鹿児島大学リポジトリ

EFFECTIVE SIGNIFICANCE LEVEL, WHEN AIC IS

EMPLOYED IN PRE-TEST ESTIMATION

著者

INADA Koichi, KASAGI Fumiyoshi

journal or

publication title

鹿児島大学理学部紀要. 数学・物理学・化学

volume

15

page range

11-18

別言語のタイトル

予備検定にAICを用いたときの有効な有意水準につ

いて

URL

http://hdl.handle.net/10232/6396

EFFECTIVE SIGNIFICANCE LEVEL, WHEN AIC IS

EMPLOYED IN PRE-TEST ESTIMATION

著者

INADA Koichi, KASAGI Fumiyoshi

journal or

publication title

鹿児島大学理学部紀要. 数学・物理学・化学

volume

15

page range

11-18

別言語のタイトル

予備検定にAICを用いたときの有効な有意水準につ

いて

URL

http://hdl.handle.net/10232/00007013

Rep. Fac. Sci., Kagoshima Univ., (Math., Phys. & Chem.) No. 15, p. 1ト18, 1982

EFFECTIVE SIGNIFICANCE LEVEL, WHEN AIC

IS EMPLOYED IN PRE-TEST ESTIMATION

By ●

Koichi Inada* and Fumiyoshi Kasagi**

(Received August 31, 1982)

Abstract

In the present paper we evaluate the effective significance level when AIC is

●

employed in pre-test estimation. Numerical values of the significance level are presented in Table 1-3.

1. Introduction

The estimation after preliminary test of significance has been studied by various authors. The earlier works include, among others, Bancroft [3], Asano [2] and Kitagawa [6]. The central issue in this type of problem is how to determine the significance level of the preliminary test. We cite the works of Sawa and Hiromatsu [7]. Hirano [5] applied AIC (Akaike's information criterion [1]) to determine when to pool and when not to pool. AIC is equal t0 -2logeL(p,)+2k, where L({L) is the

maximum likelihood and k is the number of unknown parameters.

Assume we have two models Ho and Hl9 which we ought to determine before the

esimator is numerically calculated, and we ought to select the model which has smaller value of AIC, and upon selecting the model we compute the maximum likelihood

estimate assuming this model. When this principle is employed, it is clear we will arrive

the maximum likelihood estimate under the model chosen, out of two, by the preliminary likelihood ratio test. The consideration on distribution does not determine the critical value but the likelihood function and the difference in the numbers of parameters determine it, and thus the effective significance level of the preliminary

●

test is determined弧tomatically.

The purpose of this paper is to evaluate the effective significance level. Assuming

that the correlation is known, we consider in §2 estimation of mean vector and inァ3 that of one component of the means, upon introducing certain types of pair of model.

● ●

In §4, estimation of correlation coefficient is considered.

Department of Mathematics, Faculty of Science, Kagoshima University, Kagoshima, Japan. ** National (〕ardiovascular (っenter Research Institute, Osaka, Japan.

12 K. Inada and F. Kasagi 2.Estimationofbivariatenormalmean Let(vl),v2,.,..,yMbearandomsamplesfromabivariatenormaldistri-///¥,.,-.../In¥ butionN(ju,A)withtliemeanvector//--(^1)andthecovariancematrixA-a2(?). 2.10nesidedcase Fromthenatureofthedataitisknownthat;te1-//2-0or(/^≧Oand/u2≧0)where theinequalityisstrictforatleastone. ● Herewehavetwoalternativemodelsaboutmeans; ModelHo:/i1-u2-0, ModelH^.ux≧Oanda2≧Owheretheinequalityisstrictforatleastone. ThenwewoulddeterminetousetheprocedureminimizingAICasthetestcriterion ●● ofthepreferencebetweenModelHoandModelHvWedenoteAICunderHoandunder HxasAIC(Ho)andAIC(H^respectively. Case1.Whena2andpareknownparameters,wemakeuseofthefollowingrelation. AIC{Ho)-AIG{Hx)<0 (1) E∃

w%2<4, X≧0, Y≧0

or蒜(Y-px)*<ア, X<0, T-pねO

n

ovI=^(X-pY)2<i, Y<0, X-pY ≧ 0

or X-pY<O, Y-px<o

1

where %2-7二才(X2-2pXY+Y2) and (X,Y) is the sample mean

vector. That is to say, if (1) is satisfied we would accept Model Ho, and the estimator of /ms the zero vector. On the other hand if (1) is not satisfied, the estimator (L of ft is as follows;

(2)

(3)

if X≧O and Y≧0,

if X<O and Y-pX≧0,

if X-py≧O and Y<0,

if X-pY<O and Y-pX<0.

When Ho is true, the significance level α of the test criterion stated above is exactly ●

α - l-PriAICiHJ-AICiHjXO IH｡)

6-Hit-cos-V) .  ∞ __ォat｡ iー_

'フ妄けり2dx.

Effective Significance Level IK

In Table 1 values α are tabulated for various values of p.

Case 2. When a2 is an unknown parameter and /> is a known parameter, after some

calculations, we arrive at the simple result as follows; AIC(｣To) -AIC(ffl) < 0 (X,Y) ∑ -w)′

芸(x{,yA ∑-1<Xi,TA′

1-1

<l-e,-2/n, X≧0, Y≧0

n(O,Y-px) j:-i(O,Y-PXY

MS^^^^^^^m

yi iXiJi) ∑-1{XiJi)′ 1-1 n(X-pT,O) ∑-1{X- >r,o)′

芸(xuyA ∑-'(XtJi)′

H:-印

or X-pY<0,

Y-<l-e-*'n, X<0, Y-pX≧0

<l-e-2/サ, X-pY≧O, y<o

where E-(三g)･

● ●

As the estimation procedure of ju is similar to Case 1, we can also obtain the significance

㌔

level α as follows;

(5) α - 1-去COs-1^一 i'トー /サ(-去,葦主上去cos-^-/t))71-^ォ/サ(l,n-1)

where Ia(n, m) -B(n, m) α

j x^il-x)1"-1 dx

O 1

and B(n, m) -巨nJ(トx)

-1 dx.

0

2.2 Two sided case

In this section we would discuss the two sided case.

Model Ho: ^-^-0,

Model Hx′: (pl≧O and ju2≧0) or (〝1≦O and 〝2≦0)

where at least one of the inequalities &re strict in both cases.

Case 1. When a2 and p are known parameters, we have the following relation.

AIC{Ho) -AIC{H^) <0

w%2<4, X≧0, Y≧0

/｢㌃

(6)   ⇔

VF諌

{x-pY)>-2, X<0, Y>O, ¥X¥^¥Y¥

(Y-pX)<2, X<0, Y>0, ¥Xl<iri

It!

′ ㌃

1作二戸

K. Inada and F. Kasagi

(Y-pX)>-2, X>0, Y<0, ¥X暮≦IYl

(x-p?)<2, X<0,?<0, ¥X¥>¥?¥.

Therefore our estimator声of [i is as follows;●

(7) 声-if (2) is satis丘ed, 'o'

if (2) is not satis丘ed,

(章when X and Y are of the same sign,

, 0 , when X andアare of the different sign and ￨X￨^￨Y¥,

(y-- xj when X and Y are of the different sign and ¥X¥<-YI･

● ●

The signi丘cance level α is given by

(8) α-2[ e ^tt- cos-1/)) 27T 1 r∞ ｢∞

277膏二手J J

2  2 1

+宿

∫ Jexpト

2(l-/>2)

(♂-2pxy+y*)} dxdy].

Values of α are also tabulated in Table 1.

Case 2. When a2 is an unknown parameter and /> is a known parameter, we have the

following relation. ● AIG{H｡) - AIG{Hl′)<0 n(x,Y) ∑-¥Y,Y)′ S (*<T<) ∑-KXiJA′ i=1 (9) ⇔

<l-e,-2/ and {(X≧0,Y≧o) or (X≦0, Y≦o))

n(O,Y-pX) ∑-¥O,?-px)′ s (*, *ォ) ∑-1{XiJi)′ 1-1

n(Y-pX,O) ∑-HO,Y-pX)'

S (*ォT<) ∑-HXtYtY 1-1

<l-e-2/* and {(X+Y≧0, Y≦o)

or (x+Y≦0, Y≧o))

<l-e-2/ and {(X+Y≦0,X≧o)

or (X+Y≧0, X≦o)}.

● ●

As the estimation procedure of 〟 is similar to Case 1, we can also get也e sigm丘cance

level α as follows;

Effective Significance Level Table 1 p   α(one-sided) α(two-sided) (ド α(one-sided) α(two-sided) . 0325     . 0650 . 0366    . 0732 . 0399     ･0798 . 0427    . 0854 . 0453    . 0906 . 0477    . 0954 . 0500    . 0998 . 0522    . 1042 . 0544    . 1082 t t H C ! 一 C O   ^   I O   ォ O t -  0 0   0 3 ●         ●         ●         ●         ●         ●         ●         ●         ●         ● o o o o o o o o o o . 0566    . 1122 . 0588    . 1158 .0609    . 1190 . 0631    . 1222 . 0654    . 1250 . 0679    . 1276 . 0704    . 1298 . 0733    . 1318 . 0767    . 1336 . 0807    . 1346 15

3. Estimation of one component of the bivariate normal mean

In this section we won･ld Consider the estimation of 〝 If we can known the value of //1? it is natural that we should use this knowledge to estimate [x2.

3.1 0ne sided case

Our aim is to estimate /j12 where two alternative models are given as follows; ●

Model Hn: A*i-O,

Model Hx¥ [*!>().

Then the estimator (x2 of //2 is as follows; ●

(ll)       v>望-i…

Y-pX if Ho is accepted,

if Hl is accepted.

Case 1. When a2 and p are known parameters, whether Model Ho is accepted or not

is judged through AIC similary asァ2. Then we have the following relation.

●

AIC(Ho) - AIG(Hj)<0

(12)

/ ㌃ x

α

< / 頂

Therefore the significance level α of this test criterion is exactly ●

(13)

α

i V甘)

=0.07864-･.

Case 2. When α2 is an unknown parameter and 〟 is a known parameter, we have

the following relation. ●

AIC{Ho) -AIC{Hx)ォS

(14)  ⇔ <2(サーl)

-V｢㌃ x

/ s {{xi-XT-M&i-X) (Y-THiYi-Y)*)

2(n-1 V言辞二手,

16 K. Inada and F. Kasagi

where t桝is distributed acOording to the t-distribution with m degrees of freedom. The estimator /22 of fi2 is the same as (ll) and the significance level α is exactly

(15)  a - 1-Pr&( -!)< V2(n-1) Yel"1-! }  く

Values of αネre tabulated in Table 2 for various values of n. It should be noted

that under Hn

α - lim Pr(AIC(H｡トAIC{Hx) ≧ 0￨#n)

tiSコ声E =0.07864 -･. Tabl丑 2

α ノ軒下Jei/サ-l n  α ノ2(n-1)ノ両

. 1100      1. 3309 10 . 0929      1. 3759 15 . 0878      1. 3894 20 . 0854      1. 3958 25 . 0840      1. 3996 30  . 0831     1. 4021 35  . 0825     1. 4039 40  . 0820     1. 4052 5げ  . 0813     1. 4070 100  . 0800     1. 4106

3.2 Two sided case

Our aim is to estimate ju2 where two alternative models are given as follows; ●

Model Ho: ^-0,

Model Hx′: pl幸0,

When a2 is an unknown parameter and p is a known parameter, we have the following

●

relation.

(16)

AIC{Ho)-AIC{H^) <b

⇔悔(n_1)¥< ^軒コうV評言-1.

Therefore the significance level α is exactly

●

(17)   α - ¥-Pr{¥un_l) l < /軒=力作諦=丁)

- 1-IX-.-W÷,サーl).

4. Estimation of correlation coefficient

Lettherandomsamples tribution^rMf^m ¥[x2J¥paicr2剖

(｣)>-...戟)

be taken from a bivariate normal dis-where /uv 〝2, ox and a2 are unknown prameters. Our

aim is to estimate the correlation coe瓜cient

Model Ho¥ p-0,

Effective Significance Level

Then the preliminary test estimator p of p is ●

β-0 if Ho is accepted,

r if Hl is accepted,

● ナ‡

∑ (xi-X) {Yi-Y)

i-l

By simple computations, we have

(19)       AIC{Ho)-AIC{Hx) <Q

⇔回</

1-e-2/n

When Ho is true, the significance level α is exactly ●

(20)     α - 1-Pr(鶴-z¥ < y五二百/節仁子)

- l-h-e2′n(-i,穿) ･

Next when two models are given as follows;

●

Model Ho′: p-po

Model Ex′: p幸p｡  where po (幸0) is the known constant. 仇e preliminary test estimator β of 〟 lS●

(21)

β-i

po if Ho'is accepted,

r if fft′ is accepted.

By tedious computations we get the following relation.

● (22)   AIC{Ho′トAICiH, ′) <0 ォ* pL<r<pu where - ip｡-a-p｡*) *′ ye2/有-1 } ,

e2/"+ 1-e2/>02

PL=二 Pu=

e2/"+(l-e2/ w

{/O｡+ a-zvv/n V盲申こす).

17

When #｡ is true, the significance level α is

●

α - l-Pr{pL<r<pu).

We can get values of α by referring to the table by David [4]. These are shown in Table 3, including the case of p0-0. It should be noted that under HQ in both cases

18

23

K. Inada and F. Kasagi

α - lim Pr{AIC(H｡)-AIC(HJ≧O IH｡)

nう∞ - lim Pr {AIC(H｡′トAICiH,′)≧0￨ #｡′) nー∞ =0.1572-･. 6 Table 3 10   15 O   < M   * r H   < ｣ >   o O c 3 3 ●         ●         ●         ●         ●         ● O O O O O O C O c o o o c o C 一 q ハ 一 c o m f n c o m r n 1 1 1 1 1 1 ● ● ● ● ● ● O O O O O O C O b -b -CO -CO -CO -CO -CO -CO 1 1 1 1 1 1 ● ● ● ● ● ● ペ3 0 0) 05 OO b-0 0   0 0   ! >   t -  t -  t -1   1   1   1   1   1 ● ● ● ● ● ● <^> <X> サO サO'生 Q一 〇〇 〇〇 〇O OO 00 00 1 1 1 1 1 1 ● ● ● ● ● ● c o c o   < x >   1 0 -< *   i -i O 5   0 5   O i O 5   C 7 5   C 3 2 1   1   1   1   1   1 ●         ●         ●         ●         ●         ● 0   0   0 3   0 0   < ゥ   r H C < 1   C < I t -I t -I i -H r -I Q 一 ( M   < M   < M O 一 q 一 ● ● ● ● ● ● 0   0 5   O O N r H c ! 一 ^   C O C O C O C O c O S t M * ｣ ]   琶   q m S S l   囲 ● ● ● ● ● ● ^   < D I O C O o O   ォ D t -  サ ^   サ > . t -  C D   < ^ > ( M   ( M   < M C q W   ( M ● ● ● ● ● ● 3 Q一O <^> <35"** t -  ｣ -  t -  C O I O I O C O C O c O c O C O C O

The authors are deeply indebted to Professor A. Kudo of Kyushu University for

his helpful advices and encouragements.

References

[1] Akaike, H. (1973). Infbrmation 仇eory and an extension of the maximum likelmood principle, 2nd International Symposium on Information Theory, B.N. Petrov and F. Csaki,

●

eds., Akademiai Kiado, Budapest, 267-281.

[2] Asano, C. (1960). Estimation after preliminary tests of significance and their applications to biometrical researches, Bull. Math. Statist., 9,ト14.

[3] Bancroft, T.A. (1944). On biases in estimation due to the use of preliminary tests of significance, Ann. Math. Statist., 15, 190-204.

●

[4] David, F.N. (1938). Tables of the ordinates and probability integral of the distribution of the correlation coe侃cient in small samples, Cambridge University Press, London. [5] Hirano, K. (1978). On level of significance of the preliminary test in pooling means, Ann.

Inst. Statis. Math., 30, A, 1-8.

[6] Kitagawa, T. (1963). Estimation after preliminary tests of significance, Univ. Calif. Pub. Statist., 3, 147-186.

[7] Sawa, T. and Hiromatsu, T. (1973). Minimax regret significance points for a preliminary test in regression analysis, Econometrica, 41, 1093-1101.