Nonparametric density estimation for linear processes with infinite variance

Transcription

1 Ann Inst Stat Mat 2009) 61: DOI /s x Nonparametric density estimation for linear processes wit infinite variance Tosio Honda Received: 1 February 2006 / Revised: 9 February 2007 / Publised online: 21 September 2007 Te Institute of Statistical Matematics, Tokyo 2007 Abstract We consider nonparametric estimation of marginal density functions of linear processes by using kernel density estimators. We assume tat te innovation processes are i.i.d. and ave infinite-variance. We present te asymptotic distributions of te kernel density estimators wit te order of bandwidts fixed as = cn 1/5, were n is te sample size. Te asymptotic distributions depend on bot te coefficients of linear processes and te tail beavior of te innovations. In some cases, te kernel estimators ave te same asymptotic distributions as for i.i.d. observations. In oter cases, te normalized kernel density estimators converge in distribution to stable distributions. A simulation study is also carried out to examine small sample properties. Keywords Linear processes Kernel density estimator Domain of attraction Stable distribution Noncentral limit teorem Martingale central limit teorem 1 Introduction Let {X i } i=1 be a linear process defined by X i = b j ɛ i j, i = 1, 2,..., 1) j=0 were {ɛ i } i= is an i.i.d. process, b 0 = 1, and b j c 0 j β, j = 1, 2,...,and c 0 is a positive constant. a j a j means tat a j/a j 1as j. Te marginal density T. Honda B) Graduate Scool of Economics, Hitotsubasi University, 2-1 Naka, Kunitaci, Tokyo , Japan onda@econ.it-u.ac.jp

2 414 T. Honda function of {X i } i=1 is denoted by f x). We will specify te conditions on {ɛ i} and β later in tis section. In tis paper, we estimate te marginal density function f x) by kernel density estimators and present te asymptotic properties wen ɛ 1 as infinite variance. Te asymptotic distributions depend on bot te tail beavior of ɛ 1 and β. A lot of autors ave examined te asymptotic properties of kernel density estimators of marginal density functions of dependent observations. Most of tem considered density estimation for mixing processes by imposing assumptions on joint density functions and te order of mixing coefficients until about te early 1990s. See Fan and Yao 2003) for a review of te results for strongly mixing processes. However, it is difficult to ensure tat te order of mixing coefficients satisfies te assumptions unless te coefficients b j decay sufficiently fast. See Doukan 1994) for sufficient conditions for linear processes to be strongly mixing. Terefore attention as been focused on te asymptotic properties of kernel density estimators for subordinated Gaussian processes {GX i )}, were {X i } is a stationary Gaussian process, and linear processes, especially subordinated Gaussian processes and linear processes wit long memory since te late 1980s or te early 1990s. Note tat Hall and Hart 1990) pointed out tat te asymptotic properties depend on te degree of long memory wen we estimate te marginal density functions of linear processes wit long memory. Tis is true of subordinated Gaussian processes wit long memory. As for subordinated Gaussian processes wit long memory, tere are, for example, Ceng and Robinson 1991),Csörgő and Mielniczuk 1995), and Ho 1996). See also te references terein. Tey examined te asymptotic properties of kernel density estimators by exploiting Hermite expansions. Ho 1996) proved tat kernel density estimators beave asymptotically in te same way as for i.i.d. observations wen te degree of long memory does not exceed a level, or we can say wen te degree of long memory is weak. He proved it by evaluating te moments. Hidalgo 1997) studied te asymptotic properties of kernel density estimators for linear processes by appealing to Appell expansions. Except for Gaussian cases, very restrictive conditions are necessary to verify te validity of Appell expansions and te paper does not mention tose conditions at all. In Gaussian cases, Appell expansions coincide wit Hermite expansions. See Sect. 6 of Giraitis and Surgailis 1986) and pp of Surgailis 2004) for te conditions. Teoretical studies for linear processes wit long memory ave developed since te seminal papers, Ho and Hsing 1996, 1997). Tey applied te martingale decomposition metod to examine te properties of subordinated linear processes wit long memory. See Koul and Surgailis 2002) for bot Hermite expansions for Gaussian cases and te martingale approac for linear processes of Ho and Hsing 1996, 1997). Recently several autors considered te asymptotic properties of kernel density estimators for linear processes wit sort memory or long memory by using te martingale approac initiated by Ho and Hsing 1996, 1997), for example, Honda 2000), Wu and Mielniczuk 2002), Bryk and Mielniczuk 2005), and Scick and Wefelmeyer 2006). See also te references terein. Especially Wu and Mielniczuk 2002) fully examined te asymptotic properties of kernel density estimators for linear processes. However, all of tem assumed ɛ 1 as finite variance and tat te distribution of ɛ 1 satisfies some restrictive assumptions, for example, te existence of te bounded and

3 Nonparametric density estimation 415 Lipscitz continuous density function. Under tose conditions, Wu and Mielniczuk 2002) proved tat kernel density estimators beave asymptotically in te same way as for i.i.d. observations in te cases of sort memory and weak long memory and tat kernel density estimators beave asymptotically in te same way as te sample means wen te degree of long memory exceeds a level. It is well known tat standardized sample means of linear processes wit long memory converge in distribution to te standard normal distribution wen ɛ 1 as finite variance. Te standardization is different from tat for linear processes wit sort memory. See Teorem of Taniguci and Kakizawa 2000). Note tat Hallin and Tran 1996) considered kernel density estimation for linear processes wit sort memory by appealing to truncation arguments. Tey assumed tat ɛ 1 as finite variance and β>4. From a teoretical point of view, te marginal density function of {X i }, f x), exists witout finite variance or te bounded density function of ɛ 1. It is strange tat teoretical studies of kernel density estimators are limited to te cases were ɛ 1 as finite variance and te bounded density function. Besides recently a lot of attention is paid to eavy tailed time series data. Terefore it is important to investigate te asymptotic properties of kernel density estimators in te cases were ɛ 1 does not ave finite variance. We examine kernel density estimators in te cases were ɛ 1 does not ave finite variance by exploiting te results of Hsing 1999), Koul and Surgailis 2001), Surgailis 2002), and Pipiras and Taqqu 2003). We treat te asymptotic properties in a compreensive way. We briefly mention te results of te above papers later in tis section. In addition, C2 below also allows unbounded or discontinuous density functions of ɛ 1. Wen ɛ 1 does not ave finite variance, te asymptotic distributions depend on bot te tail beavior of ɛ 1 and β. Wen te effect of te eavy tail of ɛ 1 and dependence among observations does not appear, te asymptotic distributions are te same as for i.i.d. observations. Wen te effect of te eavy tail of ɛ 1 and dependence among observations appears, te asymptotic distributions are stable distributions. Hereafter we sall call te effect tat of α and β. In order to see te differences between asymptotic properties and small sample properties, we carried out a simulation study and te result is given in Sect. 3. We describe te conditions on {ɛ i }.LetGx) denote te distribution function of ɛ 1. C1: Suppose tat 0 <α<2. Ten lim x x α Gx) = lim x x α 1 Gx)) = c 1 > 0. E{ɛ 1 }=0wen 1 <α<2. C2: Letting φθ) denote te caracteristic function of ɛ 1,weave φθ) < C1 + θ ) δ for some positive δ. C stands for generic positive constants wose values cange from place to place and are independent of te sample size n. C1 implies E{ ɛ 1 r } <, 0 < r <α, and E{ ɛ 1 r }=, r α, 2) and tat te distribution of ɛ 1 belongs to te domain of attraction of a symmetric α-stable distribution. Te caracteristic function of te α-stable distribution S α σ,η,µ) as te form of

4 416 T. Honda { exp{ σ α θ α 1 iηsignθ) tanπα/2)) + iµθ}, α = 1 exp{ σ θ 1 + 2iηsignθ) log θ /π) + iµθ}, α = 1, 3) were i stands for te imaginary unit. It is called symmetric if η = µ = 0. See Samorodnitsky and Taqqu 1994) for more details about stable distributions. C2 is necessary for te existence and regularity conditions of bot f x) and joint density functions of some random variables. It is because C2 guarantees te desirable properties of te caracteristic functions of tose random variables. See P1-3 in Sect. 2. We fix x 0 and estimate f x 0 ) by te kernel density estimator f ˆx 0 ) defined below. f ˆx 0 ) = 1 ) K, 4) i=1 were is a bandwidt and K u) is a kernel function. We take = c 2 n 1/5 for some positive c 2 because of simplicity of presentation and partly because tis is te optimal order wen f x) is twice differentiable at x 0 and te effect of α and β does not appear. A comment on te effect of bandwidts is given in Sect. 2. We assume tat K u) is a symmetric bounded density function wit compact support. We examine te asymptotic properties of f ˆx 0 ) E{ f ˆx 0 )} in te following tree cases. Case 1: 1 <α<2 and 1/α < β < 1 Case 2: 1 <α<2 and 1 <β<2/α Case 3: 0 < α < 2 and 2/α < β. Wen 1 <α<2, we ave by te von Bar and Esseen inequality tat E{ X 1 r } C j βr < j=1 for any r suc tat βr > 1 and 1 < r <α. Note tat X 1 as infinite variance. X 1 is well defined in Case 3, too. See te proof of Teorem 2.2 of Pipiras and Taqqu 2003). Some autors say tat {X i } as long memory in Case 1. Koul and Surgailis 2001) deals wit Case 1 and te weak convergence of empirical distribution functions is proved. Te asymptotics of M-estimators of linear regression models are also examined tere. Surgailis 2002) deals wit Case 2. Te asymptotic properties of empirical distribution functions of X i and partial sums of HX i ), were Hx) is any bounded function, are given tere. Hsing 1999) and Pipiras and Taqqu 2003) examined te asymptotic properties of partial sums of HX i ), were Hx) is any bounded function, in Case 3. Tose papers are crucial to our results and tose papers are also based on te martingale decomposition metod of Ho and Hsing 1996, 1997). We ave to obtain necessary teoretical results to deal wit te cases of infinite variance oter tan Cases 1 3 above. Tey are te cases were 0 < α < 1 and 1/α < β < 2/α. Te approaces of Koul and Surgailis 2001) and Surgailis 2002)

5 Nonparametric density estimation 417 crucially depend on te assumption of α>1. We need anoter metod to deal wit te cases and it is a subject of future researc. Peng and Yao 2004) applied Hsing 1999), Koul and Surgailis 2001), and Surgailis 2002) to nonparametric estimation of trend functions, i.e., nonparametric regression wit fixed design. Tere is a close similarity between nonparametric regression wit random design and kernel density estimation. However, nonparametric estimation of trend functions and kernel density estimation are different problems. In Case 1, te effect of te eavy tail of ɛ 1 and te dependence among observations, wic we call tat of α and β, appears wen β < 1/α + 2/5. In Case 2 wen αβ < 5/3. However, we see no effect of α and β in Case 3. We repeat tat te asymptotic distributions are te same as for i.i.d. observations wen te effect of α and β does not appear. In Peng and Yao 2004), te effect of α and β always appears. Te paper is organized as follows. In Sect. 2, we decompose f ˆx 0 ) E{ f ˆx 0 )} into two components and give a euristic argument of te asymptotic asymptotic properties of f ˆx 0 ) E{ f ˆx 0 )}. Ten te main teorems of tis paper are presented. We state te result of a simulation study in Sect. 3. Te main teorems are proved in Sect. 4. Te proofs of tecnical lemmas are confined to Sect Te asymptotic distributions We state te main results of tis paper in Teorems 1 3. First we give definitions and notations. Ten necessary properties of density functions are described. We present te asymptotic distributions of kernel density estimators after a euristic argument. Te proofs of te teorems are deferred to Sect. 4. Let d and p stand for convergence in distribution and convergence in probability, respectively. We omit n and a.s. for brevity. We rewrite X i as X i = X + X, 5) were j 1 X = b l ɛ i l and X = b l ɛ i l. l=0 We denote te distribution functions of X and X by F j x) and F j x), respectively. C2 and Lemma 1 of Giraitis et al. 1996) imply te existence of te density functions and we denote tem by f j x) and f j x), respectively. We state necessary properties of density functions, wic can be verified by using C1 and C2. P1 and P2 are derived by following te proof of Lemmas 1 2 of Giraitis et al. 1996). P3 is part of Lemma 4.2 of Koul and Surgailis 2001). Tere exists a positive integer s 1 for wic P1, P2, and P3 old. P1: f s x) is twice continuously differentiable and f s x) and all te derivatives up to te second order are uniformly bounded for s s 1. Note tat we can take s =. P2: X 1,s1, X i,s1 +i 1) as te bounded joint density function for any i 2. l= j

6 418 T. Honda P3: Wen 1 <α<2, 1 < r <α, and rβ >1, tere exists a constant C suc tat f x) f s x) C s 1/r β uniformly in x for any s s 1. Note tat C depends on α, β, and r. Wen we use P3 in te proofs of Lemmas 2 and 4, r is specified and satisfies te conditions in P3. Before we state Teorems 1 3, we give a euristic argument of te asymptotics of f ˆx 0 ). We need to decompose f ˆx 0 ) E{ f ˆx 0 )} into two components for te argument. Let be s 2 a large positive integer and put s 0 = s 1 + s 2. We will be more specific about s 2 in te proofs of Teorems 1 3. We write S i for te σ -field generated by {ɛ j j i}. f ˆx 0 ) E{ f ˆx 0 )}=S a + S b, 6) were S a = 1 = 1 n S b = 1 = 1 n i=1 i=1 i=1 [ K [ 1 K [ i=1 ) { ) Xi x Si s0}] 0 E K ) ] K ξ) f s0 x 0 + ξ X i,s0 )dξ ) { )}] Si s0} E K K ξ) f s0 x 0 + ξ X i,s0 )dξ 1 {K E [ { E K 7) )}]. 8) Te domain of integration is, ) wen it is omitted. Remember tat X = j 1 l=0 b lɛ i l and tat f j x) is te density function of X. Similar expressions can be found in 3) 5) of Wu and Mielniczuk 2002). In Wu and Mielniczuk 2002) and Bryk and Mielniczuk 2005), s 0 = 1 and te Lipscitz continuous density function of ɛ 1 is assumed. Tey applied te martingale central limit teorem to S a. A tecnique is devised to avoid suc assumptions on ɛ 1 in tis paper. Te asymptotic properties of S a are examined in Lemma 1 below. We investigate te asymptotic properties of S b in Sect. 4 by using te results of Hsing 1999), Koul and Surgailis 2001), Surgailis 2002), Pipiras and Taqqu 2003), and Surgailis 2004). Te asymptotic distribution of 6) depends on wic of S a and S b is stocastically larger. We put = c 2 n γ γ > 0) only in tis euristic argument. In eiter case, we ave S a = O p ) 1/2 ) and E{ ˆ f x 0 )} f x 0 ) 2 2 f x 0 )ν, 9)

7 Nonparametric density estimation 419 were ν = u 2 K u)du. Te asymptotic properties of S b are independent of and depend only on α and β. In addition we ave S b = O p n 1/α β ) in Case 1, S b = O p n 1+1/αβ) ) in Case 2, S b = O p n 1/2 ) in Case 3. Te stocastic order is exact in all te above expressions. Ten 9) and a simple calculation imply tat we cannot improve te rate of convergence of f ˆx 0 ) by coosing γ oter tan 1/5. If tere are tree parameters, α, β, and γ, tings will be notationally complicated and te paper will be longer. Tus we present te teorems wit γ = 1/5. Note tat tere is no teoretical difficulty for tree parameters, α, β, and γ. Wen S b is stocastically larger tan S a, te effect of α and β appears in te asymptotic properties of f ˆx 0 ). Since te asymptotic properties of S b are independent of and depend only on α and β, we ave no optimal bandwidt and we can coose larger bandwidts witout affecting te asymptotic properties of f ˆx 0 ). Wen S a and S b ave te same stocastic order, we can say tat te effect of α and β still appears. However, we ave no result on te joint distribution of S a and S b and we do not refer to tis case in tis paper. Wen E{ ɛ 1 2+δ } < for some positive δ and = c 2 n 1/5, te effect of dependence among observations does not appear in te case of β>9/10 in contrast to Case 2 below. Here we state te main results of tis paper. Case 1 Wen β is smaller tan 1 1/α + 2/5), te effect of α and β appears in te asymptotic properties. Wen α is smaller tan 5/3, te effect of long memory always appears. Teorem 1 Suppose tat C1 and C2 old and tat 1 <α<2and 1/α < β < 1. Ten we ave 1/α β< 2/5: f ˆx 0 ) E{ f ˆx 0 )}) d N0,κf x 0 )), 1/α β> 2/5: n β 1/α f ˆx 0 ) E{ f ˆx 0 )}) f d x 0 )c A Z, were κ = K 2 u)du, Ɣ2 α) cosαπ/2) 1 c A = c 0 2c 1 1 α 1 0 t s) β + dtds ) 1/α x + = x 0, and Z is a random variable wose distribution is S α 1, 0, 0). Te asymptotic joint distributions of te kernel density estimators at different points are independent in case of 1/α β< 2/5and degenerate in case of 1/α β> 2/5, respectively. Case 2 Wen 1 <β<5/3α), te effect of α and β appears in te asymptotic properties. Wen α is larger tan 5/3, te effect does not appear.,

8 420 T. Honda Teorem 2 Suppose tat C1 and C2 old and tat 1 <α<2and 1 <β<2/α. Ten we ave 1/αβ) < 3/5: f ˆx 0 ) E{ f ˆx 0 )}) d N0,κf x 0 )), 1/αβ) > 3/5: n 1 1/αβ) d f ˆx 0 ) E{ f ˆx 0 )}) c 1 c0 α/σ αββ αβ )) 1/αβ) c + f L + + c f L ), were L + and L are mutually independent random variables wose distributions are S αβ 1, 1, 0), c + f = c f = 0 0 f x 0 t) f x 0 ))t 1 1/β dt, f x 0 + t) f x 0 ))t 1 1/β dt, σ αβ = Ɣ2 αβ) cosπαβ/2) /αβ 1). Te asymptotic joint distributions of te kernel density estimators at different points are independent in case of 1/αβ) < 3/5 and degenerate in case of 1/αβ) > 3/5, respectively. Case 3 In tis case, we see no effect of α and β in te asymptotic properties. Teorem 3 Suppose tat C1 and C2 old and tat 0 <α<2and 2/α < β. Ten we ave f ˆx 0 ) E{ f ˆx 0 )}) d N0,κf x 0 )). Te asymptotic joint distributions of te kernel density estimators at different points are independent. Wen te effect of α and β does not appear, we can define te asymptotically optimal bandwidt in te same way as for i.i.d. observations. By combining Teorems 1 2, we know te effect of α and β appears in te following cases. 1 <α<5/3 1/α < β < 1 in Case 1 and 1 <β<5/3α) in Case 2 5/3 <α<2 1/α < β < 1/α + 2/5 incase1 Ten we can only say tat larger bandwidts will improve small sample properties and it may be ard to conduct statistical inference. Te same problem appens for linear processes wit finite variance and long memory. However, it is important to know te statistical properties of suc often used estimators as kernel density estimators. 3 Simulation study We carried out a simulation study to examine te small sample properties. Te result is presented in Tables 1, 2, 3 below. In tis simulation study ɛ 1 follows a standard symmetric α-stable distribution, S α 1, 0, 0). We estimate f x 0 ) by using te Epanecnikov kernel.

9 Nonparametric density estimation 421 We took α = 1.2, 1.5, 1.8, β = 0.9, 1.3, 1.7, 2.1,, { c0 j + 1) b j = β, 0 j , 1000 j We mean i.i.d. observations by β = and c 0 is cosen so tat X 1 also follows S α 1, 0, 0). Wetried = 0.2, 0.3, 0.4 to see te effect of bandwidts. We conducted te simulation study by using R2.3.1 and te fbasics package. Te sample size is 200 and eac entry of Tables 1, 2, 3 are based on 2,000 repetitions. In Tables 1, 2, 3, mean, var, and mse stand for te sample means, te sample variances, and te sample mean squared errors of te repetitions, respectively. Te values of β are on te left margins of te Tables 1, 2, 3. Te true values of f x 0 ) are as follows: α = 1.2 : f 0.0) = , f 0.75) = , f 1.5) = α = 1.5 : f 0.0) = , f 0.75) = , f 1.5) = α = 1.8 : f 0.0) = , f 0.75) = , f 1.5) = Teorem 1 tells tat te effect of α and β appears in te asymptotic properties in te cases of α, β) = 1.2, 0.9), 1.5, 0.9), 1.8, 0.9). Teorem 2 tells tat te effect appears in te asymptotic properties in te case of α, β) = 1.2, 1.3). We obtained te following implications from Tables 1, 2, Te variance is more serious tan te bias in eac pair of α, β). Tus larger bandwidts will be better. 2. Te effect of α and β is seen in te cases of α, β) = 1.2, 0.9), 1.2, 1.3), 1.5, 0.9), 1.8, 0.9). Tis is conformable wit Teorems 1, 2. Especially te effect is remarkable in te case of α, β) = 1.2, 0.9). Even wen te effect is seen, larger bandwidts seem to perform better contrary to Teorems 1, Te effect of α and β rapidly disappears as β becomes larger. 4 Proofs of teorems We prove Teorems 1 3 in tis section. Te proofs of all te lemmas are postponed to Sect. 5. We begin wit Lemma 1 wic deals wit S a in 6) and 7). We reproduce S a ere for reference. S a = 1 = 1 n i=1 i=1 [ K [ 1 K ) { E K ) ) Xi x Si s0}] 0 ] K ξ) f s0 x 0 + ξ X i,s0 )dξ

10 422 T. Honda Table 1 α = 1.2 x Mean Var Mse Mean Var Mse Mean Var Mse Mean Var Mse Mean Var Mse Table 2 α = 1.5 x Mean Var Mse Mean Var Mse Mean Var Mse Mean Var Mse Mean Var Mse

11 Nonparametric density estimation 423 Table 3 α = 1.8 x Mean Var Mse Mean Var Mse Mean Var Mse Mean Var Mse Mean Var Mse Remember s 0 = s 1 + s 2 and tat s 1 is fixed in P1 3. In te proofs of te teorems, we take a large s 2 and temporarily fix it. Ten we let n tend to. Tus we can take n = ks 0 for simplicity of presentation witout affecting te asymptotic properties. Since te summands in S a do not form martingale differences, we cannot apply te martingale central limit teorem directly and we need Lemma 1. We furter decompose S a into four components. were S a = N 1l = 1 E N 2l = 1 E k N 1l + l=1 s 1 +l 1)s 0 i=1+l 1)s 0 { K s 1 +l 1)s 0 i=1+l 1)s 0 { K k N 2l + l=1 [ K k N 3l + l=1 ) ) S1+l 1)s0 s1}] [ { E K k N 4l, 10) l=1, 11) ) S1+l 1)s0 s1} ) Si s0}], 12)

12 424 T. Honda N 3l = 1 E N 4l = 1 E ls 0 i=s 1 +1+l 1)s 0 { K ls 0 i=s 1 +1+l 1)s 0 { K [ K ) ) S1+l 1)s0}], 13) [ { ) E K S1+l 1)s0} ) Si s0}]. 14) N 1l and N 2l consist of s 1 terms, respectively and N 3l and N 4l consist of s 2 terms, respectively. N 1l, N 2l, N 3l, and N 4l are S s1 +l 1)s 0 -, S 1+l 1)s0 -, S 1+ls0 -, and S 1 s2 +ls 0 - measurable, respectively. In addition, E{N 1l S s1 +l 2)s 0 }=E{N 2l S 1+l 2)s0 } = E{N 3l S 1+l 1)s0 }=E{N 4l S 1 s2 +l 1)s 0 }=0. 15) Lemma 1 Suppose tat C1 and C2 old. Ten we ave k l=1 k ) Var N 2l = O l=1 d N 1l N 0, s ) 1 κ f x 0 ), s 0 ) k ) 1, Var N 4l = O n l=1 k l=1 ) 1, 16) n d N 3l N 0, s ) 2 κ f x 0 ). 17) s 0 Remark 1 Take an arbitrary positive integer m. Ten te proof of Lemma 1 in Sect. 5 and standard arguments imply tat k l=1 N 3l for x 01,...,x 0m are asymptotically mutually independent if x 0k = x 0l k = l). We go on to te proofs of Teorems 1 3. We investigate te asymptotic properties of S b in 8) for Cases 1 3 in Propositions 1 3, respectively. Ten by combining Lemma 1 and Propositions 1 3, we derive te asymptotic distributions of f ˆx 0 ) E{ f ˆx 0 )}. Case 1 Te proof of Proposition 1 below is based on te arguments in Koul and Surgailis 2001). Especially te proof of Lemma 2 is a modified and simplified argument of tose of Koul and Surgailis 2001). Proposition 1 Suppose tat C1 and C2 old and tat 1 <α<2and 1/α < β < 1. Ten we ave n β 1/α S b d f x 0 )c A Z. Remark 2 Te proof of Proposition 1 after tat of Teorem 1 implies tat Z in Proposition 1 comes from te sample mean of X 1,...,X n. Terefore two n β 1/α S b for any pair of x 01, x 02 ) are asymptotically degenerate.

13 Nonparametric density estimation 425 We prove Teorem 1. Proof of Teorem 1 First let 1/α β + 2/5 be smaller tan 0. Ten Proposition 1 implies tat S b = o p 1). By taking a sufficiently large s 2 in Lemma 1, we can make s 2 /s 0 and s 1 /s 0 arbitrarily close to 1 and 0, respectively. Tese yield te convergence in distribution of ˆ f x 0 ) E{ ˆ f x 0 )}) d N0,κf x 0 )). Next let 1/α β + 2/5 be larger tan 0. By Lemma 1,weaven β 1/α S a = o p 1). Tus te convergence in distribution of te latter case follows from Proposition 1. Hence te proof of te teorem is complete. Proof of Proposition 1 We adopt te notations of Koul and Surgailis 2001) and prove te proposition by following te arguments tere. Provided tat n β 1/α S b + 1 n K ξ) f x 0 + ξ)dξ n s 0 j= i=1 j+s 0 ) Proposition 1 follows from 1.9) of Koul and Surgailis 2001), n β 1/α 1 i=1 j=s 0 b j ɛ i j d c A Z. b i j ɛ j = o p 1), 18) We will establis 18). As in Koul and Surgailis 2001), we represent S b as S b = 1 = 1 i=1 n s 0 j=s 0 E{K i S i j } E{K i S i j 1 }) j= i=1 j+s 0 ) Ten left-and side of 18) is rewritten as were K ξ) n β 1/α 1 n s 0 E{K i S j } E{K i S j 1 }). 19) j= i=1 j+s 0 ) U i,i j ξ) dξ, 20) U ξ) = f j x 0 + ξ b j ɛ i j X +1 ) f j x 0 + ξ b j u X +1 )dgu) + f x 0 + ξ)b j ɛ i j. 21)

14 426 T. Honda We define R n ξ) by n s 0 R n ξ) = n β 1/α 1 j= i=1 j+s 0 ) ten by Jensen s inequality, we obtain for any r 1, { E K ξ)r n ξ)dξ r } We evaluate E{ R n ξ) r } in Lemma 2 below. U i,i j ξ), 22) K ξ)e{ R n ξ) r }dξ. 23) Lemma 2 For any positive M and r suc tat 1 < r = α/1 + η) and 0 <η< αβ 1)/2, we ave lim sup E{ R n ξ) r }=0. n ξ <M 18) follows from 20), 22), 23), and Lemma 2. Hence te proof of te proposition is complete. Case 2 Teorem 2 follows from Lemma 1 and Proposition 2 below. Te proof of Proposition 2 below is based on te arguments in Surgailis 2002). We adopt te notations of Surgailis 2002). Especially te proof of Lemma 4 is a modified and simplified argument of tose in Surgailis 2002). We applied te argument on pp of Surgailis 2004), te proof of 3.6) tere, to te proof of Lemma 3. Proposition 2 Suppose tat C1 and C2 old and tat 1 <α<2 and 1 <β<2/α. Ten we ave n 1 1/αβ) d c1 c α ) 1/αβ) S b 0 σ αβ β αβ c + f L + + c f L ). Remark 3 Te proof of Lemma 3.1 of Surgailis 2002) implies tat L + and L come from n 1/αβ) n i=1 ɛ i 0) 1/β E{ɛ i 0) 1/β }) n 1/αβ) n i=1 ɛ i 0 1/β E{ ɛ i 0 1/β }), and respectively. As in Teorem 2.1 of Surgailis 2002), L + and L are common to every x 0. It also follows from te proof of Lemma 3.1 of Surgailis 2002) tat te result of Proposition 2 does not depend on s 0. Te proof of Proposition 2 is given after Teorem 2 is proved.

15 Nonparametric density estimation 427 Proof of Teorem 2 First let 1/αβ) be smaller tan 3/5. Ten Proposition 2 implies tat S b = o p 1). By taking a sufficiently large s 2 in Lemma 1, we can make s 2 /s 0 and s 1 /s 0 arbitrarily close to 1 and 0, respectively. Tese yield te convergence in distribution of f ˆx 0 ) E{ f ˆx 0 )}) d N0,κf x 0 )). Next let 1/αβ) be larger tan 3/5. By Lemma 1, weaven 1 1/αβ) S a = o p 1). Tus te convergence in distribution of te latter case follows from Proposition 2. Hence te proof of te teorem is complete. Proof of Proposition 2 We begin te proof wit several definitions. H n, t) = K ξ) f x 0 + ξ t) f x 0 + ξ))dξ, T n = H n, b j ɛ i ) E{H n, b j ɛ i )}), i=1 j=s 0 T n = H n, b j ɛ i j ) E{H n, b j ɛ i j )}). i=1 j=s 0 We prove te following two lemmas in Sect. 5. Lemma 3 Lemma 4 T n T n = o p n 1/αβ) ). ns b T n = o p n 1/αβ) ). We consider T n since Lemmas 3, 4 imply tat ns b = T n + o p n 1/αβ) ).Byte Taylor series expansion, H n, b j ɛ i ) = f x 0 b j ɛ i ) f x 0 ) ξ ) + 2 K ξ) ξ η) f x 0 + η b j ɛ i ) f x 0 + η))dη dξ 0 = f x 0 b j ɛ i ) f x 0 ) + 2 H n, b j ɛ i ), 24) were H n, u) is clearly defined. H n, u) is a continuously differentiable bounded function and te derivative is also bounded. In addition, lim H n, u) = 1 ξ 2 K ξ)dξ f x 0 u) f x 0 )). n 2

16 428 T. Honda By applying te arguments in Lemma 3.1 of Surgailis 2002) to H n, u), we can sow tat for any r <αβ, r E H n, b j ɛ i ) j=s 0 < C r, were C r depends on r. Tis means tat 2 H n, b j ɛ i ) in 24) is negligible in n 1/αβ) T n and te result of Proposition 2 follows from Lemma 3.1 of Surgailis 2002). Hence te proof of Proposition 2 is complete. Case 3 Te proof of Proposition 3 below is a modified and simplified argument of part of te arguments in Hsing 1999) and Pipiras and Taqqu 2003). We adopt te notations of te papers. Since S b = O p n 1/2 ) from Proposition 3 below and we can treat S a in te same way as in Teorems 1, 2, it is easy to prove te result of Teorem 3. We omit te proof of Teorem 3. Proposition 3 Suppose tat C1 and C2 old and tat 0 <α<2 and 2/α < β. Ten we ave Proof Wat we ave to prove is n 1/2 S b = O p 1). ne{sb 2 }=O1). 25) Let ɛ be a random variable wic is distributed as ɛ 1 and independent of {ɛ j } j=. Ten we ave P b j ɛ i b j ɛ 1) C b j α and E{ b j ɛ i b j ɛ 2 I b j ɛ i b j ɛ < 1)} C b j α. 26) See 3.35) and 3.36) of Pipiras and Taqqu 2003) about 26). S b is written as S b = 1 H n X i,s0 ), 27) n were H n ζ ) = i=1 K ξ) f s0 x 0 + ξ ζ) E{ f x 0 + ξ X 1 )})dξ. As in te proofs of Propositions 1, 2, we define U ξ). In Case 3, it is defined by U ξ) = f j x 0 + ξ b j ɛ i j X +1 ) E{ f j x 0 + ξ b j ɛ i j X +1 ) S i j 1 }

17 Nonparametric density estimation 429 = f j x 0 + ξ b j ɛ i j X +1 ) f j x 0 + ξ b j ɛ X +1 ))dgɛ). 28) As in te proofs of Propositions 1, 2, U ξ) is S i j -measurable and we ave E{U ξ) S i j 1 }=0. In addition, H n X i,s0 ) is written as H n X i,s0 ) = K ξ) j=s 0 U ξ)dξ. 29) By 27) and 29), we ave ns b = H n X i,s0 ) = i=1 K ξ) U ξ)dξ. 30) i=1 j=s 0 Te properties of U ξ), 30), Jensen s inequality, and te Caucy Scwarz inequality imply tat E{nS b ) 2 } K ξ) i=1 j=s 0 j =s 0 E{U 2 ξ)})1/2 E{U 2 i, j ξ)})1/2 dξ, 31) were i = i j + j. Provided tat for any positive M, sup E{Ui, 2 j ξ)} C b j α, 32) ξ M 25) follows from 31). Te integrand in 28) is bounded by C1 ɛ i j ɛ)b j )). Terefore by 26), we get E{U 2 ξ)} CE{ ɛ i j ɛ)b j 2 I ɛ i j ɛ)b j < 1)}+P ɛ i j ɛ)b j 1)) C b j α. Hence 32) is establised and te proof of te proposition is complete. 5 Proofs of tecnical lemmas In tis section we prove tecnical lemmas. Te proofs of Lemmas 2 and 4 are modified and simplified arguments of tose in Koul and Surgailis 2001) and Surgailis 2002),

18 430 T. Honda respectively. We applied te argument on pp of Surgailis 2004), te proof of 3.6) tere, to te proof of Lemma 3. Proof of Lemma 1 We write K i for K X i x 0 )/) in te proof for notational simplicity. Note tat for any s s 1, 1 E{K i S i s }= K ξ) f s x 0 + ξ X i,s )dξ 33) and tat te above expression is a bounded continuous function of X i,s. Hence 16) follows from 15) by appealing to te properties of martingale differences. In fact, k ) Var N 2l = i=1 k i=1 E{N 2 2l } and E{N 2 2l } C K,s 1 s 2 1 n 2, were C K,s1 depends on 33). Note tat s 2 is tentatively fixed for N 4l. We prove te latter of 17) byusing15) and applying te martingale central limit e.g. Teorem of Cow and Teicer 1988)). Te former can be treated in te same way and te proof is omitted. Since N 3l C/, we ave only to sow tat 15) and P2 imply tat k l=1 k E{N3l 2 S 1+l 1)s 0 } l=1 E{N 2 3l S 1+l 1)s 0 } p s 2 s 0 κ f x 0 ). 34) k ls 0 1 = E{K i 2 S 1+l 1)s 0 } l=1 i=s 1 +1+l 1)s 0 k ls 0 ls E { } K i1 K i2 S 1+l 1)s0 = l=1 i 1 =s 1 +1+l 1)s 0 i 2 =i 1 +1 k ls 0 l=1 i 1 =s 1 +1+l 1)s 0 E { K i2 S 1+l 1)s0 } ls 0 i 2 =s 1 +1+l 1)s 0 1 E { K i1 S 1+l 1)s0 } k ls 0 1 E{K i 2 S 1+l 1)s 0 }+O p ). 35) i=s 1 +1+l 1)s 0 l=1

19 Nonparametric density estimation 431 By P1 and te ergodic tereom, we obtain k ls 0 1 E{K i 2 S 1+l 1)s 0 } i=s 1 +1+l 1)s 0 l=1 = 1 n k ls 0 l=1 i=s 1 +1+l 1)s 0 p s 2 s 0 κ f x 0 ) K 2 ξ) f i 1 l 1)s0 x 0 X i,i 1 l 1)s0 )dξ + O p ) since E{ f s x 0 X i,i s )}= f x 0 ).34) follows from 35) and 36). 36) Proof of Lemma 2 We adopt te notations of Koul and Surgailis 2001), for example, U k), Mk), Dk), χ k) k), and W. Note tat we can do witout 2.1) tere. We fix a positive γ suc tat 1/r 1/α)/β < γ < α r)/r) 1 1/rβ)). Ten r1 + γ)<α. {M j,n ξ)} defined in 37) form a martingale difference sequence wit respect to {S j }. M j,n ξ) = U i,i j ξ). 37) i=1 j+s 0 ) Ten by te von Bar and Esseen inequality, we ave E{ R n ξ) r } 2 n s 0 j= E{ M j,n ξ) r }. 38) To evaluate E{ M j,n ξ) r }, we decompose U ξ) into tree components, U 1), U 2), and U 3). Hereafter we suppress te dependence on ξ for notational convenience. U 1) = f j x 0 + ξ b j ɛ i j X +1 ) f j x 0 + ξ b j u X +1 )dgu)+ f j x 0 + ξ X +1 )b j ɛ i j 39) U 2) = b j ɛ i j f x 0 + ξ) f x 0 + ξ X +1 )) 40) U 3) = b j ɛ i j f x 0 + ξ X +1 ) f j x 0 + ξ X +1 )). 41) Ten M j,n = 3 k=1 M k) j,n, were Mk) j,n = i=1 j+s 0 ) U k) i,i j.

20 432 T. Honda Provided tat E{ M j,n ξ) r } C i=1 j+s 0 ) r i j) β1+γ), 42) 38), 42) and some calculation as on p. 321 of Koul and Surgailis 2001) imply te result of Lemma 2. We will establis 42). We give some definitions before we consider M 1) 1) j,n and U. χ 1) = I b j v 1, b j ɛ i j 1), χ 2) = I b j v > 1, b j ɛ i j 1), χ 3) = I b j ɛ i j > 1). By using tem, we ave 3 E{ M 1) j,n r } C D k) j,n, 43) k=1 were j,n = E D k) i=1 j+s 0 ) χ k) i,i j U 1) i,i j r. We sow tat all of D k) j,n are bounded by C n i=1 j+s0 )) i j) β1+γ) ) r. We deal wit D 1) j,n and D3) j,n. D2) j,n can be treated in te same way as D3) j,n. We can represent U 1) as were Since b W 1) j ɛ i j = b j u b W 2) j ɛ i j = b j u U 1) = W 1) W 2), 44) ) f j x 0 + v + ξ X +1 )dv dgu), ) f j x 0 + ξ X +1 )dv dgu). f j x 0 + v + ξ X +1 ) f j x 0 + ξ X +1 ) C v γ for v 1, we ave U 1) χ 1) C b j ɛ i j b j u ) v γ dv dgu) C b j 1+γ 1 + ɛ i j 1+γ ). 45)

21 Nonparametric density estimation 433 By 44), 45) and Minkowski s inequality, we get D 1) j,n C C i=1 j+s 0 ) i=1 j+s 0 ) b i j 1+γ E{1 + ɛ j r1+γ) }) 1/r Next we deal wit D 3) j,n. By exploiting 44), we ave D 3) j,n C 2 l=1 k=1 2 E r i j) β1+γ). 46) i=1 j+s 0 ) r i,i j I b i j ɛ j > 1), 47) W l,k) r were b W 1,1) j ɛ i j = b j u f j x 0 + v + ξ X +1 )dv I b j ɛ i j > 1)I ɛ i j > u )dgu), ) b W 1,2) j ɛ i j = f j x 0 + v + ξ X +1 )dv b j u I b j ɛ i j > 1)I ɛ i j u )dgu), ) b W 2,1) j ɛ i j = f j x 0 + ξ X +1 )dv b j u I b j ɛ i j > 1)I ɛ i j > u )dgu), ) b W 2,2) j ɛ i j = f j x 0 + ξ X +1 )dv b j u I b j ɛ i j > 1)I ɛ i j u )dgu). ) Noticing tat W 1,1) =C b j ɛ i j 1+γ I b j ɛ i j > 1), W 1,2) =C b j u 1+γ I b j ɛ i j > 1)dGu) C b j 1+γ, W 2,1) =C b j ɛ i j 1+γ I b j ɛ i j > 1), W 2,2) =C b j u 1+γ I b j ɛ i j > 1)dGu) C b j 1+γ,

22 434 T. Honda we get D 3) j,n C i=1 j+s 0 ) b i j 1+γ r C i=1 j+s 0 ) i j) β1+γ) r 48) Combining 43), 46), and 48), we ave We deal wit M 2) j,n. E{ M 1) j,n r } C i=1 j+s 0 ) r i j) β1+γ). 49) E{ M 2) j,n r } E{ ɛ j r r }E b i j f x 0 + ξ) f x 0 + ξ X i,i j+1 )) i=1 j+s 0 ) r CE b i j X i,i j+1 i=1 j+s 0 ) r C b i j E{ X i,i j+1 r }) 1/r i=1 j+s 0 ) r C i j) 2β+1/r C i=1 j+s 0 ) i=1 j+s 0 ) r i j) β1+γ). 50) See te definition of r and γ about te last line of 50). Finally, by using P3, weave E{ M 3) j,n r } C C i=1 j+s 0 ) i=1 j+s 0 ) i j) β i j) β+1/r r i j) β1+γ). 51) Hence 42) is proved for every k and te proof of te lemma is complete. r

23 Nonparametric density estimation 435 Proof of Lemma 3 We prove Lemma 3 by using te results given in Surgailis 2002, 2004). Especially te proof of 3.6) of te latter. However, we do not deal wit te uniformity and we can do witout 2.3) of Surgailis 2002) in te proofs of Lemmas 3, 4 ere. We write V n for te difference of T n and T n and represent V n as were V n = T n T n = A n k) + B n k), 52) k=1 k=1 A n k) = H n, b j ɛ n+1 k ) E{H n, b j ɛ n+1 k )}), j=k k+n 1 B n k) = H n, b j ɛ 1 k ) E{H n, b j ɛ 1 k )}). j=k We coose a positive numbers r suc tat 1 < rβ,0< 1 + r1 β) < r/αβ) and 1 < r <α. It is not difficult to ceck te existence of r. Ten by te von Bar and Esseen inequality, we obtain ) E{ V n r } 2 E{ A n k) r }+ E{ B n k) r }. 53) k=1 Provided tat E{ A n k) r }=on r/αβ) ) k=1 and k=1 k=1 te result of te lemma follows from 53). We will prove 54). First we deal wit A n k). Since E{ B n k) r }=o n r/αβ)), 54) H n, 0) = 0, H n, x) C, and H n, x) C, we ave H n, b j ɛ n+1 k ) C1 ɛ n+1 k j β )). 55) By exploiting 55), we obtain E[ H n, b j ɛ 1 ) E{H n, b j ɛ 1 )} r ]} Cj βr. 56) As in te argument on p. 337 of Surgailis 2004), we ave r E{ A n k) r } C j β Cn 1+r1 β) = o n r/αβ)). k=1 k=1 j=k Hence te former of 54) is establised.

24 436 T. Honda Next we consider B n k).56) and te argument for A n k) imply tat E{ B n k) r } Cn 1+r1 β) {u 1 β 1 + u) 1 β } r du = o n r/αβ)). k=1 Te latter of 54) is also establised. Hence te proof is complete. 0 Proof of Lemma 4 We adopt te notations of Surgailis 2002), for example, U k), M k) n, j, and V nj. We coose two positive numbers, λ and r, suc tat 2 1 2β 1 + 1/αβ) < r <α and αβ r αβ2βr 1 r) <λ<1 2 αβ αβ3 2βr). Te existence of λ and r is proved in Surgailis 2002). [a] stands for te largest integer wic is less tan or equal to a. We represent ns b T n as were V 1n = V 2n = [n λ ] i=1 j=s 0 i=1 j=[n λ ]+1 ns b T n = V 1n + V 2n, 57) K ξ)u ξ)dξ, K ξ)u ξ)dξ, U ξ) = f j x 0 + ξ b j ɛ i j X +1 ) f j+1 x 0 + ξ X +1 ) f x 0 + ξ b j ɛ i j ) + E{ f x 0 + ξ b j ɛ i j )}. If E{ V n1 2 }=on 2/αβ) ) and E{ V n2 r }=o n r/αβ)), 58) te result of te lemma follows from 57). We will establis 58). As in te proof of Lemma 2, we rewrite V n1 and V n2 as n s 0 V n1 = K ξ) M 1) dξ and V n2 = K ξ) n, j ξ) j=1 [n λ ] n 1 [n λ ] M 2) n, j ξ) j= dξ, 59) were M 1) n, j ξ) = i=1 j+s 0 ) U i,i j ξ)i i j [n λ ])

25 Nonparametric density estimation 437 and M 2) n, j ξ) = i=1 j+s 0 ) U i,i j ξ)i i j > [n λ ]). {M l) n, j ξ)} form martingale difference sequences wit respect to {S j}. 58) follows from 59), Jensen s inequality and te von Bar and Esseen inequality if we ave sown tat for any positive M, uniformly on { ξ M}, n s 0 j=1 [n λ ] n 1 [n λ ] j= i=1 j+s 0 ) i=1 j+s 0 ) Provided tat on { ξ M}, E{Ui,i 2 j ξ)i i j [nλ ])}) 1/2 E{ U i,i j ξ) r I i j > [n λ ])}) 1/r 2 = o n 2/αβ)), 60) r = o n r/αβ)). 61) E{ U ξ) r } Cj 1 2rβ, j s 0, 62) some calculation as on pp of Surgailis 2002) implies 60) and 61). In te calculation we use te fact tat E{ U ξ) 2 } CE{ U ξ) r } since U ξ) is uniformly bounded in, and ξ. We will establis 62). Hereafter we suppress te dependence on ξ. We decompose U into tree components. were U = U 1) + U 2) + U 3), 63) U 1) = f j x 0 + ξ b j ɛ i j X +1 ) f j+1 x 0 + ξ X +1 ) f j x 0 + ξ b j ɛ i j ) + E{ f j x 0 + ξ b j ɛ i j )}, U 2) = f j x 0 + ξ b j ɛ i j ) f j x 0 + ξ) f x 0 + ξ b j ɛ i j ) + f x 0 + ξ), U 3) = E{U 2) }. First we evaluate U 1), wic is written as U 1) = ) b j ɛ i j f j x 0 + ξ + z X +1 ) f j x 0 + ξ + z))dz dgu). b j u

26 438 T. Honda Since b j ɛ i j b j u f j x 0 + ξ + z X +1 ) f j x 0 + ξ + z))dz C b j u + b j ɛ i j ) X +1, we ave E{ U 1) r } Cj 1 2βr. 64) Next we deal wit U 2), wic is also written as U 2) = b j ɛ i j 0 Using P3 and te above expression, we obtain f j x 0 + ξ + z) f x 0 + ξ + z))dz. U 2) C b jɛ i j j 1/r β. 65) 65) implies tat E{ U 2) r } Cj 1 2βr. 66) Finally, by Jensen s inequality, we obtain U 3) r Cj 1 2βr. 67) 62) follows from 64), 66), and 67). Hence te proof of te lemma is complete. Acknowledgments muc. Te autor appreciates elpful comments of te associate editor and te referee very References Bryk, A., Mielniczuk, J. 2005). Asymptotic properties of density estimates for linear processes: application of projection metod. Journal of Nonparametric Statistics, 17, Ceng, B., Robinson, P. M. 1991). Density estimation in strongly dependent non-linear time series. Statistica Sinica, 1, Cow, Y. S., Teicer, H. 1988). Probability teory 2nd ed). New York: Springer. Csörgő, S., Mielniczuk, J. 1995). Density estimation under long-range dependence. Te Annals of Statistics, 28, Doukan, P. 1994). Mixing: properties and examples. Lecture Notes in Statistics, Vol. 85. New York: Springer. Fan, J., Yao, Q. 2003). Nonlinear time series. New York: Springer. Giraitis, L., Koul, H. L., Surgailis, D. 1996). Asymptotic normality of regression estimators wit long memory errors. Statistics & Probability Letters, 29, Giraitis, L., Surgailis, D. 1986). Multivariate Appell polynomials and te central limit teorem. In E. Eberlein, M. S. Taqqu Eds.), Dependence in probability and statistics pp ). Boston: Birkäuser. Hall, P., Hart, J. D. 1990). Convergence rates in density estimation for data from infinite-order moving average process. Probability Teory and Related Fields, 87,

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