Central limit theorem for asymmetric Kernel functionals

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Economies Department

Central Limit Theorem

for Asymmetric Kernel Functionals

M

a r c e l o

F

e r n a n d e s

ECO No. 2000/1

EUI WORKING PAPERS

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European University Institute 3 0001 0033 3256 8 © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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E U R O P E A N U N IV E R S IT Y IN S T IT U T E , F L O R E N C E ECONOM ICS D EPA R TM EN T

WP 3 3 0

EUR

EUI Working Paper E C O No. 2000/1

Central Limit Theorem

for Asymmetric Kernel Functionals

Ma r c e l o Fe r n a n d e s © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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No part of this paper may be reproduced in any form without permission o f the author.

© 2 0 0 0 M arcelo Fernandes Printed in Italy in March 2 0 0 0

European University Institute

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C entral Lim it Theorem for A sym m etric

Kernel Functionals

Marcelo Fernandes

February 2000

A b stra ct

Asymmetric kernels are quite useful for the estimation of den sity functions which have bounded support. Gamma kernels are designed to handle density functions whose supports are bounded from one end only, whereas beta kernels are particularly conve nient for the estimation of density functions with compact sup port. These asymmetric kernels are non-negative and free of boundary bias. Moreover, their shape varies according to the loca tion of the data point, thus also changing the amount of smooth ing. This paper extends the central limit theorem for degenerate U-statistics in order to compute the limiting distribution of cer tain asymmetric kernel functionals.

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1 Introduction

Fixed kernels are not appropriate to estimate density functions whose supports are bounded in view that they engender boundary bias due to the allocation of weight outside the support in the event th at smoothing is applied near the boundary. A proper asymmetric kernel never assigns weight outside the density support and therefore should produce better estimates of the density near the boundary. Indeed, Chen (1999a,b) showed th at replacing fixed kernels with asymm etric kernels increases substantially the precision of density estimation close to the boundary. In particular, beta kernels are particularly appropriate to estim ate densities with com pact support, whereas gamma kernels are more convenient to handle density functions whose supports are bounded from one end only. These asymmetric kernels are non-negative and free of boundary bias. Moreover, their shape varies according to the location of the d ata point, thus also changing the amount of smoothing.

The aim of this paper is to build on Hall’s (1984) central limit theorem for degenerate U-statistics in order to derive asymptotics for asymmetric kernel functionals. The motivation is simple. It is often the case th at one must derive the limiting distribution of density functionals such as

where the support A is bounded. Examples abound in econometrics and statistics. Indeed, a central limit theorem for the density functional (1) is useful to study the order of closeness between the integrated square error and the mean integrated squared error in the ambit of non-parametric kernel density estimation. Although there are sharp results for non- param etric density estimation based on fixed kernels (Bickel and Rosen blatt, 1973; Hall, 1984), no results are available for asymm etric kernel density estimation.

Furthermore, goodness-of-fit test statistics are usually driven by second-order asymptotics (e.g. Ait-Sahalia, 1996 and Ai't-Sahalia, Bickel

(1) © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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and Stoker, 1998), so that density functionals such as (1) arise very naturally in th at context. Consider, for instance, one of the goodness-of- fit tests advanced by Fernandes and Grammig (1999) for duration models gauges how large is

A( / , 0) = r w(x)[Te(x) - r f (x ) ff( x ) d x , (2)

JO

where w(x) is a trimming function and T/(■) and r fl(-) denote the non- and paxametric hazard rate functions, respectively. It follows from the functional delta method th at the asymptotic behaviour of (2) is driven by the leading term of the second functional derivative, namely

J o P ( z ) [ / (2 0- / ( z ) ] 2 dx = 2^ W^ r - F ( x ) ~ / ( x ) ]2dx.

Note th at duration data are non-negative by definition, hence it is con venient to utilise gamma kernels to avoid boundary bias in the density estimation.

The remainder of this paper is organised as follows. In section 2, I review the properties of beta and gamma kernels. In sections 3 and 4 , 1 apply Hall’s (1984) central limit theorem for degenerate U-statistics to derive the limiting distribution of gamma and beta kernel functionals, respectively.

2 Asymmetric kernels

Let X i , . . . , Xt be a random sample from an unknown probability density function / defined on a bounded support A. In what follows, I consider that A is either bounded from one end or com pact. W ithout loss of generality, I assume th at A = [0, oo) in the first case, whereas A = [0,1] in the second context. Finally, assume that the density function / and its second order derivative are bounded and uniformly continuous on the real line. © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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Instead of the usual non-parametric kernel density estimator

f( x ) =

where K is a fixed kernel function and h is a smoothing bandwidth, consider the asymmetric kernel estimator

f( x ) = K A(Xt),

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where K A(-) corresponds either to the gamma kernel

T. , x uxtbex p (-u /b ) Tf .

k x/m A u) ~ Y {x /b + \ )if/b I ^U6 ^O,00:i} or to the beta kernel

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ux!b(\ —

K x/b+i,(1-* )/t+ i(u ) = € [0 ,1 ]} (5)

according to the support under consideration.

Chen (1999a,b) showed th at both estimators are boundary bias free in view th at the bias is of order O(b) both near the boundaries and in the interior of the support. The absence of boundary bias is due to the fact that asymmetric kernels have the same support of the underlying density, and hence no weight is assigned outside the density support. The trick is that asymmetric kernel functions are flexible enough to vary their shape (and thus the amount of smoothing) according to the location of x within the support.

On the other hand, the asymptotic variance of asymm etric kernels is of higher order O (T~lb~1) near the boundaries than in the interior, which is of order O (T ~lb~l/2^j. Nonetheless, this has negligible im pact on the integrated variance, thus it does affect the mean integrated square error. Furthermore, it is possible to show that the optimal bandwidth b, =

O = O (hi), where h. is the optimal bandwidth for fixed kernel

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estimators. Accordingly, both beta and gamm a kernel density estimators achieve the optimal rate of convergence for the mean integrated squared error of non-negative kernels.1 Lastly, a unique feature for the gamma kernel estimator is that its variance decreases as x increases, though at the expense of an upping in the bias.

3 Gamma kernel functionals

The asymptotic behaviour of gamma kernel functionals of the form (1) is derived using U -statistic theory. For this reason, I utilise a decompo sition which forces a degenerate U-statistic to emerge. Let rr(x , X) =

tpl'2(x)K xlb+l<b(Xt), rT(x ,X ) = rT(x ,X ) - E x [rT(x,X )] and f u denote the integral over the support of u. Then,

I = _l 2 dx = 1 s , t J x T (x ,A ,)d x = I\ + I l + 13 + h , where Ji = / rT(x ,X t)rT(x ,X ,)dx 1 3 < t J x h = f i Y , J x rU x ,X t)d x h = \ & xM * , * ) ] d * u = X t)Ex[rT{x

The first term stands for a degenerate U -statistic and will contribute with the variance in the limiting distribution. The second term will

1 Non-negative kernels define the d ass of second order kernel functions. Higher order kernels may bring about some bias reduction at the expense of assuming negative values (see Muller, 1984, for a list).

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contribute with the asymptotic mean, hence it may be interpreted as some sort of asymptotic bias of the functional estim ator depending on the context. The third and the fourth terms are, in turn, negligible under a proper choice of the bandwidth

b.

Suppose the bandwidth

b

is such that

Tb9/<4 shrinks to zero as sample size T grows. This assumption implies some degree of undersmoothing in view that Chen (1999b) has shown th at the optimal bandwidth for gamma kernels is of order O ( r ~ 2/b^.

I start by deriving the first two moments of r r ( x , X ) . Note th at

Ex (rT(x,X)) = v l' 2(x) j x Kx/b+l,b(X )f(X )dX = <pl/2(x)E( [f(fl],

where ( ~

Q(x/b

4 -1,

b).

The mean and variance of a

G(n, v

) are simply

fiv and jiv2, respectively. Therefore, applying a Taylor expansion yields

E ( [f(C)\ = f ( E c) + ^f"(x)V( + o(b)

= f ( x + b) + ^ f"(x)(x + b)b + o(b)

= f( x ) + b / ' ( x ) + i / " ( x ) ] + o (6).

It is noteworthy th at the last expression demonstrates th at the gamma kernel estimation of the density function has a uniform bias of order 0(b).

Put differently, the order of magnitude of the bias does not depend on the position of x, th at is, whether it is close to the origin or in the interior of the support. To sum up, E x [rT(x,X )} = ipx/2(x )}(x ) + 0 (b ), which implies that rT(x ,X ) = 0(b).

The second moment of r r (x ,X ) is computed in similar way. It follows from Chen’s (1999b) derivation of the variance of the gamm a kernel estimator that

Ex [t2t(x, X)] = <p(x)Jx K l /b+lJb(X )f(X ) dX = 'P(z)Bb(x)E1,[f(Tj)], where B b(x)

T(2x/b + \)/b

22*/o + ir2(x/& + 1) © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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and 77 ~ Q(2x/b + 1, b). Hence applying a Taylor expansion yields

E n[f(v)} = f { E , ) + \ n * ) V n + o(b)

= f(2 x + b) + ^ f"(x)(2x + b)b + o(b) = f(x ) + }'{x)x + b [f'(x) + f"(x)x] + o(b) = f( x ) + f'(x )x + 0(b).

It follows then that

E { h ) = ± j ' E x [i& x,X )\ d i

= f j x <P{x)Bb{x){f{x) + f'(x )x +0(6)]dx

=

^

Jx

<fiix )B b(x )f(x )d x + 0 ( 1 /T ).

For b small enough, Chen (1999b) approximates B b(x) according to the behaviour of x/b. The motivation stems from the fact th at, in the interior of the support, x /b grows without bound as b shrinks to zero, whereas x /b converges to some non-negative constant c in the boundary. The decomposition dictates that

( if x /b —> oc

B b(i:) ~ <

i if x /b - *

c,

which implies th at B b(x) is higher near the origin. Nonetheless, I show th at there is no impact whatsoever in E(12).2

Let 6 = 61-£, where 0 < e < 1. Then,

E { h ) = ^ J ^ ( x ) B b(x )f(x )d x + 0 (\ /T )

2 This result is analogous to Chen’s (1999b) result concerning the variance of the gamma kernel estimator. In particular, the variance mounts as x approaches the boundary, but this increase does not affect the integrated variance of the estimator.

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= ^ f + f <fi(x)Bb(x )f(x )d x + 0 ( 1 /T ) T Jo Ji

= 2^ n T

I

6-1/V 1/V ( a :) / ( x ) d i + 0 ( T -16- e) = J x V(x )x~l/2f(x )d x + o ( T - lb - l/2)

provided that t is properly chosen and E jy>(x)x“ 1/2] is finite. Therefore, it ensues that

L— 1/4

T b ^ E ( h ) = t — E [z -> 'V (x )] •

Notice also that

v { h ) = y,e

J

7j.(x, X)da — E 2

T 3

J

77.(1, .Y )dzj

J

j t

(

x

,X )

dxj -

J^Er%.(x,X)

d ij = o ( T -3r ‘) .

T 3E

Thus, V (Tb^AI 2) = T 2b1/2V (I2) = O ( j ^ f e - 1/ 2) , which is of order o (l) given the assumption on the bandwidth. Thus, by Chebyshev’s inequal ity,

T61/4/ 2 - — = E [ x -1/2^ ( x }] = op(l ).

The fact th at b = o (t- 4/9) also ensures th at the third and fourth terms are negligible if properly normalised. Indeed, it follows from

h =

^

l

£ 2[rr (x, X)]dx = (ft2) = O (h2)

that T bl^ I2 = O (Tb9^^, which is o (l) by assumption. Furthermore,

h = 2(7^ ) E j f ?t(x, Xt)E x rT(x, X )dx © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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and hence

E(I<) = 2{Tt 1} J x E x [rT(x ,X )]E x [rT(x,X )}dx = 0

given th at f T(x ,X ) has zero mean. Besides,

E x { l f T(x, X t)E x [rT(x, X ) ] d x} 2 = O (b2) ,

which implies th at E (I4) = O (T _1fc2) and therefore

E ( T61/4/4)2 = T 2bx'2E ( / 42) = O (Tb5/2) = o (l).

Afresh, it stems from Chebyshev’s inequality th at T V/4/ 4 = op(l). Finally, recall that / , = E » < t ^ r ( A t, A a), where

HT(Xt,X .) = j s f ' M x ,X t) fT(x ,X ,)dx.

Then, I x is a degenerate U -statistic in view th at HT(Xt, X ,) is symmetric, centred, and E [HT{Xt, A3)|X3] = 0 almost surely. To see why, note that

E [H T(Xt,X,)\X,} = ^ f x rT(x ,X ,) E [ fT(x ,X t)\X,]dx = J x rT(x ,X ,)E [fT (x ,X t)]dx

in view of the independence between X t and X ,. It suffices then to observe th at fr (x , Xt) has by construction zero mean. Thereby, I apply Hall’s (1984) central limit theorem for degenerate U-statistics, which states that if

E x ^ i E l ' l H T i X u X J H T i X u X ^ + l E x ^ W i X u X , ) }

m ( X u X ,)} U

as sample size grows, then

h JL + N ^0, y E XuXl [h2(Xu X 2) ] ) . © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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Tedious algebra shows that (6) holds. Indeed, the two terms of the nu merator are of order 0 (T ~ l2b~2) and 0 ( T _96_3/2), respectively, whereas the denominator is of order 0 (T ~ ab~l ). In what follows, I demonstrate the last assertion as a by-product of the derivation of the asymptotic variance above.

Let V„ = ^ E XuX, [H l(X lt X 2)}, then

V„ = 2 J XiX2\Jx f T (x ,X i ) fT(x ,X 2) d x ^ f ( X 1,X 2)d (X u X 2)

= 2 J ]Jx f T(x, X ) fT(y, X ) f( X )d x ]2 d (x , y)

= 2 [ <p{x)<p(y)E2x { [A'

x

/6+1,6(AT) - E K(x,

j

,)]

J x ty

v

1

X [^v/fc+l,»(^0 -

E K(v,b)

]} d (i,y ),

where E K{tl,b) = E x [ ^ u/i,+li6( J f ) ] . Then, it ensues th at

vh = 2 j y V{x)v (y) [fx K ,M » { X ) K „ m Jb(X )d F (X )] \ x , y) + 0 (b 2)

due to the fact that

EK(x,b)EK(y,b) f x K x/b+l,b(X )E K M d F (X ) ' J x E K^ b)K y/b+lfi(X )d F (X ) [ EK(x,b)EK(y,b)dF(X ) 0 (6' ) . Let g(X ) = f ( X ) K x/b+l'„(X), then VH = 2J <fi{x)ip(y) \jx ff(X )d tf y/k+1,6( X)]2d(x, y) + 0 (b 2). © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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Applying a Taylor expansion gives

J x g(X)dKv/b+ltb(X)

= £c(v/6+1,6)[S (A)]

This means that

ri- , v \i , 9"{y)Vdy/b+i,b){X ) m = 9[E(Hs/b+iMx )\+ 2 + 0 (°) = g(y + b) + ^g"{y)(y + b)b + o{b)

= fl(y) + b g'(y) + ^g"(y)y + o(6) = g{y) + 0(b).

Vh = 2 <p{x)<p(y) [f(y )K x/b+ììb{y)]2d(x,y) + 0 ( 6 2)

= 2 j x

^ ( x )

\ j y iP (y)f‘2(y)K l/t,+i,b(y)dy

\ da: + ° ( fc2)

~

2 Jx

^ ( X) [ / /l(l/)d ^x/6+l,6(ì/)j d i + 0 ( 6 2) ,

where h(j/) = ‘fi(y )f2(y)K x/b+i_ib(y). Afresh, by Taylor expanding, it yields

J^h(y)dKx/b+llb(y) = Eç(x/b+i,b)[h(y)i

h \Ee(x/b+\,b)(y)\ + 2h"(x }Vg(x/b+\,b)(y) + o(6) h( x + 6) +

i/i"(x)(x

+ 6)6 + o(6) h (x ) + 6

/i'(i)

+

^/i"(i)xj

+ o(6) h(x) + 0(6). Therefore, Vh = 2

f

tp(x)h(x)dx +0(6) Jx = 2

f

<p2( x ) f2(x)K x/b+l'b(x)dx + 0 ( 6 ) JX

= 2 j x

<fi2(x )f(x )K x/b+lib(x)dF{x) + 0 (6 ) . © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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Notice however that using the same technique it is possible to show that f x f ( X ) K 2x/Mtb(X )d X = K x/b+i}b(x )f(x ) + 0(b). Hence, it follows th at VH = 2 £ i p 2(x )f(x )K x/b+lib(x)dF (x) + 0(b) = 2 jT v 2(x) [ jx

f(X )K l/MJb(X)dx\

d F (x ) +

0(b)

= 2

J

ip2(x)B b(x )[f(x) + 0(6)]d F (x ) + 0(6) = 2 [ <p2(x)B b(x )f(x )d F (x ) + 0 (6 ) . JX

By decomposing the integral according to <5 = 61-e, it yields

V/r =

f

+

f

2ip2(x)B b(x )f(x )d F (x ) + 0(b) J o J s h~1/2 *oo = — / ip2(x)x~1/2f(x )d F (x ) + 0(6_£) y /n J s h -1/2 , = —7^ [ < p 2(x)x~l/2f(x )d F (x ) + 0 (b~ 1/2) y/H J x

for a properly chosen e and finite E ^2(x ) x _1^2j. Finally, this implies that E\uX2 [H$.(XuXi)} = 0 (T ~ 8b~1) and that

\

Tb1'4!

"

b

2 ^ E

[I_1/V W ]

N

(o,

-±=E

[^2(x )x -/2/(x )]) .

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4 Beta kernel functionals

I derive the asymptotic behaviour of beta kernel functionals using the same approach as before, that is, I consider the decomposition / =

h + h + h + h - The only difference is th at rT(x, X ) represents now

ipl/2(x)K x/b+Ui . x)/b+l(Xt). Again, the first term stands for a degen erate U-statistic and contributes with the asym ptotic variance, whereas

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the second term provides the asymptotic mean. The third and the fourth terms are, afresh, negligible under proper normalisation provided that the bandwidth b is of order o (T~4/ 9^. Once more, this assumption implies some degree of undersmoothing in view that Chen (1999a) has shown that the optimal bandwidth for beta kernels is of order O (t ~2/<5).

The limiting distribution of beta kernel functionals is perfectly anal ogous to th at derived for gamma kernels. The only distinction stems from the consideration of the upper bound, which engender a correction in versely proportional to the square root of x ( l — x) instead of x. More precisely, I show in the sequel that

Tb1/4I - <P(x)

,y/x(l - x ) _

<p2(x )f(x ) \ V I (1 - x ) \ J

(

8

)

I start by noting th at the expectation and variance of a B (fi,v)

are v /(fi + v) and l*v/[(n + v)2(fi + v 4- 1)], respectively. It is then straightforward to derive the first two moments of rT(x ,X ). Indeed,

E x [rT{x,X)\ = <pl'2(x) J x K x/b+Ul_x)/b+l(X )f(X )d X

= <pl/2(x)E ( {f( C)],

where £ ~ B (x /b + 1 , ( 1 — x )/b + 1). Therefore, the mean and variance of £ are v ( (1 — x )/b 4 -1 1 — x + 6 x /b + 1 4- (1 - x )/b 4 -1 (x/b + 1)[(1 - x )/b 4-1] (1 /6 4- 2 )2( l / 6 4- 3) 1 4 -2 6 = x ( l — x)b 4- O (b2Sj ,

respectively. Applying a Taylor expansion yields

£<[/( 0 ] = / (£ c) + ^/"(x)M< + o(6) = / + ^ " ( x )x (1 ~ x )b + o(b) © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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= / ( * ) + / ' ( * ) - — b 2bX + “ x ìb + ^ = f( x ) + / ' ( x ) ( 1 - + i / " ( x ) x ( l - x)6 + o(6)

= / ( * ) +

= f{ x ) + 0 (b),

f'(x )(l —2x ) ^ / " ( x ) x ( l - x ) 6 + o(6)

which implies th at the beta kernel estimation of the density function has a uniform bias of order 0(b). To sum up,

E x [rT(x,X )} = p1/2( x ) / ( x ) + 0 ( 6 ) ,

which implies th at fr (x , X ) = 0 (6 ).

Now I turn to the second moment of r r (x ,X ), namely

E x [r2T(x,X )} = <p(x) £ K l/b+ltil_x]/b+i(X )f(X )d X = <p(x)Ab(x)En[f(ri)],

where

B [2x /6+ 1,2(1- x )/6 + l] B2[x/6 + 1, (1- x)/6 + 1]

and T)~ B (2 x /b + 1 ,2 (1 — x)/6 + 1). The mean and variance of »j are

V , 2(1 - x ) /6+ l 2x/6 + 1 + 2(1 — x)/6 + 1 (2x/6 + 1)[2(1 — x)/6 + 1] (2 /6 + 2)2(2 /6 + 3) 2(1 — x ) + 6 2(1 + 6) ^ x (l - x )6 + 0 (62)

respectively, hence applying a Taylor expansion yields

E v(f(r,)} = f ( E v) + ^f"(x)Vv + o(6) 13 © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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=

**f(x) + /'(*)( 1**

-

* x )2 2{il bb)

+

~f"(x)x(l

-

x)b

+ o(b)

=

f ( x) + \ f ' ( x ) ( l - 2 x ) + ^ f " ( x ) x ( l - x )

6 + 0(6) =

f(x ) + f ' ( x ) ( l - 2 x )

2

= f(x ) + 0(b).

Then, it follows that

£(« -

f Ex [r fa , X ) ] dx

<p(x)Ab(x)[f(x)

+ 0 (6 )]d x

<p(x)yU (x)/(x)]dx + 0 ( 1 / T ) .

For 6 small enough, Chen (1999a) showed th at Ab(x) may be ap proximated according to the location of x within the support. More precisely, x/6 and (1 — x)/6 grows without bound as 6 shrinks to zero in the interior of the support, whereas either x/6 or (1 — x)/6 converges to some non-negative constant c in the boundaries. The approximation is such that

which implies that Ab(x) is of larger order near the boundary. Nonethe less, I show th at there is no impact whatsoever in E ( I 2).3

Let S = 61-e, where 0 < e < 1. Then,

3 T his result is analogous to Chen’s (1999a) result concerning the variance of the beta kernel estimator. In particular, the variance mounts as x approaches the boundary, but this increase does not affect the integrated variance of the estimator.

E (h)

=

^ Ji <p(x)Ab(x)f(x)dx + 0 ( l / T )

= 4 / + /

+ [ <

fi(x )Ab(x)f(x)dx + 0 ( 1 /T)

1 J o J s J l - 6 © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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1

= 2' j i ï r f i b l,2[ ^ - x ) ] l/2ip(x)f(x)dx + 0 { r lb ()

k -1/2 r !

= Jo <Pix )[x ( 1 - x)]~l/2f(x )d x + o (T - lb~l/2)

as long as e is properly chosen and E |<p(i)/^/i(l — i ) j is finite. There fore, it ensues that

f t - 1/-»

Tbl/4E (I2) = ~ = E <f(x ) ,\/x (l ~ x)

Notice also that

w * ) = f s E j ^ ( x , X )d ij - ^ 3 E 2 J j $ ( x , X)dxj =

^ E ^ r $ ( x , X ) d x

j

J^Er$(x,X )

dx]

= o ( r - 3b~l) .

Thus, V(Tb^4I 2) = T 2bl'2V (I2) = O ( T ^ f r 1/ 2) , which is of order o (l) given the assumption on the bandwidth. Thus, by Chebyshev’s inequal ity,

Tbl/4I2 - <P(*)

.\Jx(l-x).

Op(l).

Applying exactly the same techniques used in the gam m a context, it is straightforward to demonstrate th at the third and fourth terms are negligible under proper normalisation. Indeed, the fact th at the band width is such th at b = o (t~4/9^ suffices to guarantee th at Tbl/4I3 = o (l) and T b1/4U = op(l ). Lastly, it is evident given the previous discussion that Ii = X)j<t H r(Xt, X s), where HT(Xt,X .) = ^ J j T(x ,X t)rT(x ,X a)dx, © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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is a degenerate U-statistic. Let VH = ^-EXux , [H$(XU A'2)], then

Vh = 2J ^ [fx fT (x ,X 1) fT(x ,X 2)d x ^ f ( X u X2)d(X u X2)

= 2 f x y [Jx M m , X ) fT(y, X ) f( X ) d x] 2 d (x , y)

- 2 f xyV(x )<p(y)Ex {[tfx/&+i,(i-x)/6+i(A-) - £'/r(x,6)]

x [-ft'v /6+l,(i-v)/6+ l ( A ') - £ * - ( „ ,6J ] } d ( z , i f ) ,

where E x(u,b) = E x [ifti/6+it(i-t0/6+i ( X ) ] . As before, it turns out that

Vh * 2 ^ (x )^ (j/) [ £ K i+ l^ +l (X )K i+ l^ +l ( X ) d F ( X ) ] ’ d (x, y)

due to the fact that all other terms are of order O (62). Let g(X ) = f( X ) K x/b+i^i_x)/b+i(X ) and write

V" ~ 2 L y [Jx s (X )d K v/t>+i ,(l-y)/6+l ( A ) ]2d (x ,y ). It follows from a Taylor expansion that

f x 9{X ) dAy/t+i,(!_„)/(,+! (X ) ~ E B{y/b+W-v)/b+l)[g(X)] = 9 [ £ f l ( y / 4 + l ,( l - | / ) / 6 + l ) W ] + 2®” ( ® ) ^ ( v / f r + l ,( l - » ) / * + l ) ( - ^ 0 + °(b) - y + b\ i g"{y)y(l - y)b 91 1

+

26 = 9(y) + 0(b), + ■ o{b)

which implies that

Vh - [/ (j/ )-^ x / t+ i,(i-x )/ 6 + i(y )]2 d (x ,j/ )

- 2 / ¥»(x) f <p{y)f2{y)K l/b+ul- x)/6+i(i/)dj/di •'£ ./y - 2 / <p(x) / /i(y)d^l/j+1,(1_l)/t+1(j/)dx, © The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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where h(y) = ifi(y)f2(y)K x/b+i^^x)/b+i(y). Applying another Taylor ex- pansion gives forth that

h ( y ) d K x/b+i,(i-x)/b+\

(

V

)

= £b(x/<H-1,(1—X)/IH-1) [Ml/)] = h [-^ B (x / 6 + l,(l—x )/ 6 + l)(2/)] + 2 h"(x )VB(xlb+\,(\-x)lb+\){y) + o(b) = h ^ 2^ ) + ^ " ( x ) z ( l ~ x )b + °(b) = h ( x ) + 0 ( b ) . Therefore, Vh — 2

j

<p(x)h(x)dx - 2 [ <p2( x ) f2(x)K x/b+ i,(i_*)/6+ i ( i ) d i J x - 21 <p\x) [ jx H X ) K l/b+ w _x)lb+1 (A -)d x ] d F (x ) - 2 [ <p2(x)Ab(x )[f{x) + 0(b)]dF (x) J x - 2 [ ip2(x)Ab(x )f(x )d F (x ). J x

B y decomposing the integral according to S = bl~(, it yields

Vh — / + / + f 2ifi2(x)Ab(x )f(x )d F (x ) J o J S J l —6

5 -1 /2 f i - 6

~ _{V 7T J {}L V ~ X)1 1 2f{x )d F (x )

- h~ ^ j 0 - x )]“1/2/(a :)d F (x )

provided that e is properly chosen and E [^2(x )[x (l - x )]~ 1/2] is finite. Applying Hall’s central limit theorem for degenerate U-statistics com pletes then the proof.

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References

Ait-Sahalia, Y . (1996). Testing continuous-time models of the spot in terest rate, Review o f Financial Studies 9: 3 85-426.

Ait-Sahalia, Y ., Bickel, P. J. and Stoker, T. M. (1998). Goodness-of- fit tests for regression using kernel methods, Princeton University, University of California at Berkeley, and Massachusetts Institute of Technology.

Bickel, P. J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates, Annals o f Statistics

1: 1071-1095.

Chen, S. X . (1999a). B eta kernel estimators for density functions, Com putational Statistics and Data Analysis. Forthcoming.

Chen, S. X . (1999b). Probability density function estimation using gam m a kernels, School of Statistical Science, La Trobe University. Fernandes, M. and Grammig, J . (1999). Non-parametric specification

tests for conditional duration models, European University Institute and University of Frankfurt.

Hall, P. (1984). Central limit theorem for integrated squared error mul tivariate nonparametric density estimators, Journal o f Multivariate Analysis 1 4 : 1-16.

Muller, H. G. (1984). Smooth optimum kernel estim ators of densities, regression curves and modes, Annals o f Statistics 1 2 : 766-774. ©

The Author(s). European University Institute. produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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