Scietiae Mathematicae Japoicae Olie, e-004, 65 89 65 SECOND ORDER ASYMPTOTIC PROPERTIES OF A CLASS OF TEST STATISTICS UNDER THE EXISTENCE OF NUISANCE PARAMETERS Keichiro TAMAKI Received May 7 004; revised Jue 10, 004 Abstract. Uder the existece of uisace parameters, we cosider a class of tests S which cotais the likelihood ratio, Wald ad Rao s score tests as special cases. To ivestigate the ifluece of uisace parameters, we derive the secod order asymptotic expasio of the distributio of T S uder a sequece of local alteratives. This result ad cocrete examples illumiate some iterestig features of effects due to uisace parameters. Optimum properties for a modified likelihood ratio test proposed i Mukerjee [8] are show uder the criteria of secod order local maximiity. 1. Itroductio. I multivariate aalysis, the secod order asymptotic powers of various test statistics have bee ivestigated by Hayakawa [5], ad Harris ad Peers [4]. Uder the absece of uisace parameters, results o optimality are ow kow for the likelihood ratio (LR) test i terms of secod order local maximiity ad Rao s score (R) test i terms of third order local average power (Mukerjee [9]). Uder the existece of uisace parameters, Eguchi [3] studied the effect of the composite ull hypothesis from a geometric poit of view. Mukerjee [8] suggested a test that is superior to the usual LR test with regard to secod order local maximiity. The test proposed i Mukerjee [8] is motivated from the priciple of coditioal likelihood ad also from that of adjusted likelihood. I time series aalysis, uder a set-up ivolvig a ukow scalar parameter, Taiguchi [1] cosidered the problem of secod order compariso of tests. He worked with a large class of tests that cotais LR, R ad Wald s (W) tests as special cases. Taiguchi [13] showed that the local powers of all the modified tests which are secod order asymptotically ubiased are idetical up to 1/. Also Taiguchi [14] cosidered the problem of third order compariso of tests, ad suggested a Bartlett-type adjustmet for the tests i the class ad the, o the basis of such adjusted versios, explored the poit-by-poit maximizatio of third order power. Bartlett-type adjustmet procedure has bee elucidated i various directios. Cordeiro ad Ferrari [] gave a geeral formula of Bartlett-type adjustmet to order 1 for the test statistic whose asymptotic expasio is a fiite liear combiatio of chi-squared distributio with suitable degrees of freedom. Kakizawa [6] cosidered the extesio of Cordeiro ad Ferrari s [] adjustmet to the case of order k, where k is a iteger k 1. Rao ad Mukerjee [10] compared various Bartlett-type adjustmets for the R statistic. Rao ad Mukerjee [11] addressed the problem of comparig the higher order power of tests i their origial forms ad ot via their bias-corrected or Bartlett-type adjusted versios. I this paper, uder the existece of uisace parameters, we cosider the secod order properties of a class of tests S which cotais LR, R ad W tests as special cases. If uisace parameters are preset, sesitivity of test statistics to perturbatio of the uisace parameters becomes importat. It is show that the powers ad sizes of T S are equally 000 Mathematics Subject Classificatio. 6F05; 6H15; (6N99). Key words ad phrases. local maximiity; local power; local ubiasedess; uisace parameters; secod order.
66 K. TAMAKI sesitive to perturbatio of the uisace parameter. I Sectio 3 we compare the secod order local power. It is see that the local average powers of all T S are idetical. It is show that optimality properties hold for a modified test of the LR test i terms of secod order local maximiity. Sectio 4 provides a decompositio formula of local powers for LR, R ad W test statistics uder local orthogoality for parameters. The decompositio cosists of the sum of the three parts; oe is the local power for the case of kow uisace parameters, aother represets sesitivity to perturbatio of uisace parameters ad the other part ca be iterpreted as a effect of uisace parameters i test statistics. I Sectio 5, we discuss the local ubiasedess of T S. The results ad their examples illumiate some iterestig features of effects due to uisace parameters. The proofs of theorems are relegated to Sectio 6.. Asymptotic expasio of a class of tests. Let X = (X 1,...,X ) be a collectio of m-dimesioal radom vectors formig a stochastic process. Let p (x ; θ), x R m, be the probability desity fuctio of X, where θ = (θ 1,...,θ p+q ) Θ a ope subset of R p+q. Let θ 1 =(θ 1,...,θ p ) be the p-dimesioal parameter of iterest ad θ =(θ p+1,...,θ p+q ) be the q-dimesioal uisace parameter. We cosider the problem of testig the hypothesis H : θ 1 = θ 10, where θ 10 =(θ0 1,...,θp 0 ), agaist the alterative A : θ 1 θ 10. For this problem we itroduce a class of test S which cotais LR, W ad R tests as special cases. I the presece of uisace parameters, the powers ad sizes of T S are affected by the true but ukow uisace parameter. Therefore we ivestigate the ifluece of perturbatio by the sequece of local alteratives θ = θ 0 + c 1 ε where θ 0 =(θ 10,θ 0 ), θ 0 =(θ p+1 0,...,θ p+q 0 ) ad ε =(ε 1,...,ε p+q ). As i Li [7], we shall use Greek letters {α,β,γ,...} as idices that ru from 1 to p + q, the set of Eglish letters {i,j,k,...,q} as idices that ru from 1 to p, ad the set of {r,s,t,...,z} as idices that ru from p +1top + q. The idices i, r ad α will serve two purposes, first to deote a typical term i a sum ad secod to idicate the rage of a sum. For example, a α X α = p+q α=1 a αx α, a i X i = p i=1 a ix i ad a r X r = p+q r=p+1 a rx r. We make the followig assumptios: (A-1) l (θ) = log p (X ; θ) is cotiuously four times differetiable with respect to θ. (A-) The partial derivative α = / θ α ad the expectatio E θ with respect to p (x ; θ) are iterchageable. (A-3) For a appropriate sequece {c } satisfyig c + as +, the asymptotic momets (cumulats) of Z α (θ) =c 1 α l (θ), Z αβ (θ) =c 1 [ α β l (θ) E θ { α β l (θ)}], possess the followig asymptotic expasios E θ {Z α (θ)z β (θ)} = I (αβ) (θ)+o(c ), E θ {Z α (θ)z βγ (θ)} = J α,βγ (θ)+o(c ), E θ {Z α (θ)z β (θ)z γ (θ)} = c 1 K α,β,γ(θ)+o(c 3 ), ad J-th-order (J ) cumulats of Z α (θ) ad Z αβ (θ) are all O(c J+ ). (A-4) (i) I (αβ) (θ) is cotiuously two times differetiable with respect to θ.
SECOND ORDER ASYMPTOTIC PROPERTIES 67 (ii) J α,βγ (θ) ad K α,β,γ (θ) are cotiuously differetiable fuctios. (A-5) (i) I(θ) ={I (αβ) (θ)} is positive defiite for all θ Θ. (ii) L(θ) ={ c α β l (θ)} is positive defiite almost surely for all θ Θ. Let ˆθ =(ˆθ 1,...,ˆθ p+q ) be the global maximum likelihood estimator of θ ad θ = ( θ p+1,..., θ p+q ) be the restricted maximum likelihood estimator of θ give θ 1 = θ 10. The partitio θ =(θ 1,θ ) iduces the followig correspodig partitios ( ) ( ) ( ) ( ) ˆθ1 ε1 I11 (θ) I ˆθ =, ε =, I(θ) = 1 (θ) L11 (θ) L, L(θ) = 1 (θ). ˆθ ε I 1 (θ) I (θ) L 1 (θ) L (θ) Let g(θ) ={g αβ (θ)} = ( I11 (θ) I 1 (θ) 0 I (θ) ), where I 11 (θ) =I 11 (θ) I 1 (θ){i (θ)} 1 I 1 (θ). We cosider the trasformatio W i (θ) =Z i (θ) I (ir) (θ)g rs (θ)z s (θ), W r (θ) =Z r (θ), W αβ (θ) =Z αβ (θ) J γ,αβ (θ)i γδ (θ)z δ (θ), where I αβ (θ) ad g αβ (θ) are the (α, β) compoet of the iverse matrix of I(θ) ad g(θ), respectively. Heceforth we use the simpler otatios Z α, W α, I (αβ), K α,β,γ, etc. if Z α (θ), W α (θ), I (αβ) (θ), K α,β,γ (θ), etc. are evaluated at θ = θ 0. Ay fuctio evaluated at the poit θ = ˆθ will be distiguished by the additio of a circumflex. Similarly ay fuctio evaluated at the poit θ 1 = θ 10, θ = θ will be distiguished by the additio of a tilde. For the testig problem H : θ 1 = θ 10 agaist the alterative A : θ 1 θ 10, we itroduce the followig class of tests: (1) S = {T T = g ij W i W j + c 1 a 1g iα g jβ W αβ W i W j +c 1 giα g rs W αr W i W s + c 1 a ijk W iw j W k c 1 g iα g jβ g rs K α,β,r W i W j W s c 1 giα g rt g su (K α,r,s + J α,rs )W i W t W u + c 1 ai 3 W i + o p (c 1 ), uder H, where a 1,a ijk ad a i 3 are oradom costats}. This class S is a very atural oe. We ca show that famous tests based o the maximum likelihood estimator belog to S. Example 1. (i) The likelihood ratio test LR = (ˆl l ) belogs to S. I fact, from Bickel ad Ghosh [1], the expasio for the r-th compoet of c 1 (ˆθ θ ) is give by () c 1 (ˆθ r θ r )=η r + c 1 ĝ rs Ẑ sα η α + 1 c 1 ĝ rs ( ˆK s,α,β + Ĵs,αβ[3])η α η β + o p (c 1 ), where η i = c 1 (ˆθ i θ0 i ), ηr = ĝ rs Î (si) η i ad Ĵα,βγ[3] = Ĵα,βγ + Ĵβ,γα + Ĵγ,αβ. Expadig LR i a Taylor series at θ = ˆθ ad otig (), we obtai ( ) (ˆl l 1 (3) )=ĝ ij η i η j c 1 Ẑαβη α η β c 1 3 ˆK α,β,γ + Ĵα,βγ η α η β η γ + o p (c 1 ).
68 K. TAMAKI By Taylor expasio aroud θ 0, (4) ĝ ij = g ij + g ik g kα g jl g lβ (K α,β,γ + J α,βγ + J β,γα )(ˆθ γ θ γ 0 )+o p(c 1 ). Furthermore, the stochastic expasio of c 1 (ˆθ α θ0 α ) is give by (5) c 1 (ˆθ α θ0 α )=g βα W β + c 1 I αβ g δγ W βγ W δ 1 c 1 Iαα g ββ g γγ (K α,β,γ + J α,β γ )W βw γ + o p (c 1 ). Isertig (4) ad (5) i (3) ad otig Ẑαβ = W αβ + o p (1), we have (ˆl l )=g ij W i W j + c 1 g iα g jβ W αβ W i W j +c 1 g iα g rs W αr W i W s 1 3 c 1 giα g jβ g kγ K α,β,γ W i W j W k c 1 giα g jβ g rs K α,β,r W i W j W s c 1 giα g rt g su (K α,r,s + J α,rs )W i W t W u + o p (c 1 ). Hece, LR belogs to S with the coefficiets a 1 =1,a ijk = g iα g jβ g kγ K α,β,γ /3 ad a i 3 =0. Similarly, we ca get results (ii) (v): (ii) Wald s test W 1 =ĝ ij η i η j belogs to S with the coefficiets a 1 =,a ijk = g iα g jβ g kγ J α,βγ ad a i 3 =0. (iii) A modified Wald s test W = g ij η i η j belogs to S with the coefficiets a 1 =, a ijk = g iα g jβ g kγ (K α,β,γ + J α,βγ ) ad a i 3 =0. (iv) Rao s score test R 1 = ĝ ij Zi Zj belogs to S with the coefficiets a 1 =0,a ijk = g iα g jβ g kγ (K α,β,γ +J α,βγ ) ad a i 3 =0. (v) A modified versio of Rao s score test R = g ij Zi Zj belogs to S with the coefficiets a 1 =0,a ijk = 0 ad a i 3 =0. Furthermore, it is show that modified versios of the four tests W 1,W,R 1 ad R which are based o the observed iformatio belog to S. Let {l ij (θ)} = L 11 (θ) = L 11 (θ) L 1 (θ){l (θ)} 1 L 1 (θ) ad {l ij (θ)} be the (i, j) compoet of the iverse matrix of L 11 (θ). (vi) A modified versio of Wald s test W 3 = ˆl ij η i η j belogs to S with the coefficiets a 1 =1,a ijk = g iα g jβ g kγ J α,βγ ad a i 3 =0. (vii) A modified versio of Wald s test W 4 = l ij η i η j belogs to S with the coefficiets a 1 =1,a ijk = g iα g jβ g kγ (K α,β,γ +J α,βγ ) ad a i 3 =0. (viii) A modified versio of Rao s score test R 3 = ˆl ij Zi Zj belogs to S with the coefficiets a 1 =1,a ijk = g iα g jβ g kγ (K α,β,γ +J α,βγ ) ad a i 3 =0. (ix) A modified versio of Rao s score test R 4 = l ij Zi Zj belogs to S with the coefficiets a 1 =1,a ijk = g iα g jβ g kγ J α,βγ ad a i 3 =0.
SECOND ORDER ASYMPTOTIC PROPERTIES 69 (x) The test LR =LR+c 1 giα g rs ( K α,r,s + J α,rs ) Z i proposed i Mukerjee [8] belogs to S with the coefficiets a 1 =1,a ijk = g iα g jβ g kγ K α,β,γ /3 ad a i 3 = giα g rs (K α,r,s + J α,rs ). Li [7] compared the sesitivities of LR, W ad R statistics to uisace parameters. I the oe-parameter case, Taiguchi [14] discussed the third order asymptotic properties of a class of tests S 1. Rao ad Mukerjee [11] studied a wider class S ( S 1 ) which eables us to compare the various Bartlett-type adjustmets available for the members of S 1. Our class S cotais S 1 ad S, hece the class S is sufficietly rich. Remark 1. Test statistics i Example 1 are based o the maximum likelihood estimator. From () ad (3), these statistics ca be writte as (6) T =ĝ ij η i η j + c 1 b 1Ẑαβη α η β + c 1 (b ˆK α,β,γ + b 3 Ĵ α,βγ )η α η β η γ + c 1 b 4 ĝ rs ( ˆK α,r,s + Ĵα,rs)η α + o p (c 1 ), where the coefficiet (b 1,b,b 3,b 4 ) R 4. For these statistics, (7) b 1 = 1, b = 1/3, b 3 = 1, b 4 =0, for LR, b 1 = 1, b = 1/3, b 3 = 1, b 4 =1, for LR, b 1 =0, b =0, b 3 =0, b 4 =0, for W 1, b 1 =0, b = 1, b 3 =, b 4 =0, for W, b 1 = 1, b =0, b 3 =0, b 4 =0, for W 3, b 1 = 1, b = 1, b 3 = 3, b 4 =0, for W 4, b 1 =, b = 1, b 3 = 3, b 4 =0, for R 1, b 1 =, b =0, b 3 = 1, b 4 =0, for R, b 1 = 1, b = 1, b 3 = 3, b 4 =0, for R 3, b 1 = 1, b =0, b 3 =0, b 4 =0, for R 4. Isertig (4) ad (5) i (6), we obtai (8) T = g ij W i W j + c 1 (b 1 +)g iα g jβ W αβ W i W j +c 1 giα g rs W αr W i W s + c 1 giα g jβ g kγ {b K α,β,γ +(b 3 +1)J α,βγ }W i W j W k c 1 g iα g jβ g rs K α,β,r W i W j W s c 1 g iα g rt g su (K α,r,s + J α,rs )W i W t W u + c 1 b 4 g iα g rs (K α,r,s + J α,rs )W i + o p (c 1 ). The class S i (1) is motivated from (8). First, we give the secod-order asymptotic expasio of the distributio fuctio of T S uder a sequece of local alteratives. This result ca be applied to the i.i.d. case, multivariate aalysis ad time series aalysis. Let G µ,ν (z) is the distributio fuctio for a o-cetral chi-square variate with degree of freedom µ ad o-cetrality parameter ν. Theorem 1. The distributio fuctio of T S uder a sequece of local alteratives θ = θ 0 + c 1 ε has the asymptotic expasio P θ0+c 1 ε [T<z]=G p, (z)+c 1 3 j=0 m j G p+j, (z)+o(c 1 ),
70 K. TAMAKI where m 3 = 1 6 K α,β,γd α d β d γ + 1 aijk g ii g jj g kk di d j d k, m = 1 aijk g ii g jj g kk di d j d k + 1 Bαβ K α,β,γ d γ + 1 aijk [3]g ilg jk d l, m 1 = 1 J α,βγd α d β d γ 1 (K α,β,r + J α,βr + J β,αr )d α d β (d r ε r ) (9) 1 Bαβ K α,β,γ d γ 1 aijk [3]g ilg jk d l 1 grs (K α,r,s + J α,rs )d α + 1 ai 3 g ijd j, m 0 = 1 6 (K α,β,γ +3J α,βγ )d α d β d γ + 1 (K α,β,r + J α,βr + J β,αr )d α d β (d r ε r ) + 1 grs (K α,r,s + J α,rs )d α 1 ai 3g ij d j, =g ij ε i ε j, d α = g ij g jα ε i, a ijk [3] = aijk + a jki + a kij ad ( ) 0 0 {B αβ } = {I αβ } 0 (I ) 1. Secod, we cosider the sesitivity of T S to the chage ε i the uisace parameter. Test statistics that are less sesitive to such chages are geerally more desirable because their sizes ad powers are less affected by the estimatio of the uisace parameter. The we have Theorem. (i) For T S, the sesitivity of the distributio fuctio of T to uisace parameters is give by P θ0+c 1 ε [T<z] P θ 10+c 1 ε 1,θ 0 [T<z] = 1 c 1 (K α,β,r + J α,βr + J β,αr )d α d β ε r {G p+, (z) G p, (z)} + o(c 1 ). (ii) If (10) g ii g i α g jj g j α (K α,β,r + J α,βr + J β,αr )=0, is satisfied, the the distributio fuctio of T S is asymptotically idepedet of ε with a error o(c 1 ). Remark. Note that r g ij (θ) =g ii (θ)g i α (θ)g jj (θ)g j α (θ){k α,β,r (θ)+j α,βr (θ)+j β,αr (θ)}. If g ij (θ) is idepedet of θ, the the coditio (10) holds. Remark 3. I the case of i.i.d. observatios, Li [7] gave factorizatios of LR, W ad R test statistics as quadratic forms ad compared desity fuctios of these factors. The he showed that the powers ad sizes of these statistics are equally sesitive to uisace parameters. Form (i) i Theorem, we ca see that the powers ad sizes of all T S are equally sesitive to uisace parameters. Hece, our results agree with that of Li [7].
SECOND ORDER ASYMPTOTIC PROPERTIES 71 Example. Suppose that X i, i = 1,..., are i.i.d. N 1 (µ, σ ). radom variables distributed as (i) If θ 1 = σ ad θ = µ, the g 11 (σ,µ)=(σ 4 ) 1. Hece, the coditio (10) holds. (ii) If θ 1 = µ ad θ = σ, the g 11 (µ, σ )=(σ ) 1. Hece, the coditio (10) does ot hold. Example 3. Cosider the oliear regressio model (11) X t = α + β cos(t 1)λ + u t, t =1,...,, where θ 1 = β, θ =(α, λ), λ =πl/ (l a iteger), {u t } is a sequece of i.i.d. N(0,σ ) radom variables. The it follows that I(θ) = 1/(σ ) 0 β/(4lσ ) (1) 0 1/σ β/(lσ ). β/(4lσ ) β/(lσ ) β (8π l 3)/(1l σ ) For our model (11) we calculate g 11 (θ). From (1) g 11 (θ) = 1 σ 3 4σ (8π l 15) which implies that the coditio (10) does ot hold. 3. Compariso of power. Takig ε 1 = 0 i (9), it ca be see that all T S have sizes α + o(c 1 ). Hece, it would be meaigful to compare T S i terms of power up to o(c 1 ). From Theorem 1, we ca see that there is o test which is secod order uiformly most powerful i S. Thus we attempt to compare the tests i S o the basis of their secod order power. First, we derive the explicit formula to compare the local power of T S. Note that the first order powers of all T S are idetical ad idepedet of ε. Write the power fuctio of T S uder θ 0 +c 1 ε as P T (ε) =P 1 (ε 1 )+c 1 P T (ε)+o(c 1 ). From Theorem 1, we ca state Theorem 3. For T 1 ad T S with the coefficiet (a 11,a ijk 1,ai 31) ad (a 1,a ijk,ai 3), respectively, where P T1 T (ε) P (ε) = m j {G p+j, (z) G p+j+, (z)}, j=0 m = 1 (aijk 1 aijk )g ii g jj g kk di d j d k, m 1 = 1 (aijk 1 [3] aijk [3])g ilg jk d l, m 0 = 1 (ai 31 ai 3 )g ijd j. Note that m, m 1 ad m 0 are idepedet of ε. From Theorem 3 we have
7 K. TAMAKI Corollary 1. For T 1 ad T S with the coefficiet (a 11,a ijk 1,ai 31 ) ad (a 1,a ijk,ai 3 ), respectively, P T1 (ε 1, 0) P T (ε 1, 0) = m j{g p+j, (z) G p+j+, (z)}, j=1 where m, m 1 ad m 0 are the same as Theorem 3. Example 4. Suppose that X i, i =1,..., are i.i.d. radom variables distributed as ( )) 1 ρ N (µ,. ρ 1 The parametric orthogoality holds. If θ 1 = ρ ad θ = µ, the (13) g 11 (ρ, µ) = 1+ρ (1 ρ ), K 6ρ +ρ3 1,1,1(ρ, µ) = (1 ρ ) 3, J 1,11(ρ, µ) = K 1,1,1 (ρ, µ), g (ρ, µ) = 1+ρ, K 1,,(ρ, µ) = (1 + ρ), J 1,(ρ, µ) =0. For test statistics T 1 ad T i (7) with the coefficiet (b 11,b 1,b 31,b 41 ) ad (b 1,b,b 3,b 4 ), respectively, m 3ρ + ρ3 = (1 ρ ) 3 {(b 1 b ) (b 31 b 3 )}(ε 1 ) 3, m 1 = 3 3ρ + ρ 3 (1 ρ )(1 + ρ ) {(b 1 b ) (b 31 b 3 )}ε 1, m 0 = 1 (1 + ρ) (b 41 b 4 )ε 1. Based o the above we ca compare the secod order power amog W i (i =1,, 3, 4), R i (i =1,, 3, 4), LR ad LR. (i) (i)-1 If ρ>0 ad ε 1 > 0, the (ε) =P R1 R3 (ε) =P (ε) <PW <P W1 (ε) =P W3 (ε) =P R4 (ε). P W4 (i)- If ρ<0 ad ε 1 > 0, the (ii) LR versus LR, P W1 (ε) =P W3 (ε) =P R4 (ε) =P R1 <P W4 (ε) =P R (ε) <PLR (ε) (ε) <PLR (ε) <P W (ε) =P R (ε) (ε) =P R3 (ε). P LR 1 (ε) P LR (ε) = (1 + ρ) ε 1{G p, (z) G p+, (z)} implies, for ε 1 ρ = 1. > 0, P LR (ε) <P LR (ε) ad P LR ( ε 1,ε ) <P LR ( ε 1,ε ) uless
SECOND ORDER ASYMPTOTIC PROPERTIES 73 From (13) i Example 4, it is see that cumulats g 11, K 1,1,1 ad J 1,11 ted to as ρ ±1, ad g ad K 1,, ted to as ρ 1. Hece, we eed to ispect secod order power fuctios if ρ is close to ±1. Note the relatio (14) G p, (z) G p+, (z) =f p+, (z), where f p, (z) is the probability desity fuctio of o-cetral chi-square variate with p degree of freedom ad o-cetrality parameter. From (13) ad (14) it follows that secod order powers of all test statistics i Example 1 coverge to 0 as ρ ±1at each fixed ε 1. I Figure 1, we plotted P LR (solid lie), P LR (dotted lie), P R1 (dashed lie) ad P W1 (dash-dotted lie) of Example 4 with α =0.05, ε 1 = 1 ad 1 <ρ<1. Figure 1 illustrates that secod order powers of these statistics coverge to 0 as ρ ±1. I Figure, we plotted P LR (solid lie), P LR (dotted lie), P R1 (dashed lie) ad P W1 (dash-dotted lie) of Example 4 with α =0.05, ε 1 =0.1 ad 1 <ρ<1. We ca see that the extreme poits is close to ±1 i compariso with Figure 1. Figures 1 ad are about here. Example 5. Let {X t } be a Gaussia MA(1) process with the spectral desity f θ (λ) = σ π 1 ψeiλ. If θ 1 = ψ ad θ = σ, the, g 11 (ψ, σ 1 )= 1 ψ, K 1,1,1(ψ, σ 6ψ )= (1 ψ ), J 1,11(ψ, σ 4ψ )= (1 ψ ), (15) g (ψ, σ )= 1 σ 4, K 1,,(ψ, σ )=J 1, (ψ, σ )=0. Note that g (K 1,, + J 1, ) = 0. For test statistics T 1 ad T i (7) with the coefficiet (b 11,b 1,b 31,b 41 ) ad (b 1,b,b 3,b 4 ), respectively, m = ψ (1 ψ ) {3(b 1 b ) (b 31 b 3 )}(ε 1 ) 3, m 1 = 3ψ (1 ψ ) {3(b 1 b ) (b 31 b 3 )}ε 1, m 0 =0. Based o the above we ca compare the secod order power amog W i (i =1,, 3, 4), R i (i =1,, 3, 4), LR ad LR. (i) If ψ>0 ad ε 1 > 0, the P W4 (ε) =P R1 (ii) If ψ<0 ad ε 1 > 0, the P W1 (ε) =P W3 (ε) =P R3 (ε) <PR (ε) <PLR (ε) =P LR (ε) =P W (ε) <P W1 (ε) =P W3 (ε) =P R4 (ε). (ε) =P R4 <P W4 (ε) =P R1 (ε) <PLR (ε) =P LR (ε) =P W (ε) <P R (ε) (ε) =P R3 (ε).
74 K. TAMAKI From (15) i Example 5, it is see that cumulats g 11, K 1,1,1 ad J 1,11 ted to as ψ ±1. Hece, we eed to examie secod order powers if ψ is close to ±1. From (14) ad (15) it follows that secod order powers of all test statistics i Example 1 coverge to 0asψ ±1 at each fixed ε 1. I Figure 3, we plotted P LR (solid lie), P R1 (dashed lie) ad P W1 (dotted lie) of Example 5 with α =0.01, ε 1 =6.5 ad 1 <ψ<1. From Figure 3 we observe that secod order powers of these statistics coverge to 0 as ψ ±1. I Figure 4, we plotted P LR (solid lie), P R1 (dashed lie) ad P W1 (dotted lie) of Example 5 with α =0.01, ε 1 =0.65 ad 1 <ψ<1. We ca see that the extreme poits is close to ±1 i compariso with Figure 3. Figures 3 ad 4 are about here. Example 6. Let {X t } be a Gaussia AR(1) process with the spectral desity f θ (λ) = σ 1 π 1 ρe iλ. If θ 1 = ρ ad θ = σ, the g 11 (ρ, σ )= 1 1 ρ, K 1,1,1(ρ, σ 6ρ )= (1 ρ ), J 1,11(ρ, σ ρ )= (1 ρ ), (16) g (ρ, σ )= 1 σ 4, K 1,,(ρ, σ )=J 1, (ρ, σ )=0. Note that g (K 1,, + J 1, ) = 0. For test statistics T 1 ad T i (7) with the coefficiet (b 11,b 1,b 31,b 41 ) ad (b 1,b,b 3,b 4 ), respectively, m = ρ (1 ρ ) {3(b 1 b ) (b 31 b 3 )}(ε 1 ) 3, m 1 = 3ρ (1 ρ ) {3(b 1 b ) (b 31 b 3 )}ε 1, m 0 =0. Based o the above we ca compare the secod order power amog W i (i =1,, 3, 4), R i (i =1,, 3, 4), LR ad LR. (i) If ρ>0 ad ε 1 > 0, the (ii) If ρ<0 ad ε 1 > 0, the P W (ε) <P LR (ε) =P LR (ε) =P W1 (ε) =P W3 (ε) =P W4 (ε) = P R1 (ε) =P R3 R4 (ε) =P (ε) <PR (ε). P R (ε) <PLR (ε) =P LR (ε) =P W1 (ε) =P W3 (ε) =P W4 (ε) = P R1 (ε) =P R3 R4 (ε) =P (ε) <PW (ε). From (16) i Example 6, it is see that cumulats g 11, K 1,1,1 ad J 1,11 ted to as ρ ±1. Hece, we eed to examie secod order powers if ρ is close to ±1. From (14) ad (16) it follows that secod order powers of all test statistics i Example 1 coverge to 0 as ρ ±1 at each fixed ε 1.
SECOND ORDER ASYMPTOTIC PROPERTIES 75 I Figure 5, we plotted P LR (solid lie), P R (dashed lie) ad P W (dotted lie) of Example 6 with α =0.01, ε 1 = 3 ad 1 <ρ<1. From Figure 5 it is see that secod order powers of these statistics coverge to 0 as ρ ±1. I Figure 6, we plotted P LR (solid lie), P R (dashed lie) ad P W (dotted lie) of Example 6 with α =0.01, ε 1 =0.8 ad 1 <ρ<1. We ca see that the extreme poits is close to ±1 i compariso with Figure 5. Figures 5 ad 6 are about here. Next we cosider the criterio of average power P T (ε 1,ε )+P T ( ε 1,ε ). The from Theorem 1 it is easily see that for each T S, P T (ε 1,ε )+P T ( ε 1,ε )=(K α,β,r + J α,βr + J β,αr )d α d β ε r {G p, (z) G p+, (z)}. It is, therefore, clear that the average powers of all T S are idetical up to c 1. However, eve i this situatio, with a more detailed aalysis it is possible to compare tests i S i a meaigful way uder suitable choice of criterio. Uder the absece of uisace parameters, Mukerjee [9] showed that LR statistic is optimal i terms of secod-order local maximiity. However, i the presece of uisace parameters, optimality properties do ot geerally hold for LR test i terms of secod-order local maximiity. We ca see the optimality of LR statistic i terms of secod-order local maximiity. For each fixed, let Pε T ( ) = mi P T (ε), PLR ε ( ) = mi P LR (ε), where the miimum is take over ε 1 such that g ij ε i ε j =. The we ca get the followig result. Theorem 4. For T S whose coefficiets do ot satisfy z(a ijk [3]g jk + g iα B βγ K α,β,γ )+ (p +){a i 3 giα g rs (K α,r,s + J α,rs )} = 0 (the coefficiets of LR satisfy a ijk [3]g jk = g iα B βγ K α,β,γ ad a i 3 = giα g rs (K α,r,s + J α,rs )), there exists a positive 0 such that Pε T ( ) <Pε LR ( ), wheever 0 < < 0. Example 7. (i) I Example 4, W 1,W 3 ad R 4 are most powerful i Example 1 except LR at each fixed ε 1 > 0 ad ρ>0 with a error o(c 1 ). Hece, we compare W 1 ad LR tests i terms of secod-order local maximiity. Note that the coditio (10) holds. From Theorem 1 ad Example 4, { 3ρ + P LR ρ3 1 (ε) = (1 ρ ) 3 (ε 1) 3 3 G 5, (z) G 3, (z)+ } 3 G 1, (z), { 3ρ + P W1 ρ3 (ε) = (1 ρ ) 3 (ε 1) 3 3 G 7, (z)+g 5, (z) G 3, (z)+ } 3 G 1, (z) (3ρ + ρ 3 ) + (1 ρ )(1 + ρ ) ε 1{ G 5, (z)+g 3, (z)} 1 + (1 + ρ) ε 1{G 3, (z) G 1, (z)}, where = (ε 1 ) (1 + ρ )/(1 ρ ).Ifρ =1/, 1 ad α =0.05, the Pε LR ( ) = P { (1 LR ρ ) 1/ } (1 + ρ ),ε 1/, Pε W1 ( ) = P W1 { (1 ρ ) 1/ } (1 + ρ ),ε 1/.
76 K. TAMAKI Thus we ca see wheever 0 < 1. P W1 ε ( ) <Pε LR ( ), (ii) If ρ<0 ad ε 1 > 0, the R 1,R 3 ad W 4 are most powerful i Example 1 except LR with a error o(c 1 ). Hece, we compare R 1 ad LR tests i terms of secod-order local maximiity. The { 3ρ + P R1 ρ3 4 (ε) = (1 ρ ) 3 (ε 1) 3 3 G 7, (z) G 5, (z) G 3, (z)+ } 3 G 1, (z) 4(3ρ + ρ 3 ) + (1 ρ )(1 + ρ ) ε 1{G 5, (z) G 3, (z)} 1 + (1 + ρ) ε 1{G 3, (z) G 1, (z)}. If ρ = 1/, 1 ad α =0.05, the { (1 ρ ) 1/ } Pε LR ( ) = P LR (1 + ρ ),ε 1/, Pε R1 ( ) = P R1 { (1 ρ ) 1/ } (1 + ρ ),ε 1/. Thus we ca see wheever 0 < 1. P R1 ε ( ) <Pε LR ( ), Example 8. (i) I Example 5, W 1,W 3 ad R 4 are most powerful i Example 1 at each fixed ε 1 > 0 ad ψ>0 with a error o(c 1 ). Hece, we compare W 1 ad LR test i terms of secod-order local maximiity. For MA(1) model i Example 5, the coditio (10) holds. From Theorem 1, we obtai P LR (ε) = ψ (1 ψ ) (ε 1) 3 {G 5, (z) G 3, (z)+g 1, (z)}, ψ P W1 (ε) = (1 ψ ) (ε 1) 3 { G 7, (z)+g 5, (z) G 3, (z)+g 1, (z)} + 3ψ 1 ψ ε 1{ G 5, (z)+g 3, (z)}, where = (ε 1 ) /(1 ψ ). If ψ =1/, 1 ad α =0.01, the we have } Pε LR ( ) = P {(1 LR ψ ) 1/ 1/,ε, } Pε W1 ( ) = P W1 { (1 ψ ) 1/ 1/,ε. Hece, wheever 0 < 1. P W1 ε ( ) <Pε LR ( ),
SECOND ORDER ASYMPTOTIC PROPERTIES 77 (ii) If ψ<0ad ε 1 > 0, the R 1,R 3 ad W 4 are most powerful i Example 1 at each fixed ε 1 > 0 ad ψ>0 with a error o(c 1 ). Hece, we compare R 1 ad LR test i terms of secod-order local maximiity. From Theorem 1, we get P R1 (ε) = ψ (1 β ) (ε 1) 3 {G 7, (z) G 5, (z) G 3, (z)+g 1, (z)} + 6ψ 1 ψ ε 1{G 5, (z) G 3, (z)}. If ψ = 1/, 1 ad α =0.01, the } Pε LR ( ) = P { (1 LR ψ ) 1/ 1/,ε, } Pε R1 ( ) = P R1 { (1 ψ ) 1/ 1/,ε. Hece, wheever 0 < 1. P R1 ε ( ) <Pε LR ( ), 4. Effect of uisace parameters. I this sectio, we cosider the case where the uisace parameter θ = θ 0 is kow. Let θ 1 =( θ 1,..., θ p ) be the maximum likelihood estimator of θ 1 uder θ = θ 0. Ay fuctio evaluated at the poit θ 1 = θ 1, θ = θ 0 will be distiguished by the additio of a horizotal bar. The the correspodig statistics with that i Example 1 are give by (17) LR 0 =LR 0 =( l l ), W 10 = Ī(ij)τ i τ j, W 0 = I (ij) τ i τ j, W 30 = L (ij) τ i τ j, W 40 = L (ij) τ i τ j, R 10 = Īij 0 Z iz j, R 0 = I ij 0 Z ij iz j, R 30 = L 0 Z iz j, R 40 = L ij 0 Z iz j, where τ i = c 1 ( θ i θ0 i ), {L (ij)(θ)} = L 11 (θ), ad I ij 0 (θ) ad Lij 0 (θ) are the (i, j) compoet of the iverse matrix of I 11 (θ) ad L 11 (θ), respectively. The stochastic expasios of test statistics i (17) are give by T 0 = I ij 0 Z iz j + c 1 (b 1 +)I0 ik Ijl 0 W kl Z iz j + c 1 I0 ii I jj 0 Ikk 0 {b K i,j,k +(b 3 +1)J i,j k }Z iz j Z k + o p (c 1 ), where the coefficiet (b 1,b,b 3 ) is the same as i (7) ad W ij = Z ij J k,ij I kl 0 Z l. Hece, we cosider the followig class of tests: S 0 = {T 0 T 0 = I ij 0 Z iz j + c 1 a 1I0 ik Ijl 0 W kl Z iz j + c 1 aijk Z iz j Z k + o p (c 1 ), uder H, where a 1 ad a ijk are oradom costats}. For simplicity we assume the local parametric orthogoality at θ = θ 0, amely (A-6) I ir =0 i =1,...,p, r = p +1,...,p+ q.
78 K. TAMAKI The the class S ca be writte as S = {T T = I ij 0 Z iz j + c 1 a 1 I0 ik I jl 0 W klz i Z j +c 1 I ij 0 grs W jr Z i Z s + c 1 aijk Z iz j Z k c 1 Iik 0 Ijl 0 grs K k,l,r Z i Z j Z s c 1 I ij 0 grt g su (K j,r,s + J j,rs )Z i Z t Z u + c 1 a i 3Z i + o p (c 1 ), uder H, where a 1,a ijk ad a i 3 are oradom costats}. Thus the compariso betwee T ad T 0 with the same coefficiet will illustrate what ifluece uisace parameters exert o the performace of test statistics. The we have the followig theorem. Theorem 5. (i) Uder (A-6), for T S ad T 0 S 0 with the same coefficiet, the distributio fuctios of T are decomposed ito (ii) If (18) P θ0+c 1 ε [T<z]=P θ 10+c 1 ε 1,θ 0 [T 0 <z] + 1 c 1 (K i,j,r + J i,jr + J j,ir )ε i ε j ε r {G p+, (z) G p, (z)} + 1 c 1 {I (ij)a j 3 grs (K i,r,s + J i,rs )}ε i {G p+, (z) G p, (z)} + o(c 1 ). (19) (0) K i,j,r + J i,jr + J j,ir =0, g rs (K i,r,s + J i,rs )=0, are satisfied, the the distributio fuctio of T S with a i 3 =0is equal to that of T 0 S 0 with the same coefficiet as T up to order c 1 Remark 4. The coditio (19) agree with (10) i Theorem uder (A-6). If the coditio (0) holds, the LR test is secod order asymptotically ubiased. Therefore, the third term of the right had i (18) ca be iterpreted as secod order local bias i the usual likelihood ratio test (see Mukerjee [8]). I Sectio 5, we will observe that this term ca also be iterpreted as a effect of uisace parameters i test statistics. Thus, we provide a decompositio formula of local powers for test statistics uder local orthogoality for parameters. Example 9. This example relates to the ratio of idepedet expoetial meas. Let p(x 1,x ; µ 1,µ )=(µ 1 µ ) 1 exp{ (µ 1 1 x 1 + µ 1 x )}, x 1,x > 0. (i) If θ 1 = µ 1 /µ ad θ =(µ 1 µ ) 1/, the parametric orthogoality holds ad g 11 (θ) = (θ 1 ) / ad g (K 1,, + J 1, ) = 0. Hece, the coditios (19) ad (0) hold. (ii) If θ 1 =(µ 1 µ ) 1/ ad θ = µ 1 /µ, the parametric orthogoality holds ad g 11 (θ) = (θ 1 ) ad g (K 1,, + J 1, )=(θ 1 ) 1. Hece, the coditio (19) holds ad (0) does ot hold. Example 10. Let {X t } be a Gaussia ARMA(1, 1) process with the spectral desity f θ (λ) = σ 1 ψe iλ π 1 ρe iλ..
SECOND ORDER ASYMPTOTIC PROPERTIES 79 (i) If θ 1 = σ ad θ =(ρ, ψ), the parameter orthogoality holds, [ ] (1 ρ g 11 (σ,ρ,ψ)=(σ 4 ) 1, I (σ,ρ,ψ)= ) 1 (1 ρψ) 1 (1 ρψ) 1 (1 ψ ) 1, K 1,, (σ,ρ,ψ)= σ 1 ρ, J 1,(σ,ρ,ψ)= σ 1 ρ, K 1,3,3 (σ,ρ,ψ)= σ 1 ψ, J 1,33(σ,ρ,ψ)= σ 1 ψ, K 1,,3 (σ,ρ,ψ)= σ 1 ρψ, J 1,3(σ,ρ,ψ)= σ 1 ρψ. Hece, the coditio (19) hold, ad g rs (K 1,r,s +J 1,rs )=σ shows that the coditio (0) does ot hold. (ii) If θ 1 =(ρ, ψ) ad θ = σ, the parameter orthogoality holds, [ ] I 11 (ρ, ψ, σ (1 ρ )= ) 1 (1 ρψ) 1 (1 ρψ) 1 (1 ψ ) 1, ad g 33 (K i,3,3 + J i,33 ) = 0. Hece, the coditios (19) ad (0) hold. 5. Ubiased test. We discuss the local ubiasedess of T S. Uder the absece of uisace parameters, LR test is locally ubiased. However, uder the existece of uisace parameters, LR test is ot geerally locally ubiased. From Theorem 1, amog the test statistics i Example 1, LR test is the oly oe which is secod order asymptotically ubiased uless g ij g jα g rs (K α,r,s + J α,rs ) = 0. If g ij g jα g rs (K α,r,s + J α,rs ) = 0, the LR = ). Hece, LR test is locally ubiased. Sice T S is ot geerally ubiased, we cosider modificatio of T S to T = h(ˆθ 1 )T + c 1 Ai Zi so that T is secod order asymptotically ubiased, where h(θ 1 ) is a smooth fuctio ad A i is a oradom costat. The followig theorem asserts that this is accomplished by choosig A i ad h i (θ 1 )= i h(θ 1 ) satisfy appropriate coditios. LR + o p (c 1 Theorem 6. Suppose that h(θ 1 ) is a cotiuously two times differetiable fuctio with h(θ 10 )=1ad A i is a oradom costat. The, for T S, the modified test T = h(ˆθ 1 )T + c 1 A i Zi is secod order asymptotically ubiased if h i = h i (θ 10 ) ad A i satisfy (i) h i = 1 p+ (g ijg jα B βγ K α,β,γ + g ij g kl a jkl [3]), (ii) A i = g iα g rs (K α,r,s + J α,rs ) a i 3. For h(θ 1 ) ad A i satisfyig (i) ad (ii), respectively, i Theorem 6, from Theorem 1, we ca get the asymptotic expasio of the distributio fuctio of T. Theorem 7. Suppose that h(θ 1 ) ad A i satisfy (i) ad (ii), respectively, i Theorem 6. The, for T S, the distributio fuctio of the modified test T = h(ˆθ 1 )T + c 1 A i Zi uder a sequece of local alteratives θ = θ 0 + c 1 ε has the secod order asymptotic expasio P θ0+c 1 ε [T <z]=g p, (z)+c 1 3 j=0 m j G p+j, (z)+o(c 1 ),
80 K. TAMAKI where (1) m 3 = 1 6 K α,β,γd α d β d γ 1 (p +) g ijb αβ K α,β,γ d γ d i d j + 1 aijk g ii g jj g 1 kk di d j d k (p +) aijk [3]g ii g jkg j k di d j d k, m 1 = (p +) g ijb αβ K α,β,γ d γ d i d j 1 aijk g ii g jj g 1 kk di d j d k + (p +) aijk [3]g ii g jkg j k di d j d k, m 1 = 1 J α,βγd α d β d γ 1 (K α,β,r + J α,βr + J β,αr )d α d β (d r ε r ), m 0 = 1 6 (K α,β,γ +3J α,βγ )d α d β d γ + 1 (K α,β,r + J α,βr + J β,αr )d α d β (d r ε r ). If p = 1, the () a ijk g ii g jj g kk 1 (p +) aijk [3]g ii g jkg j k =0. I this case we observe that the coefficiets m 3, m, m 1 ad m 0 i (1) are idepedet of T S, ad hece all the powers of the modified tests T are idetical up to secod order. O the other had, if p, the uiform results are ot available. Example 11. Cosider the ARM A(1, 1) model i Example 10 (ii). For test statistics i (7), (3) a ijk g i1g j1 g k1 = b K 1,1,1 +(b 3 +1)J 1,11 ρ = (1 ρ ) (3b b 3 1), ad (4) 1 4 aijk [3]g i1g jk g 11 = 3 4 b K 1,i,j g ij g 11 + 1 4 (b 3 +1)J 1,ij [3]g ij g 11 = 3 { } 4 b 6ρ (1 ρ ) + 4ψ (1 ρ )(1 ρψ) + 1 4 (b 3 +1) 1ρ3 ψ 10ρ ψ 8ρ +4ψ + (1 ρ ). (1 ρψ)(ρ ψ) (3) ad (4) show that () does ot holds. We give factorizatios of T S as quadratic forms. By direct computatio, T S ca be factorized as T = g ij T i T j + o p (c 1 ),
SECOND ORDER ASYMPTOTIC PROPERTIES 81 where T i = W i + 1 c 1 a 1g ij g jα g kβ W αβ W k + c 1 g ijg jα g rs W αr W s + 1 c 1 g ij a jkl W kw l 1 c 1 g ij g jα g kβ g rs K α,β,r W k W s 1 c 1 g ij g jα g rt g su (K α,r,s + J α,rs )W t W u + 1 c 1 g ij a j 3 + o p(c 1 ). The the asymptotic mea of T i uder θ = θ 0 is give by E θ0 [T i ]= 1 c 1 g ij g kl a jkl 1 c 1 g ij g jα g rs (K α,r,s + J α,rs ) + 1 c 1 g ija j 3 + o(c 1 ). Similarly, we cosider factorizatios of T 0 S 0 as quadratic forms. The asymptotic mea of T i0 uder θ = θ 0, where T 0 = I ij 0 T i0t j0 + o p (c 1 ), is give by E θ0 [T i0 ]= 1 c 1 I (ij) I (kl) a jkl + o(c 1 ). Uder (A-6), A i i Theorem 6 ca be writte as c 1 A i =I ij 0 (E θ 0 [T j0 ] E θ0 [T j ]) + o(c 1 ). Note that the third term of the right had i (18) is give by E θ0 [T i ] E θ0 [T i0 ]. Therefore, this term (ad hece A i ) ca be iterpreted as a effect of uisace parameters i T S. 6. Proofs. I this sectio, we give the proofs of theorems. Proof of Theorem 1. Sice the actual calculatio procedure is formidable, we give a sketch of the derivatio. First, we evaluate the characteristic fuctio of T, ψ (ξ,ε) =E θ0+c 1 ε [exp(tt )], T S, where t =( 1) 1/ ξ. Let D(θ) ={D αβ (θ)} be the uique lower triagular matrix with positive diagoal such that D(θ)D (θ) = We cosider the trasformatio ( I11 (θ) 0 0 I (θ) Y α = D αβ W β, where D αβ (θ) is the (α, β) compoet of the iverse matrix of D(θ). Deotig L (x )=p (x ; θ 0 + c 1 ε)/p (x ; θ 0 ), we have ψ (ξ,ε) = exp{tt (x )}L (x )p (x ; θ 0 )dx (5) = E θ0 [exp{tt + log L (x )}]. ).
8 K. TAMAKI We expad log L (x ) i a Taylor series i c 1 ε, leadig to (6) log L (x )=W i ε i + g αr g rs W s ε α 1 I (αβ)ε α ε β + 1 c 1 W αβε α ε β + 1 c 1 J γ,αβg δγ W δ ε α ε β 1 6 c 1 (K α,β,γ + J α,βγ [3])ε α ε β ε γ + o p (c 1 ) = D ij ε i Y j + g αr g rs D st ε α Y t 1 I (αβ)ε α ε β + 1 c 1 W αβε α ε β + 1 c 1 J γ,αβ g δγ D δζ ε α ε β Y ζ 1 6 c 1 (K α,β,γ + J α,βγ [3])ε α ε β ε γ + o p (c 1 ). Isertig (6) i exp{tt + log L (x )} we obtai, after further expasio ad collectio of terms, (7) { exp{tt + log L (x )} = exp t } p (Y i ) + D ij ε i Y j + g αr g rs D st ε α Y t 1 I (αβ)ε α ε β i=1 { 1+c 1 q 1(Y α,w βγ ) } + o p (c 1 ), where q 1 (, ) is a polyomial. I view of the assumptio (A-3) we ca easily evaluate the asymptotic cumulats of (Y α,w βγ ). Sice E θ {Y α (θ)w βγ (θ)} = o(c 1 ), we derive the secod order Edgeworth expasio of the distributio of Y α. Thus the secod order Edgeworth expasio of the distributio of Y α is give by (8) where P θ0 (Y α <y α )= = y α y α { f(y α ) 1+ 1 6 c 1 q(y α )dy α + o(c 1 ), p+q β,γ,δ=1 } C βγδ H βγδ (y α ) { } f(y α 1 )= exp 1 p+q (y α ), (π)(p+q)/ α=1 C αβγ = D α1αg α1α D β1βg β1β D γ1γg γ1γ K α,β,γ, dy α + o(c 1 ) ad H βγδ (y α ) are the Hermite polyomials. Note that t p (y i ) + D ij ε i y j + g αr g rs D st ε α y t 1 I (αβ)ε α ε β 1 p+q (y α ) i=1 = tg ijε i ε j 1 t 1 1 p+q r=p+1 α=1 p {(1 t) 1/ y i (1 t) 1/ D ji ε j } i=1 {y r g αs g st D tr ε α }.
SECOND ORDER ASYMPTOTIC PROPERTIES 83 From (5), (7) ad (8) it follows that { } p ψ (ξ,ε) = exp t (y i ) + D ij ε i y j + g αr g rs D st ε α y t 1 I (αβ)ε α ε β (9) i=1 { 1+c 1 q 1(y γ, 0) } q(y δ )dy ζ + o(c 1 ( tgij ε i ε j ) = exp (1 t) p/ 1 t 1+c 1 ) 3 m j (1 t) j Ivertig (9) by Fourier iverse trasform we ca prove Theorem 1. Proof of Theorem. (i) Note that d α is idepedet of ε. From Theorem 1, for T S we have P θ0+c 1 ε [T<z] P θ 10+c 1 ε 1,θ 0 [T<z] j=0 + o(c 1 ). = 1 c 1 (K α,β,r + J α,βr + J β,αr )d α d β ε r {G p+,λ (z) G p,λ (z)} + o(c 1 ), which leads to (i). (ii) From d α = g ij g jα ε i, clearly (K α,β,r + J α,βr + J β,αr )d α d β ε r = g ii g i α g jj g j α (K α,β,r + J α,βr + J β,αr )ε i ε j ε r. Hece, we get (ii) i Theorem. Proof of Theorem 3 ad Corollary 1. From Theorem 1 we ca see that m 3 = 1 aijk g ii g jj g kk di d j d k + C 3, (30) m = 1 aijk g ii g jj g kk di d j d k + 1 aijk [3]g ilg jk d l + C, m 1 = 1 aijk [3]g ilg jk d l + 1 ai 3 g ijd j + C 1, m 0 = 1 ai 3 g ijd j + C 0, where C 0, C 1, C ad C 3 are idepedet of a 1, a ijk ad a i 3 ad hece are the same for all test statistics i S. Theorem 3 ad Corollary 1 follow from (30). Proof of Theorem 4. Let a ijk ad a i 3 be the coefficiets of T S. The, we ca rewrite P T (ε) =Q 1,i j k (aijk ε )εi j ε k + Q,ijr ε i ε j ε r (31) + 1 g li(g iα B βγ K α,β,γ + a ijk [3]g jk)ε l {G p+, (z) G p+4, (z)} + 1 g ij{a i 3 g iα g rs (K α,r,s + J α,rs )}ε j {G p, (z) G p+, (z)}. Note that ε i ( /λ) 1/, where λ is the smallest eigevalue of I 11. By (31) P T (ε) Ψ 1 (,a ijk ) 3/ +Ψ r ( ) ε r + 1 g li(g iα B βγ K α,β,γ + a ijk [3]g jk)ε l {G p+, (z) G p+4, (z)} + 1 g ij{a i 3 giα g rs (K α,r,s + J α,rs )}ε j {G p, (z) G p+, (z)},
84 K. TAMAKI where Ψ 1 (,a ijk )= p i,j,k =1 Q 1,i j k (aijk ) λ 3/, Ψ r ( ) = p Q,ijr λ 1. i,j=1 Hece, we obtai (3) where M( ) = Similarly, we have P T ε ( ) Ψ 1 (,a ijk ) 3/ +Ψ r ( ) ε r +M( ), [ 1 mi g ij ε i ε j = g li(g iα B βγ K α,β,γ + a ijk [3]g jk)ε l {G p+, (z) G p+4, (z)} + 1 ] g ij{a i 3 g iα g rs (K α,r,s + J α,rs )}ε j {G p, (z) G p+, (z)}. (33) Pε LR ( ) Ψ 1 (, g iα g jβ g kγ K α,β,γ /3) 3/ Ψ r ( ) ε r. From (3) ad (33), { } Pε LR ( ) Pε T ( ) Ψ 1 (,a ijk )+Ψ 1(, g iα g jβ g kγ K α,β,γ /3) 3/ Ψ r ( ) ε r M( ). Hece, for T S whose coefficiets do ot satisfy z(a ijk [3]g jk + g iα B βγ K α,β,γ )+(p + ){a i 3 giα g rs (K α,r,s + J α,rs )} = 0, there exists a positive 0 such that wheever 0 < < 0. P LR ε ( ) P T ε ( ) > 0, Proof of Theorem 5. The distributio fuctio of T 0 S 0 uder a sequece of local alteratives θ 1 = θ 10 + c 1 ε 1 has the asymptotic expasio P θ10+c 1 ε 1,θ 0 [T 0 <z]=g p, (z)+c 1 3 j=0 m j0 G p+j, (z)+o(c 1 ), where (34) ( 1 m 30 = 6 K i,j,k + 1 ai j k I (i i)i (j j)i (k k) ) ε i ε j ε k, m 0 = 1 ai j k I (i i)i (j j)i (k k)ε i ε j ε k + 1 Iij 0 K i,j,kε k + 1 aijk [3]I (il)i (jk) ε l, m 10 = 1 J i,j,kε i ε j ε k 1 Iij 0 K i,j,kε k 1 aijk [3]I (il)i (jk) ε l, m 00 = 1 6 (K i,j,k +3J i,jk )ε i ε j ε k. Note that, uder (A-6), d r = 0 ad {B αβ } = ( (I11 ) 1 0 0 0 ).
SECOND ORDER ASYMPTOTIC PROPERTIES 85 The the coefficiets m 3, m, m 1 ad m 0 i Theorem 1 ca be writte as m 3 = m 30, m = m 0, m 1 = m 10 + 1 (K i,j,r + J i,jr + J j,ir )ε i ε j ε r (35) + 1 {ai 3I (ij) g rs (K j,r,s + J j,rs )}ε j, m 0 = m 00 1 (K i,j,r + J i,jr + J j,ir )ε i ε j ε r 1 {ai 3I (ij) g rs (K j,r,s + J j,rs )}ε j. The compariso of (34) ad (35) leads to Theorem 5. Proof of Theorem 6 ad 7. Note that Z i = W i + o p (1). Expad T as (36) T = h(ˆθ 1 )T + c 1 A i Zi =(1+c 1 h i η i )T + c 1 A i W i + o p (c 1 ). Isertig (5) i (36) we obtai where T = g ij W i W j + c 1 a 1g iα g jβ W αβ W i W j +c 1 giα g rs W αr W i W s W i W j W k c 1 giα g jβ g rs K α,β,r W i W j W s giα g rt g su (K α,r,s + J α,rs )W i W t W u + c 1 a i 3 W i + o p (c 1 ), + c 1 a ijk c 1 (37) a ijk = a ijk + h l g li g jk. a i 3 = a i 3 + A i. This implies T S, ad hece a ecessary ad sufficiet coditio for its locally ubiasedess is that the coefficiets i (37) satisfy (i) a ijk [3]g il g jk + g li g iα B βγ K α,β,γ =0, (ii) a i 3 g ij g ji g iα g rs (K α,r,s + J α,rs )=0. Note that a ijk [3]g il g jk = a ijk [3]g ilg jk +(h l g l i g jk + h l g l j g ki + h l g l k g ij )g il g jk = a ijk [3]g ilg jk +(p +)h l. Solvig (i) ad (ii) with respect to h i ad A i, we obtai the relatios i Theorem 6. Theorem 7 follows from the above argumet ad Theorem 1. Ackowledgmets The author would like to express his sicere thaks to Professor Masaobu Taiguchi for his ecouragemet ad guidace. Also thaks are exteded to Professor Takeru Suzuki for may helpful suggestios ad advice throughout the period i which he supervised his research.
86 K. TAMAKI Refereces [1] Bickel, P. J. ad Ghosh, J. K. (1990). A decompositio for the likelihood ratio statistic ad the Bartlett correctio a Bayesia argumet. A. Statist. 18. 1070 1090. [] Cordeiro, G. M. ad Ferrari, S. L. P. (1991). A modified score test statistic havig chi-squared distributio to order 1. Biometrika. 78. 573 58. [3] Eguchi, S. (1991). A geometric look at uisace parameter effect of local powers i testig hypothesis. A. Ist. Statist. Math. 43. 45 60. [4] Harris, P. ad Peers, H. W. (1980). The local power of the efficiet scores test statistic. Biometrika. 67. 55 59. [5] Hayakawa, T. (1975). The likelihood ratio criterio for a composite hypothesis uder a local alterative. Biometrika. 6. 451 460. [6] Kakizawa, Y. (1997). Higher-order Bartlett-type adjustmet. J. Statist. Pla. If. 65. 69 80. [7] Li, B. (001). Sesitivity of Rao s score test, the Wald test ad the likelihood ratio test to uisace parameters. J. Statist. Pla. If. 97. 57 66. [8] Mukerjee, R. (1993). A extesio of the coditioal likelihood ratio test to the geeral multiparameter case. A. Ist. Statist. Math. 45. 759 771. [9] Mukerjee, R. (1994). Compariso of tests i their origial forms. Sakhyā. Ser. A. 56. 118 17. [10] Rao, C. R. ad Mukerjee, R. (1995). Compariso of Bartlett-type adjustmets for the efficiet score statistic. J. Statist. Pla. If. 46. 137 146. [11] Rao, C. R. ad Mukerjee, R. (1997). Compariso of LR, score ad Wald tests i a o-iid settig. J. Multivariate. Aal. 60. 99 110. [1] Taiguchi, M. (1988). Asymptotic expasio of the distributio of some test statistics for Gaussia ARMA processes. J. Multivariate. Aal. 7. 494 511. [13] Taiguchi, M. (1991). Higher Order Asymptotic Theory for Time Series Aalysis. Lecture Notes i Statistics. Vol. 68. Heidelberg: Spriger-Verlag. [14] Taiguchi, M. (1991). Third-order asymptotic properties of a class of test statistics uder a local alterative. J. Multivariate. Aal. 37. 3 38. Departmet of Mathematical Scieces School of Sciece & Egieerig Waseda Uiversity, 3-4-1, Okubo, Shijuku-ku, Tokyo, 169-8555, Japa E-mail: k-tamaki@toki.waseda.jp
SECOND ORDER ASYMPTOTIC PROPERTIES 87 secod order powers - -1 0 1-1.0-0.5 0.0 0.5 1.0 ρ Figure 1: For the bivariate ormal model with correlatio coefficiet θ 1 = ρ, both meas θ = µ ad both variaces 1 i Example 4, secod order powers of LR, LR,R 1 ad W 1 statistics are plotted. P LR (ε) (solid lie), P LR (ε) (dotted lie), P R1 (ε) (dashed lie) ad P W1 (ε) (dash-dotted lie) with α =0.05 ad ε 1 =1. secod order powers -3 - -1 0 1-1.0-0.5 0.0 0.5 1.0 ρ Figure : For the bivariate ormal model with correlatio coefficiet θ 1 = ρ, both meas θ = µ ad both variaces 1 i Example 4, secod order powers of LR, LR,R 1 ad W 1 statistics are plotted. P LR (solid lie), P LR (dotted lie), P R1 (dashed lie) ad P W1 (dash-dotted lie) with α =0.05 ad ε 1 =0.1.
88 K. TAMAKI secod order powers -0.0005 0.0 0.0005-1.0-0.5 0.0 0.5 1.0 ψ Figure 3: For MA(1) model i Example 5, secod order powers of LR, W 1 ad R 1 statistics are plotted. P LR (solid lie), P W1 (dotted lie) ad P R1 (dashed lie) with α =0.01 ad ε 1 =6.5. secod order powers -0-10 0 10 0-1.0-0.5 0.0 0.5 1.0 ψ Figure 4: For MA(1) model i Example 5, secod order powers of LR, W 1 ad R 1 statistics are plotted. P LR (solid lie), P W1 (dotted lie) ad P R1 (dashed lie) with α =0.01 ad ε 1 =0.65.
SECOND ORDER ASYMPTOTIC PROPERTIES 89 secod order powers - -1 0 1-1.0-0.5 0.0 0.5 1.0 ρ Figure 5: For AR(1) model i Example 6, secod order powers of LR, W ad R statistics are plotted. P LR (solid lie), P W (dotted lie) ad P R (dashed lie) with α =0.01 ad ε 1 =3. secod order powers -10 0 10-1.0-0.5 0.0 0.5 1.0 ρ Figure 6: For AR(1) model i Example 6, secod order powers of LR, W ad R statistics are plotted. P LR (solid lie), P W (dotted lie) ad P R (dashed lie) with α =0.01 ad ε 1 =0.8.