RATE-OPTIMAL DATA-DRIVEN SPECIFICATION TEST BY THI THUY ANH VO AND ALAIN GUAY
[Resume]
1. Introduction Consider a linear autoregressive distributed lag dynamic regression (AD) model:
( 0) ( B)Yt = c + (1) ( B ) X 1t + ....... + ( q ) + u t
where the ( j ) ( B ) =
(0)
l =0
mj
ij
B l are polynomials of order mj in lag operator B associated with the
dependant variables Yt and the q exogenous variables Xjt, C is a constant, and ut is an unobservable disturbance. The polynomial (0)(B) is assumed to have all roots outside the unit circle, and is normalized by setting 00=0. The Xjt is also assumed that to be covariance stationary with E(Xij)<. Note that
0 = ( 10 ,...., m0 0 )' ,
j = ( 0 j , 1 j ,..., m j j )' ,
j=1,
2,
3,...,q.
Then
= (c, ' 0 ,..., ' q )' is a
(m
j =0
q
j
+ 1) x1 vector consisting of all unknown coefficients in (1). The
model (1) can be estimated by (e.g) the ordinary least square (OLS) method. One of the principal conditions for the consistent of the OLS estimator for is that {ut} is serially uncorrelated. We want to test if the model (1) has a good specification. It is well known that the serial correlation of {ut} may occur due to misspecification of the model (1), such as omitting relevant variables, choosing to low lag order for Yt or the Xij, or using inappropriate transformed variables. So the hypothesis of interest is: H0: (j)=0 v.s Ha: (j) 0 for some j 0, where (j) is autocorrelation of residues order j. Tests of H0 are called lack-of-fit tests or specification tests. Hong (1996) proposed three classes of consistent one-sided tests for serial correlation of unknown form for the residual of model (1). The tests are obtained by comparing a kernel-based normalized spectral density, using quadratic norm, the Hellinger metric, and the Kullback-Leiber
2
information criterion respectively. Under the null hypothesis of no serial correlation, the three classes of Hong test statistics are asymptotically N(0,1) or equivalent. The popular Box and Pierce (1970) (BP) test is a special case of Hong tests. The BP test can be viewed as a quadratic norm based test using truncated periodogram. Hong tests may be more powerful than the later because many other kernels deliver tests better power. In other word, in the Hong tests, the weight given to autocorrelation order j ((j)) is close to unity (the maximum weight) when j is small relative to n and the larger j is, the less weight is given to (j). By contrast, the Lagrange multiplier (LM) tests of Breusch (1978), Godfrey (1978), BP (1970) test whose the statistics are LM, QT respectively give the equal weight to (j). The LM and BP statistics are the following:
^ QT = T r j2 ,
j =1
m
(0) (0)
LM=nR2, where R2 issue from the regression MA or AR of the residues.
Intuitively, this might not be the optimal weighting because for most stationary processes the autocorrelation decay to zeros as the lag increase. This difference may be used to explain the power of Hong test. In other word, the null distribution of the Hong tests remains invariant when the regressors include lagged dependent variables. By simulation study, Hong (1996) found that his tests have good power against an AR(1) process and a fractionally integrated process. In particular, they have better power than LM test as well as BP test and Ljung and Box (1978) test. However, the power of Hong tests depends on the choice of the parameter of the kernel. In this paper, we derive some new classes of test which is based on those of Hong but we chose an optimal parameter of the kernel. Like Geurre and Lavergne (2004), our data driven choice of the kernel parameter relies on a specific criterion tailored for testing purpose. The asymptotic power of a test of H0 is often investigated by deriving the asymptotic probability that the test rejects H0 against a local alternative hypothesis whose the distance from the null hypothesis convergences to zero as n . This approach is the familiar Pitman's local analysis. The convergent rate is slower than n-1/2. Our tests have the optimal rate in the sense of Horowitz and Spokoiny (2001). The advantages of our tests in comparison with Hong (1996) tests are: (1) the choice of the parameter of the kernel is not arbitrage but data-driven. Our data-driven choice of this parameter
3
relies on a specific criterion tailored for testing purpose. This choice renders the test robust and more powerful and yields an adaptive rate-optimal test; (2) the test is adaptive and rate optimal in the sense of Horowitz and Spokoiny (2001); (3) the test detects Pitman local alternatives with rate that can be arbitrary close to n-1/2. The rest of this paper includes of five sections. Section 2 presents method and test statistic. In section 3, we study the asymptotic distribution under the null hypothesis. Section 4 concentrates on the asymptotic local power of the test. Section 5 talks about Monte Carlo Evidence. 2. Method and test statistic Suppose that {ut} is a stationary real-valued process with E(ut)=0, autocovariance function R(j), autocorrelation function (j), and normalized spectral density function
+
f ( ) = (2 ) -1
j =-
( j ) cos(j ) with [-, ]
(0)
The hypotheses of interest are: H0: (j)=0 for all j 0 v.s Ha: (j) 0 for some j 0. Under the null hypothesis, f( )=f0( )=1/(2) for all [-, ]. Our test statistic is based on the difference between f( ) and f0( ). If this difference is large enough, the null hypothesis will be rejected. Let D(f1, f2) be a divergence measure for two spectral densities f 1, f2 such that D(f1,f2) 0
^ ^ and D(f1,f2)=0 if and only if f1>f2. The consistent test can be then based on D ( f n ; f 0 ) where f n
is a kernel estimator of for f. Somme D is used for measuring the difference of f from f0: Quadratic norm:
Q( f ; f 0 ) = 2 ( f ( ) - f 0 ( )) 2 d -
1/ 2
,
(0)
Hellinger metric:
H ( f ; f 0 ) = ( f 1 / 2 ( ) - f 0
[
1/ 2
( ))d
]
1/ 2
,
( 0)
Kullback-Leibler information criterion (relative entropy) :
4 I ( f ; f 0 ) = -
where (f)= [-, ]; f( >0. These measures are intuitively appealing and have their own merits. The quadratic norm delivers a computationally convenient statistic that is simply a weighted average of squared sample autocorrelations with the weights depending on the kernel. The Box and Pierce statistic can be
( f )
ln( f ( ) / f 0 ( )) 2 d ,
( 0)
^ ^ viewed as based on Q ( f n , f 0 ) with f n being a truncated periodogram. ^ f( ) is unobservable, so we have to estimate it. Let be an estimator of . Then the residual of
(1) is:
^ ^ ^ ^ ^ u t = ( 0) ( B) y t - c - (1) x1t - .... - ( q ) ( B ) X qt .
So,
n -1
(0)
^ f ( ) = (2 ) -1
-1 ^ ^ ^ ^ with ( j ) = R ( j ) / R (0) and R ( j ) = n n -1 n
j = - ( n -1)
^ ( j ) cos(j ) ,
t- j
(0)
i = j +1
^ ^ uu
t
. A kernel estimator of f( ) is given by:
^ f n ( ) = (2 ) -1
j = - n +1
k( j / p
n
^ ) ( j ) cos( j ) , [-, ],
(0)
where the bandwidth pn is an integer and pn, pn/n0. Like Hong (1996), the following conditions are imposed: Assumption A.1: k: R[-1,1] is a symmetric function that is continuous at zeros and at
all but a finite number of points, with k(0)=1 and
-
k
2
( z ) dz .
The condition that k(0)=1 and k is continuous at 0 imply that for j small relative to n, the weight given to unity (the maximum weight) and the higher j is, the less weigh is put for (j). This is reasonable because for most stationary processes, the autocorrelation decay to zeros as the lag increases. The assumption A.1 includes the Barlett, Daniell, general Turkey, and Parzen kernels are of compact support, i.e. k(z)=0 for |z|>1. For these kernel, pn is called the `the lag truncation number,' because the lags of order j>pn receive zero weight. In contrast, the Daniell and QS
5
kernels are of unbounded support; here p is not a `truncated point,' but determines the `degree of
^ smoothing' for f n . ^ ^ ^ Hong (1996) proposed the standardized versions of Q 2 ( f n , f 0 ) , H 2 ( f n , f 0 ) , Q 2 ( f n , f 0 ) , ^ I ( fn , f0 ) : ^ M 1n = ((1 / 2)nQ 2 ( f n ; f 0 ) - C n (k )) /(2 Dn (k ))1 / 2 ^ = (n k 2 ( j / p n ) 2 ( j ) - C n (k )) /(2 Dn (k ))1 / 2 ,
j =1 n -1
(0)
^ M 2 n = 2nH 2 ( f n ; f 0 ) - C n (k ) /( 2 Dn (k ))1 / 2 , ^ M 3n = nI ( f n ; f 0 ) - C n (k ) /(2 Dn ( k ))1 / 2 ,
n -1 j =1 n -1 j =1
(
(
)
)
(0) (0)
2 4 where C n (k ) = (1 - j / n)k ( j / p n ) , Dn ( k ) = (1 - j / n)(1 - ( j + 1))k ( j / p n ) . For (12)
and (13), we impose the following additional on k.
Assumption A.1(b) : ).
-
- iz k ( z ) dz < and K ( ) = (1 / 2 ) k ( z)e dz 0 for (-, -
This absolute integrability of k ensures that its Fourier transform K exists. Assumption A.1 includes the Barlett, Daniel, Parzen, and QS kernel, but rules out the truncated and general Tukey kernel. Under some regularity conditions, these statistics are asymptotic normal (0,1). If the kernel used is a truncated kernel, M1n is a standardized version of BP statistic. The fact that many kernels are better than the truncated kernel used by BP, Hong tests are more powerful. The power of this test statistic depends on the choice of p n. In this paper, we propose three new classes of tests where the choice of pn is optimal. We didn't find a paper in the literature which talks about the choice of pn of the statistics based on the spectral approach. But many adaptive rate-optimal tests are based on the maximum approach, which consists in choosing as a test statistic the maximum of standardized statistics associated with a sequence of smoothing parameters. Horowitz and Spokoiny (2001) proposed a test of a parametric model of a conditional mean function against alternatives a non parametric model. This test is also based the maximum
6
approach. Guerre and Lavergne (2004) proposed data-driven smooth tests for a parametric regression function. The smoothing parameter of these test statistics is selected through a new criterion that favors a large smoothing parameter under the null hypothesis. The advantage of this choice is that the distribution of the statistics under the null hypothesis is standard (normal distribution) and this test detects local Pitman alternatives converging to the null at a faster rate than the one detected by a maximum test. In this paper, we propose a new class of statistics which is based on the class of Hong (1996) statistics but the choice of the parameter of the kernel used is not arbitrage but data driven. The procedure of the selection of pn is based on that proposed by Guerre and Lavergne (2004). Define
^ ^ T1 pn = (1 / 2) nQ 2 ( f n ; f ) - C n (k ) ^ ^ T2 pn = 2nH 2 ( f n ; f 0 ) - C n (k ) ^ ^ T3 pn = nI ( f n ; f 0 ) - C n (k )
Let P be a set of possible values of pn and Jn be the number of the elements of P. We have:
(0) (0) (0)
P = { p min , p min + 1,....., p max } ,
(0)
where pmin and pmax are of order round(ln(lnn)) and round(lnn) respectively and J n=pmax-pmin. We see that Jn when n . On an informal ground, the approach of Guerre and Lavergne (2004) favors a baseline statistic
^ ^ Tipn 0 with lowest variance among the Tipn with i=1,2,3. In our case, the approximation of the ^ ^ standard deviation of Tipn is v pn = 2 Dn (k ) where Dn (k ) is defined above. It is easy to
demonstrate that 2 Dn (k ) obtains minimal value when pn is equal pnmin. Our statistic is the following:
^ M in ( ~ n ) = Tip* /(2 Dn0 ( k ))1 / 2 , i=1, 2, 3. p n
where Dn0 (k ) =
[
]
(0)
(1 - j / n)(1 - ( j + 1))k
j =1
n -1
4
( j / 1) and ~in satisfies p
~ = arg max T - v ^ ^ ^ ^ ^ pin ipn n pn , pn 0 = arg max Tipn - Tip n - n v pn , pn 0 , p P p P
n n
{
}
{
}
(0)
7 ^ where n>0 and v pn , pn 0 = 2 Dn (k ) + 2 Dn0 (k ) - 4 Dn0 n , the approximation of asymptotic null
^ ^ standard deviation of Tipn - Tipn 0 . Our criterion for the choice of the kernel parameter penalizes
each statistic by a quantity proportional to its standard deviation while the criteria reviewed in Hart (1997) use larger penalty proportional to the variance. 3. Asymptotic null distribution To establish the asymptotic null distribution of our test, we assume the following conditions: Assumption A.2: {ut} is identically and independently distributed (i.i.d) with E(ut) = 0, E(u2t) = 20 et E(ut4) = 4< .
1/ 2 ^ Assumption A.3: n ( - ) = O p (1).
Although most of papers suppose {ut} be normal, we assume that {ut} is i.i.d because in financial models, it is well known that {ut} has highly leptokurtic distribution. Theorem 1. a: Suppose Assumption A.1 (a), A.2, A.3 hold. Let pn , pn/n 0. Then
M 1n d N (0,1)
Theorem 1. b: Suppose Assumption A.1- A.3 hold. Let pn, pn3/n0. Then
M 2 n - M 1n = o p (1), M 3n - M 1n = o p (1), M 2 n d N (0,1), M 3n d N (0,1).
See Hong (1996) for the proof of this theorem. The asymptotic law under the null hypothesis of our new classes of tests is given in the two next theorems. Theorem 2: Suppose Assumption A.1, A.2, A.3 hold and pnmin is order of round(ln(lnn)), pmax is order of lnn. Let n with
n (1 + ) 2 ln J n ,
p ~ with Z, standard normal critical value. for some >0, then Pr( M 1n ( p1n ) Z )
(0)
The theorem 2 is proved in two main steps. Firstly, we show that
8 ^ ^ T1 p - T1 p0 P ( ~1n p1n 0 ) = P max n p n pn P v p ^ n, p n ^ ^ goes to zero. Then we show that T1 p0 / v p0 converges to a standard normal.
Theorem 3: Suppose Assumption A.1- A.3 hold. Let pn, pn3/n0. Then
( 0)
^ ^ ^ ^ ^ ^ (T1 pn - T2 pn ) / v pn , pn 0 = o p (1), (T1 pn - T3 pn ) / v pn , pn 0 = o p (1), p n P ,
p p , Pr( M 3n ( ~3n ) Z ) p with Z, standard normal p and Pr( M 2 n ( ~ 2 n ) Z )
critical value. The data choice driven of the kernel parameter favors pn0 under the null hypothesis. Indeed, since
^ ^ ^ p Tipn - Tipn 0 is order of v pn , pn 0 under H0, ~n = p n 0 asymptotically under H0 if n divers fast ^ ^ enough. Hence the null limit distribution of our tests is the one of Tip 0 / v p0 , that is standard
normal, our tests have bounded critical value. This is an advantage of our in statistics comparison avec the statistics using approaches maximum. Under the null hypothesis, our new classes of
^ ^ tests is equivalent to the classes of tests Min, i=1, 2, 3 of Hong (1996), but the fact that Ti1 pn / v p0 ^ ^ is larger than Tipn / v pn under the alternative hypothesis will do our tests more powerful at no
cost. 4. Asymptotic local power In this section, we consider Pitman local alternatives
0 Han: f n ( ) = f 0 ( ) + a n g ( ), [ - , ] ,
(0)
where an0 as n, and g : RR is a symmetric periodic (with periodicity 2) bounded
continuous function with
-
g ( )d = 0 .
0 This condition ensures that f n is a normalized
spectral density for all n sufficiently large. an tends to 0 at a rate slower than n-1/2. Define:
9 1 ^ M 1a ( p n ) = ( Q 2 ( f n , f n0 ) - C n (k ) / Dn (k ) , 2
a ^ M 2 ( p n ) = (2nH 2 ( f n , f n0 ) - C n (k ) / Dn (k ) ,
(0) (0) (0) (0) (0) (0)
[
^ M 3a ( p n ) = (nI ( f n , f n0 ) - C n (k ) / Dn (k ) ^ ^~ T1a n = (1 / 2)nQ 2 ( f n ; f n0 ) - C n (k ) , p ^ ^ T2a~n = 2nH 2 ( f n ; f n0 ) - C n (k ) , p ^ ^ T3a~n = nI ( f n ; f n0 ) - C n (k ) , p
~ and pipn satisfies
n
[
]
]
~ = arg max T a - v ^ ^a ^a ^ ^ pipn ipn n pn , pn 0 = arg max Tipn - Tip0 - n v pn , pn 0 , p P p P
n
{
}
{
}
(0)
^ where n>0 and v pn , pn 0 =
2 Dn (k ) + 4 Dn0 (k ) - 2 Dn0 n , the approximation of asymptotic null
^a ^a standard deviation of Tipn - T1 p0 .
Theorem 4: Under the assumption A.1(a), A.2, A.3 and pn, pn/n0, an=pn1/4/n1/2, we have:
M 1n d N ( (k ),1), 2 g 2 ( )
-
where
(k ( z ( p n )) =
n
2 D(k )
with
D(k ) = k 4 ( z )dz . If in addition
0
a a 3 assumption A.1(b) hold and p n / n 0 , then M 2 n - M 1n = o p (1), a M 2 n d N ( (k ),1) and M 3an d N ( (k ),1) .
M 3an - M 1an = o p (1),
See Hong (1996) for the proof of this theorem. Theorem 5:
1/2
~ ~ Suppose the assumption A.1(a), A.2, A.3 hold and pn , pn /n0, an=n-
d ^a ^ p N ( ( k ),1) where ~ n satisfies (29). If in addition assumption (lnlnn)1/4. Then T1 ~n / v pn 0 p
10
d ^a ^ ^ ^ p3 N ( (k ),1) and T3a~n / v p n 0 d N ( (k ),1) A.1(b) hold and ~n / n 0 , then T2 ~n / v pn 0 p p
where (k ( z ( ~ )) = pn
2 g 2 ( )
-
n
2 D(k ) .
Proof of this theorem is in the appendix. Since an is slower than the parameter n1/2, our tests are less efficient than parametric tests under Han. Because (k) depends on k, different kernels may deliver different power. 5. Monte Carlo evidence In this section, we present the Monte Carlo evidence of our test to demonstrate that it is more powerful than some commonly used tests in practice and our choice of the parameter of the kernel is optimal. In order to compare with the results obtained by Hong (1996), we use his data generating process
Yt = c + 1Yt -1 + 2 X t + u t
(0)
where the exogenous variable Xt = 0.8Xt-1+vt, and the vt are NID(0,3). We set = (c, 1, 2)' = (1, 0.5, 0.5)'. The sample sizes used are n=64, 128. For each n, we set the initial value of Y equal zero and generate 2n+1 observations using (25) but we discard the first n+1 observations to reduce the effects of initial value. Our simulation programs are written on matlab language. For the statistics M2n, M3n, we use the approximation method to calculate the integral. We compare our tests with those of BP, LB, and Breusch (1978), Godfrey (1978) and M 1n statistic of Hong (1996). The following kernels are used for Min statistic, i=1,2,3 of Hong (1978) and for our statistics: Daniell (DAN): k ( z ) = sin(z ) / z .
1 - 6(z / 6) 2 + 6 z / 6 3 , z 3 / 3 3/ z 6 / Parzen (PAR): k ( z ) = 2(1 - z / 6 ) , 0, otherwise;
11 1 - z , 0, . otherwise. z 1
Barlett (BAR): k ( z ) =
12
MATHEMATICAL APPENDIX Lemma 1: Under Assumption A.1, A.2, A.3 and let pnminbe order of round(ln(lnn)) and pn, pn/ n0, pn/pn0 when n0. It follows that
^ ^ i) sup P (T pn v pn Z ) - P ( N (0,1) Z ) = o p (1).
zR
(
)
ii) P
^ ^ T1 p - T1 p n 0 Z vp p ^ n, n0
2 Z2 exp(- ) + o p (1). 2 2
Proof: The theorem 1.a gives us the lemma 1.i. 1.ii can be rewritten like
n -1 ^ sup P n k 2 ( j / p n ) 2 ( j ) - C n (k ) ( 2 Dn (k ))1 / 2 Z ) - ( P( N (0,1) Z ) = o p (1). zR j =1
We have
n -1 j =1
^ ^ T1 p n - T1 p 0 = ^ v pn, pn 0
^ n k 2 ( j / pn ) - k 2 ( j / pn 0 ) 2 ( j ) - (1 - j / n) k 2 ( j / pn ) - k 2 ( j / pn 0 )
j =1
[
]
n -1
[
]
(1 - j / n)(1 - ( j + 1) / n)[k
n j =1
2
( j / pn ) - k 2 ( j / pn 0 )
]
2
n -1 n -1 n 2 n ^ ^ k ( j / pn ) 2 ( j ) - (1 - j / n)k 2 ( j / pn ) - k 2 ( j / pn ) 2 ( j ) - (1 - j / n)k 2 ( j / pn ) j =1 j =1 j =1 j =1 . =
(1 - j / n)(1 - ( j + 1) / n)[k
n j =1
2
( j / pn ) - k 2 ( j / pn 0 )
]
2
When
pn
and
pn/n0,
we
have
p n Dn (k ) D(k ) = k 4 ( z )dz ,
0
and
~ ~ p n C n (k ) D(k ) = k 2 ( z )dz. So 0
13
^ ^ T1 p n - T1 p 0 = ^ v pn , pn 0
k
j =1
n
2
^ ( j / pn ) 2 ( j ) - pnC (k ) ( pn - pn 0 )2 D(k ) -
k
j =1
n
2
^ ( j / pn 0 ) 2 ( j ) - pn 0C (k ) ( pn - pn 0 )2 D( k ) =
=
k
j =1
n
2
^ ( j / pn ) 2 ( j ) - pnC (k ) pn 2 D(k ) -
k
j =1
n
2
^ ( j / pn 0 ) 2 ( j ) - pn 0C (k ) pn 0 2 D(k ) .
( pn - pn 0 ) / pn
( pn - pn 0 ) / pn 0
We know that M 1n
*
k =
j =1
n
2
^ ( j / p n ) 2 ( j ) - p n C (k ) p n 2 D(k )
is a more compact expression of M1n and its
asymptotic distribution under the null hypothesis is the same of M1n. That means M*1n is N(0, 1). So
^ ^ T1 p n - T1 p0 o p (1)( pn - pn 0 ) = = o p (1), ^ v pn , pn 0 pn - pn 0
using pn/pn0 when n. The last equation is equivalent the following equation:
^ ^ T1 p - T1 p 0 sup P n Z (2 Dn (k ))1 / 2 Z ) - P( N (0,1) Z ) = o p (1). ^ v z R pn , pn 0
The lemma ii follows by Mill's ratio inequality. Proof of theorem 2 : Fistly, we want to demonstrate
^ ^ T1 p - T1 p 0 P ( ~1n p1n 0 ) = P max n p n goes to zero. Let be as in condition (20) of theorem p n P v p p ^ n, n0
2. We have
14 ^ ^ T1 p - T1 p0 P ( ~1n p1n 0 ) = P max n p n p n P v p p ^ n, n0 = o p (1),
using condition (20) and n when n.
2 ^ ^ T1 p - T1 p 0 n 2 exp - 1 n + ln J n P P v n 2 1+ / 2 ^ pn , pn 0 n pn
^ ^ We now demonstrate that T1 p n / v p n 0 converges to N(0,1). ^ It is easy to demonstrate that T1 pn has minimum variance when pn=pnmin. So pn0= pnmin =ln(lnn). ^ ^ When n, pnmin but pnmin/n0, so follow the theorem 1, we have T1 pn / v p0 converges to
N(0,1). This is sufficient to establish theorem 2. Proof of theorem 3:
3 Hong (1996) demonstrate that given p n / n 0
1 1 ^ ^ ^ ^ 2 H 2 ( f n ; f 0 ) - Q 2 ( f n ; f 0 ) = o p ( p 1 / 2 / n) and I ( f n ; f 0 ) - Q 2 ( f n ; f 0 ) = o p ( p 1 / 2 / n). n n 2 2
1/ 2 1/ 2 ^ ^ ^ ^ It follows that T1 pn - T2 pn = o p ( p n / n) , T1 pn - T3 pn = o p ( p n / n), p n P.
^ ^ T1 p n - T2 p 0 = ^ v pn , pn 0 no p ( p
1/ 2 n
no p ( p1 / 2 / n) n
(1 - j / n)(1 - ( j + 1) / n)[k
nn j =1
2
( j / pn ) - k 2 ( j / pn 0 )
]
2
=
/ n)
pn - pn 0 2 D(k )
= o p (1). no p ( p1 / 2 / n) n
^ ^ T1 p n - T3 p0 = ^ v pn , pn 0 no p ( p
1/ 2 n
(1 - j / n)(1 - ( j + 1) / n)[k
n j =1
2
( j / pn ) - k 2 ( j / pn 0 )
]
2
=
/ n)
pn - pn 0 2 D ( k )
.
Given pn and pn/n0, p n Dn (k ) D (k ) =
0
k 4 ( z )dz. It follows that
15 ^ ^ T1 p n - T2 p0 = o p (1). ^ v pn, pn 0 ^ ^ T1 p n - T3 p0 = o p (1). ^ v pn , pn 0
And then we demonstrate the last part of this theorem.
^ ^ T - T2 p0 P ( ~2 n p2 n 0 ) = P max 2 p n p n p n P v p , p ^ n n0 1 n 2 exp - 2 1 + / 2 + ln J n n = o p (1).
p n P
P
^ ^ T2 p - T2 p n 0 n = ^ v pn , pn 0
p n P
P
^ ^ T1 p - T1 p n 0 n + o p (1) ^ v pn , pn 0
^ ^ T3 p - T3 p0 P ( ~3n p3n 0 ) = P max n p n p n P v p , p ^ n n0
^ ^ T3 p - T3 p 0 P n n = v ^ pn , pn 0 p n P
^ ^ T1 p - T1 p 0 P n n + o p (1) v ^ pn , pn 0 p n P
p n P
P
^ ^ T1 p - T1 p 1 n 2 n 0 n exp - 2 1 + / 2 + ln J n ^ v pn, pn 0 n
= o p (1).
The second part of this theorem follows by theorem 1. Proof of theorem 5:
^ ^ Firstly, we want to demonstrate Q 2 ( f n , f n0 ) = Q 2 ( f n , f 0 ) + (k ) + o p ( ln ln n
We have:
2 ^ ^ Q 2 ( f n , f n0 ) = 2 ( f n ( ) - f 0 ( ) - a n g ( ) d
- -
n
).
2 ^ ^ = 2 ( f n ( ) - f 0 ( )) 2 - 2a n ( f n ( ) - f 0 ( )) g ( ) + a n g 2 ( ) d.
[
(
)
]
16
-1 / 2 ^ ) In the proof of the theorem 4 of Hong (1996), he finds that ( f n ( ) - f 0 ( )) g ( ) = O p (...