these con dence intervals. Assume that X is a random variable with distribution F(X |), where is an unknown parameter. For a con dence set C(X ) of , the con dence coe1cient 1 − , given by inf P [ ∈ C(X )] = 1 − , is the traditional way to report the con dence for the con dence set C(X ). If there exists a subset S of the sample space such that (i) P ( ∈ C(X ) | X ∈ S) ¿ 1 − + for all , or (ii) P ( ∈ C(X ) | X ∈ S) ¡ 1 − − for all , for some ¿ 0, then the con dence report should be considered to be greater than 1 − or less than 1 − with respect to the case (i) or case (ii), respectively, when observation X belongs to S. The subset S is called a relevant subset. The relevant subsets of Buehler (1959) was extended and formalized by Robinson (1979a) to the concept of relevant betting procedures. The notation C(X ); (X ) is used to denote the con dence interval C(X ) associated with a con dence report (X ), where (X ) states a degree of belief or level of con dence in some sense in the proposition that ∈ C(X ) after X = x has been observed. If there exists a function S(X ) (= 1; 0), whose absolute bound is 1, such that F=
E[(I ( ∈ C(X )) − (X ))S(X )] ¿ 0
for all
or E[(I ( ∈ C(X )) − (X ))S(X )] ¿ E|S(X )| for all ; where I (·) is the indicator function, then S(X ) is called a semirelevant or a relevant betting procedure, respectively. If 0 6 S(X ) 6 1, then S(X ) is called a positive semirelevant or a relevant betting procedure. On the other hand, if −1 6 S(X ) 6 0, then S(X ) is called a negative semirelevant or a relevant betting procedure. If (x) = 1 − and S(X ) is the indicator function of some set S, then the de nition of a relevant betting procedure reduces to the relevant set. Robinson (1979a) pointed out that the existence of a semirelevant betting procedure is a mild criticism of an interval estimator, and the existence of a relevant betting procedure is a criticism on a level which seems to be just serious enough to bother about. Other related literature can be found in Casella (1987) and Maatta and Casella (1987). In Section 2, some semirelevant procedures are provided for one-sided
H. Wang / Statistics & Probability Letters 59 (2002) 307–315
309
con dence intervals of the ratio of two normal variances and in Section 3, with some parameter space restriction, relevant procedures are provided for these con dence intervals. Under the same conditions in Section 3, improved con dence estimators are provided in Section 4. Moreover, the MSE comparisons of the improved estimators and con dence coe1cient are also present in Section 4. The results in Section 3 imply that the con dence coe1cient can be improved by other con dence reports under the criterion in Kiefer (1977). Kiefer (1977) points out that in place of the con dence coe1cient 1−, a better approach is to provide a data-dependent estimate of the value of the coverage function I ( 12 = 22 ∈ CXY ). Throughout this paper, we consider the squared error loss function 2 2 2 1 1 = E r(X; Y ) − I ∈ CXY ∈ CXY ; (2) L r(X; Y ); I 22 22 where r(X; Y ) is an estimator of I ( 12 = 22 ∈ CXY ). 2. Semirelevant betting procedure In this section, we show that there exist semirelevant betting procedures for the usual one-sided con dence intervals of the ratio of two variances of two independent normal distributions. Assume that Xi , i = 1; : : : ; m and Yj , j = 1; : : : ; n are two samples from normal distributions N (1 ; 12 ) and N (2 ; 22 ), respectively. The usual (1 − ) con dence intervals for 12 = 22 are : 2 2 12 (n − 1) i (Xi − X ) = 1 ¡ c2 ; CXY = (3) : c1 ¡ × : 2 2 (m − 1) 22 j (Yj − Y ) = 2 c2 satisfy P( 12 = 22 ∈ CXY )=1− if 1 and 2 are unknown. Let S12 = i (Xi −X: )2 =(m−1) where c1 and u ‘ and S22 = j (Yi − Y: )2 =(n − 1). In the following, the notations CXY and CXY are used to denote the one-sided (1 − ) con dence intervals of (3) with respect to c2 = 0 and c1 = 0. ‘ Theorem 1. For the one-sided (1 − ) con0dence interval CXY = { 12 = 22 : (S12 = 12 )=(S22 = 22 ) ¡ c2 }; there exists a negative semirelevant betting procedure : =S1 X: =S1 |6k −1 if | YX: =S 2 = V1 : Y =S2 0 otherwise
such that
2 X: =S1 1 ‘ ¿ 0 for all 1 ; 2 ; 12 and 22 ; E I ∈ CXY − (1 − ) V1 22 Y: =S2 where k is some positive constant. Proof. The left-hand side of (4) is 2 1 ‘ (1 − ) − I dF(x; : y; : s12 ; s22 ) ∈ CXY 22 |X: |=S1 = |Y: |=S2 6k
(4)
310
H. Wang / Statistics & Probability Letters 59 (2002) 307–315
=
|X: |=S1 = |Y: |=S2 6k
dF(x; : y; : s12 ; s22 )
× (1 − ) −
|X: |=S1 = |Y: |=S2 6k |X: |=S1 = |Y: |=S2 6k
‘ I ( 12 = 22 ∈ CXY ) dF(x; : y; : s12 ; s22 )
‘ C [I ( 12 = 22 ∈ CXY ) + I ( 12 = 22 ∈ CXY )] dF(x; : y; : s12 ; s22 )
;
(5)
C ‘ where CXY is the complement set of CXY . Let 2 1 ‘ A= dF(x; : y; : s12 ; s22 ) I ∈ CXY 22 |X: |=S1 = |Y: |=S2 6k
and
2 1 C B= I dF(x; : y; : s12 ; s22 ): ∈ CXY 2 : : |X |=S1 = |Y |=S2 6k 2
Hence (5) is equal to (A + B) (1 − ) −
A A+B
:
Therefore; we only need to show 1 − ¿ A=(A + B) for all 1 ; 2 ; 12 and 22 . Let 2 1 ‘ A = dF(x; : y; : s12 ; s22 ) I ∈ C √ 2 2 XY 2 |X: |= |Y: |6k
and
B =
c2 1 = 2
2
|X: |= |Y: |6k
√
c2 12 = 22
I
12 C ∈ CXY 22
dF(x; : y; : s12 ; s22 ):
‘ = { 12 = 22 : (S12 = 12 )=(S22 = 22 ) 6 c2 }. Then A 6 A and B ¿ B . Moreover; since X: ; Y: ; S12 Note that CXY 2 and S2 are independent;
2 2 ‘ 2 2 √ = ∈ C ) dF(s ; s ) dF(x; : y) : I ( 1 2 XY 1 2 A |X: |= |Y: |¿k c2 12 = 22
= ‘ C A + B ) + I ( 12 = 22 ∈ CXY )] dF(s12 ; s22 ) |X: |=|Y: |¿k √c 2 = 2 dF(x; : y) : [I ( 12 = 22 ∈ CXY 2 1
= 1 − : Moreover; A B − AB AB − AB A A = ¿ − A + B A + B (A + B)(A + B ) (A + B)(A + B ) =
A(B − B ) ¿ 0: (A + B)(A + B )
Hence (4) ¿ 0 for all 1 ; 2 ; 12 and 22 and the proof is completed.
2
H. Wang / Statistics & Probability Letters 59 (2002) 307–315
311
By the similar argument, we also can show that there exists a positive semirelevant betting procedure : =S1 X: =S1 | ¿ k; 1 if | YX: =S = 2 V2 Y: =S2 0 otherwise ‘ . for a one-sided (1 − ) con dence interval CXY u Moreover, for the one-sided (1 − ) con dence interval CXY , by a similar argument as Theorem 1, : =S1 : | 6 k; 1 if | YX: =S ∗ X =S1 2 = V1 Y: =S2 0 otherwise
and
: −1 X =S 1 = V2∗ Y: =S2 0
:
=S1 | ¿ k; if | YX: =S 2
otherwise
can be shown to be positive and negative semirelevant betting procedures respectively. Besides the above results, the following theorem provides other positive semirelevant betting procedures for one-sided con dence intervals. ‘ Theorem 2. For the one-sided (1 − ) con0dence interval CXY ; 2 S2 1 if S12 ¡ k S1 2 T1 = S22 0 otherwise;
where k is some positive constant; is a positive semirelevant betting procedure. Proof. Let H = (S12 = 12 )=(S22 = 22 ); which is Fm−1; n−1 distribution. 2
2 S1 1 ‘ l= E I ∈ CXY − (1 − ) T1 [(h 6 c2 ) − (1 − )] dF(h): 22 S22 {h¡( 22 = 12 )k } The last expression can be written as {h¡( 22 = 12 )k } I (h 6 c2 ) dF(h)
− (1 − ) : dF(h) × {h¡( 22 = 12 )k } {h¡( 2 = 2 )k } dF(h) 2
1
(i) If ( 22 = 12 ) k ¿ c2 , (6) equals {h6c2 } dF(h) − (1 − ) ¿ 0: dF(h) × {h¡( 22 = 12 )k } {h¡( 2 = 2 )k } dF(h) The last inequality is due to
2
{ h 6c 2 }
1
dF(h) = 1 − and
{h¡( 22 = 12 )k }
dF(h) 6 1.
(6)
312
H. Wang / Statistics & Probability Letters 59 (2002) 307–315
(ii) If ( 22 = 12 )k ¡ c2 , (6) equals dF(h) × ¿ 0: {h¡( 22 = 12 )k }
Combining cases 1 and 2, the proof is completed. 3. Relevant betting procedures The previous section provides semirelevant betting procedures against (1−) for one-sided (1−) con dence intervals if the parameter space is { 12 = 22 : 12 = 22 ¿ 0}. In this section, the parameter space is assumed to be bounded. Then the semirelevant betting procedures in Section 2 are shown to be relevant betting procedures in this section. Theorem 3. Assume that 0 ¡ b1 6 22 = 12 6 b2 ¡ ∞; then T1 (S12 =S22 ) is a positive betting procedure ‘ for (1 − ) con0dence interval CXY . Proof. We only need to show that (6) in the proof of Theorem 2 is strictly greater than 0. Note that in the proof of Theorem 2; (6) equals 0 only if 22 = 12 = 0 and 22 = 12 = ∞. Hence in this parameter space assumption; (6) is strictly greater than 0. Moreover, the above results can apply to the case of con dence intervals of variance. Assume that
2 ‘ 2 (m − 1)S1 ∗ CX = 1 : 6 c 12 is the usual (1−) con dence interval for 12 , where c∗ is the upper cutoL point of m2 −1 distribution. If 12 is bounded above, then according to a similar argument as in the case of the ratio of two variances, 1 if S12 ¡ k; 2 Z1 (S1 ) = 0 otherwise and
Z2 (S12 ) =
−1
if S12 ¿ k;
0
otherwise
are betting procedures for CX‘ . 4. Improved con dence estimators ‘ u In the previous section, it is shown that there exist betting procedures for CXY and CXY under some parameter space assumption. It reveals that there exists some subset of sample space such that the con dence report of those con dence intervals can be improved when the observation is
H. Wang / Statistics & Probability Letters 59 (2002) 307–315
313
in this subset. This means that there exist better con dence estimators for the coverage function of those con dence intervals. The relationship between relevant betting procedures and admissibility properties for interval estimation has been discussed in Proposition 8.2 in Robinson (1979b). Proposition (Robinson (1979a)). For an interval estimator C(X ); (X ) of a parameter , the absence of semirelevant betting procedures implies admissibility with respect to squared-error loss, and admissibility with respect to squared-error loss implies the absence of relevant betting procedures. In the proof of Theorem 3, E 12 = 22 T (S12 =S22 ) is bounded because the parameter space of 12 = 22 is bounded, thus, it is not necessary to specify the value of in the proof of existing relevant betting procedures. Applying the above proposition guarantees the existence of improved con dence estimators, but cannot specify them. In Theorem 4, the improved con dence estimators are provided under the loss function (2). Theorem 4. For 0 ¡ b1 6 22 = 12 6 b2 ¡ ∞; 2 S2 1 − + w if 21 6 k S1 S2 r = S22 1− otherwise; ‘ ) under the loss function (2); where is an estimator better than 1 − for estimating I ( 12 = 22 ∈ CXY w is a constant satis0ed ‘ I ( 12 = 22 ∈ CXY ) dF(x; : y; : s12 ; s22 ) S12 =S22 6k
2 0¡w¡ inf − (1 − ) : (7) dF(x; : y; : s12 ; s22 ) b1 6 22 = 12 6b2 S 2 =S 2 6k 1
Proof.
2
2 2 2 2 2 1 S1 1 ‘ ‘ −I ∈ CXY −E r ∈ CXY E 1−−I 22 S22 22 2 1 ‘ 2 2 2 1−−I dF(x; : y; : s1 ; s2 ) − w = − 2w ∈ CXY dF(x; : y; : s12 ; s22 ) 2 2 2 2 2 S1 =S2 6k S1 =S2 6k 2 2 1 ‘ 2 2 2 2 =−w 2 1−−I dF(x; : y; : s1 ; s2 ) + w ∈ CXY dF(x; : y; : s1 ; s2 ) : 22 S12 =S22 6k S12 =S22 6k
(8)
Eq. (8) is greater than zero for all 12 = 22 if w satis es (7). By the same argument that the right-hand side of (6) is not smaller than zero in the proof Theorem 2; we have the right-hand side of (7) is not smaller than zero. But note that in the proof of Theorem 3; we know that (6) equals zero only if 12 = 22 = 0 and 12 = 22 = ∞; thus; the right-hand side of (7) is strictly greater than zero due to the parameter space assumption. Hence there exists w ¿ 0 such that (7) holds for all 12 = 22 in this parameter space.
314
H. Wang / Statistics & Probability Letters 59 (2002) 307–315 Table 1 Assume that 0:2 6 22 = 12 6 10. m, n and 1 − are 5, 7 and 0.95, respectively. Let w and k in r(S12 =S22 ) are 0:0368 and 0:7, respectively 22 12
S 2 2
MSE r
1
S22
MSE 1−; I
1
;I
2
22
1 2 2
‘ ∈CXY
‘ ∈CXY
1
3
5
7
9
0.983
0.962
0.957
0.967
0.994
Table 2 Assume that 0:01 6 22 = 12 6 35. m, n and 1 − are 25, 12 and 0.95. Let w and k in r(S12 =S22 ) are 0:0335 and 0:1, respectively 22 12
S 2 2
MSE r
1
S22
MSE 1−; I
1
;I
2
22
1
22
‘ ∈CXY
‘ ∈CXY
5
10
20
25
30
0.995
0.976
0.957
0.955
0.980
Table 3 Assume that 5 ¡ 22 = 12 6 20. m, n and 1 − are 61, 25 and 0.95, respectively. Let w and k in r(S12 =S22 ) are 0.0304 and 0.1, respectively 22 12
S 2 2
MSE r
1
S22
MSE 1−; I
1
;I
2
22
1 22
u ∈CXY
u ∈CXY
6
8
10
15
20
0.958
0.967
0.977
0.994
0.999
Table 4 Assume that 1 ¡ 12 ¡ 8, m = 7 and 1 − = 0:9. Let w and k in r ∗ (S12 ) are 0.097 and 2, respectively 12 ∗
‘ MSE(r (S12 ); I ( 12 ∈CXY )) ‘ )) MSE(1−; I ( 12 ∈CXY
2
3
5
7
0.940
0.968
0.989
0.996
u For estimating I ( 12 = 22 ∈ CXY ) and I ( 12 ∈ CX‘ ), 2 S2 1 − + w if S12 ¿ k; ∗ S1 2 r = S22 1− otherwise
H. Wang / Statistics & Probability Letters 59 (2002) 307–315
and
r(S12 )
=
1−+w
if S12 6 k;
1−
otherwise
315
can also be shown to be estimators better than 1− under the assumptions 0 ¡ b1 6 22 = 12 6 b2 ¡ ∞ and b1 6 12 6 b2 , respectively. In Tables 1– 4, the mean squared errors (MSE) of the new estimator and 1 − are compared. The ‘ MSE of the 1 − is (1 − ) for all 12 = 22 . The MSE of r(S12 =S22 ) for estimating I ( 12 = 22 ∈ CXY ), by straightforward calculation, is ‘ ))2 E(r(S12 =S22 ) − I ( 12 = 22 ∈ CXY
=(1 − + !)2 Fm−1; n−1 (k 22 = 12 ) + (1 − )2 (1 − Fm−1; n−1 (k 22 = 12 )) −2[(1 − )(Fm−1; n−1 (c2 ) − Fm−1; n−1 (k 22 = 12 ))I (k 22 = 12 6 c2 ) +(1 − + !)Fm−1; n−1 (min(k 22 = 12 ; c2 ))] + (1 − ); where Fm−1; n−1 (x) is the cumulative F-distribution with m−1 and n−1 degrees of freedom and c2 is the (1−)th quantile of F-distribution with m−1 and n−1 degrees of freedom. The MSE of r ∗ (S12 =S22 ) and r(S12 ) can also be computed from theoretical computation by straightforward calculation. Acknowledgements The author wishes to thank Professor C.F. JeL Wu for pointing out this valuable problem. The author would also like to thank referees for helpful comments and suggestions. References Buehler, R.J., 1959. Some validity criteria for statistical inferences. Ann. Math. Statist. 30, 845–863. Casella, G., 1987. Conditionally acceptable reventered set estimators. Ann. Statist. 15, 1363–1371. Kiefer, J., 1977. Conditional con dence statements and con dence estimators. J. Amer. Statist. Assoc. 72, 789–827. Maatta, J.M., Casella, G., 1987. Conditional properties of interval estimators of the normal variance. Ann. Statist. 4, 1372–1388. Robinson, G.K., 1979a. Conditional properties of statistical procedures. Ann. Statist. 7, 742–755. Robinson, G.K., 1979b. Conditional properties of satistical procedures for location and scale parameters. Ann. Statist. 7, 756–771.