Spectral method for decovolvig a desity Marie Carrasco Departmet of coomics Uiversity of Rochester csco@troi.cc.rochester.edu Jea-Pierre Flores GRMAQ ad IDI, Uiversite Toulouse I ores@cict.fr Prelimiary versio: Jauary, 00 Commets Welcome. Itroductio Assume we observe i.i.d. realizatios, x ::: x of the radom variable X with ukow desity h ad X satises X = Y + : where Y ad are idepedet radom variables with probability desity fuctios (p.d.f.) f ad g respectively so that h = f g: The aim of this paper is to give a estimator of f assumig g is kow. This problem cosists i solvig i f the equatio h (x) = f(y)g(x ; y)dy (.) quatio (.) is a itegral equatio ad solvig (.) is typically a ill-posed problem (Tikhoov ad Arsei, 977). Decovolutio is kow to be dicult. Fa (993) gives the optimal speed of covergece for estimators of f ad its derivatives, whatever the method used. The speed of covergece of the Mea Itegrated Square rror (MIS) is particularly slow for supersmooth distributios, for istace it is (l()) ; whe g is the pdf of the ormal distributio ad f is twice dieretiable. The most popular approach to decovolutio is the use of a kerel estimator of f obtaied by applyig the Fourier iversio formula to the empirical characteristic fuctio of X: This method was iitiated by the papers of Carroll ad Hall (988) ad Stefaski ad Carroll (990), later followed We thak James Stock for suggestig this lie of research to us. We are grateful to Susae Scheach for her isightful commets. We also wish to thak the participats of CM coferece (Rochester, 00). This paper is available at www.eco.rochester.edu/faculty/carrasco.html.
by Fa (99a,b, ad 99) amog others. Decovolutio kerel estimators have bee applied i ecoometrics by e.g. Horowitz ad Markatou (996). Recetly, Pesky ad Vidakovic (999) who deal with the estimatio of a decovolutio desity usig a wavelet decompositio. The method we propose here cosists i iterpretig (.) as a itegral equatio Tf = h (.) where T is a compact operator with respect to a well-chose referece space ad therefore admits a coutable iite umber of eigevalues ad eigefuctios. We ivert (.) usig the spectral decompositio of T: We show that if we do oied assumptios o f ad g more precisely if f is smoother tha g the our estimator achieves a much faster rate of covergece tha that obtaied without oied assumptios. I particular, we show that if f ad g are the pdf of two ormal distributios ad the variace of the error (g) is smaller tha that of the sigal (f) the the rate of the MIS is ;= : For a survey o ill-posed problems i the statistical literature ad examples o decovolutio, see Carroll et al. (99) ad Va Rooi ad Ruymgaart (999). They show how to treat the decovolutio problem by solvig the ill-posed problem (.) however they do ot use the trasformatio we use here to make T i (.) compact, therefore they ivert a operator that has a cotiuum of eigevalues. Moreover, they do ot discuss the practical implemetatio of such a estimator. Va Rooi ad Ruymgaart (99) use the same approach to recover a desity i a error-i-variables model o the sphere. The article is orgaized i the followig way. I Sectio, we preset the estimator. I Sectio 3, we establish its speed of covergece. Sectio4 gives its asymptotic ormality. Sectio 5 compares our estimator with the kerel estimator.. Method I Sectio 6, we show that the operator Tf = g(x ; y)f(y)dy (.) is ot a Hilbert Schmidt operator with respect to Lebesgue measure o R ad has a cotiuous spectrum. The mai idea of this paper is to make T compact. To do so, we cosider g (x ; y) as a ctitious probability desity fuctio of Y give X deoted Y X (yx) : We choose more or less arbitrarily a desity X that will play the role of the desity ofx: We costruct a oit desity of(x Y ) that is cosistet with what precedes: (x y) = X (x) g(x ; y): Similarly, we dee m Y as the margial desity of Y m Y (y) = (x y)dx:
Now, choose a arbitrary fuctio (ot ecessarily a pdf) Y (y) that satises Assumptio. There exists a costat M > 0such that m Y (y) Y (y) <M for ay y R: This assumptio garatees that L Y (R) ) L m (R) which itself implies T Y L X (R)by the law of iterated expectatios where L Y (R)adL X (R) are deed below ad L m (R) isthespace of square itegrable fuctios with respect to m Y Y : Deote L X (R) the space: L X (R) = (x) such that (x) X (x) dx < : ad L Y (R) thespace: L Y (R) = (y) such that (y) Y (y) dy < : Both the ier product i L X (R) ad i L Y (R) will be deoted h: :i ad both the orm i L X (R) ad i L Y (R) will be deoted k:k without cofusio: We dee the operator T which associates to ay fuctio (y) of L Y (R) a fuctio of L X (R) as: So that quatio (.) ca be rewritte as (T)(x) = Y X [ (Y ) X = x] : Tf = h: (.) Now, we dee the adoit of T which associates to ay fuctio (x) of L X (R) a fuctio of L Y (R): g (x ; y) (T X (x) )(y) = Y (y) (x) dx: For coveiece, we deote XY (xy) = g (x ; y) X (x) Y (y) eve though it is ot a coditioal expectatio. I the special case where m Y T is a coditioal expectatio operator: = Y, the (T )(y) = X Y [ (X) Y = y] : Note that i geeral T6= T. Assumptio. Y domiates f: 3
If f is cotiuous with respect to (y) Lebesgue measure o R the it is eough to take Y (y) equivalet to (y): The desity f we wish to estimate has to belog to L Y (R) : Assumptio 3. We have f (y) Y (y)dy < : This coditio may seem restrictive however most of the distributios (the ormal, double-expoetial etc) cosidered i the oparametric literature are square-itegrable. Assumptio 4. We have X (x)g(x ; y) X (x) Y (y) X (x) Y (y)dxdy < : This is a suciet coditio for T to be a Hilbert-Schmidt operator ad therefore to be compact (see Duford ad Schwartz, 963, p. 30, Darolles, Flores ad Reault, 998). As a result of compactess, T has a discrete spectrum. Let 0 = be the eigevalues ad ' 0 0 the oit orthoormal basis of T ad T respectively satisfyig: h i i) T ' (Y ) = (X) 0 h i ii) T (X) = ' (Y ) 0 h i iii) T T ' (Y ) = ' (Y ) 0 h i iv) TT (X) = (X) 0: Note that may becomplex, deotes the complex cougate of : Sice g ad X are give, the eigefuctios are either kow explicitly (see xample below) or ca be estimated by simulatios as precisely as wated (see Sectio 5) so that we ca cosider them as kow. Fially to show the cosistecy, we eed the followig assumptio. h i Assumptio 5. var X (X) (X) < for all 0: Note that a suciet coditio for Assumptio 5 is that the p.d.f. h ad X belog to L that is sup h < ad sup X <. Ideed the variace equals var h X (X) (X) i = X(x) (x)h (x) dx ; X (x) (x)h (x) dx : It is eough to show that the rst term is bouded X(x) (x)h (x) dx (sup h) D (:) (:) X (:) (sup h) X (sup h)(sup X) < : 4
quatio (.) ca be approximated by a well-posed problem usig the Tikhoov regularizatio method ( I + T T ) f = T h where the pealizatio term plays the role of the smoothig parameter i the kerel estimatio. Other regularizatio methods could have bee cosidered such as the trucatio which cosists i discardig the smallest eigevalues, however we prefer to use the Tikhoov regularizatio. f becomes f (y) = The oly ukow is T h: Note that D T h ' + ' (y): (.3) (T h)(y) = [h (X) Y = y] A atural estimator of T h is give by dt h (y) = = h (x) XY (xy) dx = h i XY (Xy) X i= So that the estimator of f takes the followig form ^f(y) = X i= XY (x i y) : (.4) D + XY (x i :) ' (:) ' (y): (.5) Remark that f ca be rewritte i the followig fashio: f (y) = = = D h T ' + ' (y) D h + ' (y) h + (X i ) X (X i ) i ' (y): Hece aother expressio of ^f is give by ^f(y) = X i= + (x i ) X (x i ) ' (y): (.6) This expressio requires the estimatio of as well as that of ' however the estimatio of ca be obtaied as a by-product of that of ' without much extra calculatio as explaied i Sectio 5. 5
xample. Assume N(0, ): We set x ; y Y X (yx) =g (x ; y) = where deotes the p.d.f. of a stadard ormal. A simple choice for X is x X (x) = so that X a N (0, ) where a meas that we do as if X were distributed as a ormal. The fact that the true distributio may be totally dieret doesot matter. We eed to determie a) Y b) XY that match Y X ad X : a) Y (y) = Y X (yx) X (x) dx = y p : + b) XY (xy) = Y X(yx) X (x) Y (y) = x ; y p where = =( + ): So that we have X Y a N 0 0 : + To calculate the eigevalues ad eigefuctios, we eed to compute T T the expectatio operator associated with (~yy) = Y ~ Y Y X (yx) XY (x~y) dx = ~y ; y p + The eigefuctios of T T are the Hermite polyomials orthoormal with respect to Y associated with the eigevalues : : 6
3. Speed of covergece 3.. Rate of the MIS i the geeral case The criterio we use is the MIS with respect to Y that is MIS = ^f ; f = ^f(y) ; f(y) Y (y)dy : The criterio usually employed i the kerel literature (e.g. Stefaski ad Carroll, 990) is the MIS with respect to Lebesgue measure o R. However here ^f(y) is ot ecessarily square-itegrable o R, therefore we replace the itegratio with respect to Lebesgue by a itegratio with respect to Y (:). The rates obtaied by Stefaski ad Carroll are ot aected by the choice of the orm so that their rates will be directly comparable with ours. The MIS ca be rewritte as MIS = = ^f(y) ; f (y)+f (y) ; f(y) Y (y)dy ^f(y) ; f (y) Y (y)dy + Var+ Bias (f (y) ; f(y)) Y (y)dy because ^f = f : As i the kerel estimatio, there is a trade-o betwee the variace (decreasig i ) ad the bias (icreasig i ): Propositio 3.. Uder Assumptios to 5, we have MIS = h i var + X (X i ) (X i ) + Df ' + (3.) A + + + : Df ' This boud depeds o the speed at which the ier products D f ' coverge to zero with : 3.. Automatic selectio of the smoothig parameter The pealizatio term must be selected to miimize the MIS give i (3.). Deote ^f D a estimator of f obtaied usig a ooptimal (quite large) deoted : A estimator of f ' is give by D ^f ' = X i= + X(x i ) (x i ): 7
Let ^ ^' ad ^ = ::: B be the estimators of ' ad obtaied by the method h d i described i Sectio 5. Deote var X (X i ) (X i ) the sample variace of X (X i )^(X i ): A estimator of the MIS is give by MIS = 0 BX B ^ @ + C ^ A = h d i var X (X i ) (X i ) + 0 BX B ^ @ + C ^ A = P i= X(x i )^ (x i ) o + ^ : (3.) This espressio ca be miimized umerically with respect to to obtai the optimal smoothig parameter. 3.3. Rate i special cases To obtai D further results o the rate of the MIS, we eed D extra assumptios o the ier product f ' : Here, we ivestigate the case where f ' decays at least as fast as : The advatage of this assumptio is to deliver simple results o the rate of covergece of the MIS. It is by o meas the oly assumptio we could have made, we could impose a weaker requiremet that would, D of course, deliver a slower speed of covergece of the MIS. A suciet coditio for f ' to decay at least as fast as is the followig: Coditio A. There exists a fuctio k such that ad kkk = T k = f k (x) X (dx) < : Propositio 3.. Uder Assumptios to 5 ad Coditio A, by choosig a regularizatio parameter = d ;=3 for some d>0 we have MIS = O ;=3 : This rate is a upper boud, we might be able to derive a optimal rate i some special cases. The rate of covergece of the MIS, ;=3 is slower tha ;4=9 obtaied for the kerel estimator i presece of a double-expoetial error ad of course much faster tha [l ()] ; the rate obtaied i presece of a ormal error. Remark. Uder Coditio A, we have Var C Bias = kkk + (3.3) D k + (3.4) 8 + (3.5)
So that for = d ;= the rate of covergece of the MIS is give by MIS + : (3.6) Whe oe kows the decay rate of the, oe has immediately the rate of the MIS. Note that the rate of is idepedet of the rate of f: Cosider the case where istead of usig the Tikhoov regularizatio, we had used the trucatio that is we replace by zero ad P by P N where N plays the role of a smoothig parameter. The optimal speed of covergece of N would be dictated by the decay rateof ad therefore depeds o f ad g: Lemma 3.3. If g is eve, that is the error has a symmetric distributio aroud zero ad for X (x) = I [; ] (x) = ad Y (y) = for all y R: A suciet coditio for Coditio A to hold is f (t) g (t) dt < (3.7) where f ad g are the characteristic fuctios of f ad g respectively. Coditio (3.7) requires that f has er tails tha g : Sice the tail behavior of a pdf is related to the smoothess of the pdf, this is equivalet to require that f be smoother tha g (see Ushakov, 999, Theorem.5.4). I the case of f Laplacia, this is a very weak requiremet. I the case of f ormal (discussed i details below), it is less likely to be fullled. Aother iterpretatio of this coditio is the followig: f ca be writte as the covolutio of g ad aother distributio. Va Rooi ad Ruymgaart (99) give a good ituitio o the diculty of decovolutio. If g is smooth the h is also smooth. If f is ot a priori kow to be smooth itself, the problem of recoverig a osmooth f from a sample of smooth h is particularly hard. xample cotiued (ormal case). Cosider ormally distributed ad X ormal ad Y is the margial of (x y). Corollary 3.4. Assume Coditio A holds. = d ;= for some d>0 we have MIS = O By choosig a regularizatio parameter ;= : Result. If N(0 ) ad if f satises exp t f (t) dt < : (3.8) where f (t) is the characteristic fuctio of f the Coditio A is satised. I particular, if Y N(0 v ) ad if v > the Coditio A is satised. 9
The proof of Result isgive i Appedix. Remark. Note that i Lemma 3.3, X is bouded ad Y = while, i Result, X is ormal ad Y is the margial of (x y) : Coditio (3.8) requires that the fuctio f be supersmooth, actually it eeds to have er tails tha g: This coditio is atural because it requires i the case of ormal distributios that the variace of the sigal be larger tha the variace of the oise. 4. Asymptotic ormality Because we have iid data, a suciet coditio for asymptotic ormality ^f (y) ; ^f (y) r var ^f (y) L N(0 ) is that the Lyapouov's coditio holds (Billigsley, 995, Theorem 7.3), i.e. for some >0 ; ( ) + 0 (4.) = += [var ( )] where i = D + XY (x i :) ' (:) ' (y): (4.) The coditio (4.) is satised uder the followig assumptios. Assumptio 6. We have X = + 3 0: This coditio requires that go to zero ot too fast. It is satised i the case of a ormal error whe = d ;=, see quatio (8.4). Assumptio 7. There is a costat M idepedet of such that h X (x ) (x ) ; X i 3 <M: Assumptio 7 is trivially satised i most cases. Propositio 4.. Uder Assumptios to 7, if 0 ad,we have ^f (y) ; f (y) r var ^f (y) L N(0 ) : 0
The followig assumptio isures that var ^f (y) ca be replaced by the sample variace. Assumptio 8. There is a costat M idepedet of such that h X (x ) (x ) i 4 <M: ad X + 4 0: Lemma 4.. Uder Assumptios -8, we have X i ; ( i ) P 0 i= X i= i ; i P 0: The followig assumptio guaratees that the bias goes to zero sucietly fast so that f ca be replaced by f. Assumptio 9. P D + P f ' 0 + I the case of a ormal error, assumig Coditio A ad = o( ;= ) Assumptio 8 is satised. To guaratee that the bias becomes egligible i compariso with the variace, we have to let go to zero faster tha the optimal rate. Propositio 4.3. Uder Assumptios to 9, if 0 ad,we have p ^f (y) ; f (y) where s = P i= 5. Implemetatio s L N(0 ) i ; P i= i where i is give by (4.). I this sectio, we discuss the practical aspects of the estimatio of f whe o explicit expressio of the eigevalues ad eigefuctios is available. Below, we explai how to estimate the eigevalues ad eigefuctios. Calculatio of eigevalues ad eigefuctios. We are lookig for the solutios of T T' = ': (5.)
The coditioal expectatio T ad T could be estimated by kerel but there is a simpler way. a) To estimate the operator T, we will use importace samplig (Geweke, 988). Deote a pdf, such that it is easy to draw data from the distributio correspodig to either by iversio of the c.d.f. or by a reectio method (see Devroye, 986). The operator T ca be estimated by (T')(x) = B = BX b= '(y)g(x ; y)dy '(y)g(x ; y) (y)dy (y) '(y b )g(x ; y b ) : (y b ) where (y b ) b = ::: B is a i.i.d. sample draw from : b) The operator T ca be estimated by (T )(y) = = Y (y) B R BX b= (x) XY (xy) dx (x) X (x) g(x ; y)dx Y (y) (x b ) g(x b ; y): where (x b ) b= ::: B is a i.i.d. sample draw from X : This way we obtai estimators of T ad T that are p B -cosistet ad do ot require a choice of a kerel ad a badwidth. Therefore T T'(y) ca be approximated by Y (y) B BX b= " B BX c= '(y c )g(x b ; y c ) (y c ) # g(x b ; y): This operator has a ite rak ad has at most B eigevalues. Note that the eigefuctios are ecessarily of the form ' (y) = BX b= b g(x b ; y) : (5.) Y (y) Replacig i ' by its expressio, we see that solvig (5.) is equivalet to dig the eigevalues ad eigevectors of the B B;matrix M with pricipal elemet: M b l = B B X c= g(x l ; y c )g(x b ; y c ) : Y (y c )(y c )
Let = h Bi 0 be the th eigevector of M associated with, the ' solutio of (5.) is the th eigefuctio of T T associated with the same eigevalue : The fuctio ' ca be evaluated at all poits. Note that the ' associated with distict eigevalues are essarily orthogoal, evertheless, they eed to be omalized. Deote ^' ad ^ the estimators of ' ad. The operator TT (y) cabe approximated by B b= " BX BX c= B B X c= $ (x x c ) (y c ): (y c )g(x c ; y b )g(x ; y b ) (y b ) Y (y b ) # It is easy to verify that the eigefuctios are of the form P B c= c $ (x x c) where = h Bi 0 = ::: are agai the eigevectors of M deed above. Hece the estimators of are give by ^ (x) = BX b= b " BX l= # g(x b ; y l )g(x ; y l ) : (y l ) Y (y l ) Calculatio of ^f: I formula (.5), we eed to compute the term It ca be approximated by D D XY (x i :) ' (:) = d XY (x i :) ' (:) = B XY (x i y)' (y) Y (y)dy: BX b= X (x i ) g(x i ; y b ) ^' (y b ) (y b ): where (y b ) b = ::: B is a i.i.d. sample draw from : Hece we obtai ^f : ^f(y) = X BX i= = + D ^ d XY (x i :) ' (:) ^' (y): We could do a theory that would take ito accout the approximatio error due to the estimatio of the eigevalues ad eigefuctios. But this theory seems useless because B ca be chose arbitrarily large ad we ca make the approximatio error arbitrarily small. 6. Compariso with the kerel desity estimator 6.. Properties of the operator T i L (R ) We rst ivestigate the properties of the operator T deed by (.) if the space of referece is L (R ) : the space of the square-itegrable fuctios with respect to Lebesgue 3
measure o R : This correspods to choose X (x) = ad Y (y) = for all x ad y: First ote that the resultig operator T is ot a Hilbert Schmidt operator because R R [g(x ; y)] dxdy is ot bouded. I the followig, we show that T T has a cotiuous spectrum with t = g (t) g (t) for t R where g (t) = R e itz g(z)dz is the characteristic fuctio of the radom variable : Because X ad Y are set equal to oe, we obtai So that We wat to solve Y X (yx) = g(x ; y) XY (xy) = Y X(yx) X (x) Y (y) T [ (x)] = (T T')(y) = (x) g(x ; y)dx: g(x ; y)dx = g(x ; y): g(x ; )'()d = '(y): We take the Fourier trasform o each side: e ity dy g(x ; y)dx g(x ; )'()d = e ity '(y)dy: We apply two successive chages of variables y = x ; u ad x = v + to obtai: g (t) g (t) e it '()d = e ity '(y)dy So that there is a cotiuous spectrum with g (t) g (t) = : t = g (t) '(y) = e ity = p associated with t (x) = e itx = p associated with t : These eigefuctios are othogoormal o [; ] : To check our calculatio, we write f o this basis of eigefuctios: Df ' ' (y) = f (x) ' (x) dx ' (y) = e ;itx f(x)dx e ity dt: We apply the chage of variable u = ;t to obtai: e iux f(x)dx e ;iuy du = f (u) e ;iuy du = f(y) by the Fourier iversio formula. 4
6.. Kerel estimator We ca apply i a heuristic maer the same method as before to estimate f:i formula (.5), the elemet DT h ' = Dh ca be estimated by = X i= h (x) (x)dx (x i ): Now, we replace the sum over by a itegral over t ad the eigevalues ad eigefuctios by their expressios i (.5) to obtai ^f(y) = X l= g (t) g (t) + e ;itxl e ity dt: We applyachage of variable u = ;t ^f(y) = X l= g (u) g (u) + e iu(xl;y) du: (6.) Usig the type of regularizatio advocated by Carroll ad al. (99, xample 3..), we get ^f(y) = X f g (u)g g (u) eiu(x l;y) du: l= Now compare these two expressios with the kerel estimator (see e.g. Stefaski ad Carroll, 990). For a smoothig parameter ad a kerel K, the kerel estimator is give by ^f k (y) = X K (u) g (u=) eiu(x l;y)= du: (6.) l= The formulas (6.) ad (6.) dier oly by the way the smoothig is applied. This suggests that the kerel estimator is obtaied by ivertig a operator that has a cotiuous spectrum. Because this spectrum is give by the characteristic fuctio of g, the speed of covergece of the estimator depeds o the behavior of g i the tails. For a formal expositio, see Carroll et al (99, xample 3..). They assume i particular that the fuctio to estimate is p dieretiable ad they obtai a rate of covergece (as a fuctio of p) that is of the same order as the rate of the kerel estimator. 5
7. Coclusio It should be emphasized that our approach is fudametally dieret from that of the kerel estimatio. We approximate the fuctio to estimate by a sequece of orthoomal fuctios give by the eigefuctios of the covolutio operator. We show that this estimator is cosistet ad asymptotically ormal. We study its rate of covergece uder extra coditios relatig the smoothess of g with the smoothess of f: We show that uder these extra assumptios, the MIS achieves a fast (arithmetic) rate of covergece. A task that remais to be doe is to prove that, uder geeral assumptios, our estimator achieves the optimal rate of covergece. 8. Appedix Proof of Propositio 3.. We examie successively the terms of variace ad bias. The variace: Usig the expressio of ^f give i (.5), we have ^f(y) ; f (y) = var 4 X D + XY (x i :) ' (:) ' (y) 5 : Because the eigefuctios ' are orthoormal with respect to Y,we have ^f(y) ; f (y) Y (y)dy = + 3 with = var hd i XY (X i :) ' (:) = var XY (X i y) ' (y) Y (y)dy = var X (X i ) Y X (yx i ) ' (y)dy = var h i X (X i ) (X i ) = var h X (X i ) (X i ) i : (8.) So that the variace term ca be maored: Var = A var h i + X (X i ) (X i ) + by Assumptio 4 where A is some costat. 6
Bias: Usig (.3), f ca be rewritte as f = = = D h T ' + ' D h + ' D f ' + ' because h = Tf: We have f ; f = I ; ( I + T T ) ; T T = ( I + T T ) ; f = f D f ' + ' It follows that kf ; f k = + : Df ' Proof of Propositio 3.. From the previous sectio, we have Var A for some A>0 (8.) ad assumig Coditio A, we have kf ; f k = = + Dk T' D k + (8.3) Usig the fact that + 4 we obtai kf ; f k 4 Dk = 4 kkk : 7
Hece, we obtai a maoratio of the MIS MIS A + 4 kkk : For of order = =3, we have MIS C =3 : Proof of Corollary 3.4. We look for a equivalet of the series i (3.3) ad (3.5). For this, we use the followig result. Let f() be the elemet of a series ad assume f()is a positive ad cotiuous decreasig fuctio of : The it is easy to see that J 0 f (s) ds + f (J) JX f() Whe goes to zero, a equivalet oftheseries is give by f() J 0 0 f (s) ds + f (0) for all J : f (s) ds: I the ormal case, the eigevalues satisfy = with < so that as goes to zero: The rate for the MIS follows. Proof of Lemma 3.3. + l() (8.4) T k = f, g (x ; y) X (x) k (x) dx = Y (y) f (y), g (x ; y) k (x) dx = f (y) : (8.5) where k X k, f Y f: Deote F (g) F (k ) F (f ) the Fourier trasforms of g, k, ad f that is F (g)(t) = R e ;itx g (x) dx: (8.5) is equivalet to F (g) F (k ) = F (f ), k (x) = e F itx (f )(t) F (g)(t) dt, k (x) = e F itx (f Y )(t) X (x) F (g)(t) dt for ay x i the support of X 8
by the iversio formula. The coditio R k (x) X (x) dx < is equivalet to Take X =0:5I [; ] ad Y X (x) e F itx (f Y )(t) F (g)(t) dt dx < : (8.6) =: (8.6) is satised as soo as F (f)(t) F (g)(t) dt < : Usigachage of variables t ;t, this is equivalet to Proof of Result. f (t) g (t) dt < : T k = f, x ; y p k (x) dx = f (y), ~ (x ; y) k (x) dx = f (y) where ~ (x) = x= p : Let x = u, we have ~ (u ; y) k (u) du = f (y), (u ; y) k (u) du = f (y) where k (u) = k(u) (v) = (v p =) : Hece we have It follows that k (u) = k (x) = k = f, k = f, k (t) = exp t e ;itu exp t e ;itx= exp t f (t) : f (t) dt f (t) dt: 9
The coditio A is equivalet to e ;itx= exp t It is suciet that exp t k (x) exp ; x dx <, f (t) dt exp ; x dx < : f (t) dt < : (8.7) Hece the result. If f is the pdf of N (0 v ) quatio (8.7) is satised i v > = which is satised for a such that > v : (8.8) Sice the rate of covergece of the eigevalues is faster for small we have iterest to choose so that is as small as possible ad satises (8.8). I practice, is assumed to be kow ad v is ukow but ca be easily estimated from the data. Proof of Propositio 4. First we check that var ( ) is bouded from below. var ( )= X X + ' (y)+ <k + + k i ' (y) ' k (y) where D i = cov XY (x i :) ' (:) D XY (x i :) ' k (:) = k cov X X k : usig the same rewritig as i (8.). var ( ) is a sum of positive terms, it is bouded from below. To establish (4.) for =,we eed to show that ; ( ) 3 = 0: D Usig XY (x :) ; T h ' (:) = h X (x ) (x ) ; as We have ; ( ) 3 ; ( )= h i + X ; X ' (y): X i ca be rewritte 3 h i 3 + X ; ' X (y) 3 +cross;products: 0
The cross-products are domiated by the rst term. The result follows from Assumptios 6 ad 7. Proof of Lemma4. var ( )=O 0 @ X A : + Uder Assumptio 8, var ( ) 0 which implies the weak law of large umbers by Theorem C of Serig (980, p. 7). For the WLLN of i, we use 0 var = O @ X 4 A : + Proof of Propositio 4.3. Uder Assumptio 9, we have f ; f var = ^f P D f ' + ' var ( ) coverges to zero. By Assumptio 8 ad Lemma 4., var ( ) ca be replaced by the sample variace. 9. Refereces Billigsley, P. (995) Probability ad Measure, Wiley & Sos, New York. Carroll, R. ad P. Hall (988) \Optimal Rates of Covergece for Decovolvig a Desity", Joural of America Statistical Associatio, 83, No.404, 84-86. Carroll, R., A. Va Rooi, ad F. Ruymgaart (99) \Theoretical Aspects of Ill-posed Problems i Statistics", Acta Applicadae Mathematicae, 4, 3-40. Devroye, L. (986) No-Uiform Radom Variate Geeratio, Spriger Verlag, NY. Darolles, S., J.-P. Flores ad. Reault (998), mimeo. Duford, N. ad J. Schwartz (963) Liear Operators, Part II, Spectral Theory, Wiley & Sos, NY. Fa, J. (99a) \O the optimal rates of covergece for oparametric decovolutio problems", The Aals of Statistics, 9, No.3, 57-7. Fa, J. (99b) \Asymptotic ormality for decovolutio kerel desity estimators", Sakhya, 53, 97-0. Fa, J. (99) \Decovolutio with supersmooth distributios", The Caadia Joural of Statistics, 0, 55-69. Fa, J. (993) \Adaptively local oe-dimetioal subproblems with applicatio to a decovolutio problem", The Aals of Statistics,, 600-60.
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