Moderate deviations for deconvolution kernel density estimators with ordinary smooth measurement errors

Statistics and Probability Letters 80 (2010) 169–176

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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

Moderate deviations for deconvolution kernel density estimators with ordinary smooth measurement errors Weixing Song Department of Statistics, Kansas State University, Manhattan, 108F, Dickens Hall KS, 66502, United States

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info

Article history: Received 18 May 2009 Received in revised form 9 October 2009 Accepted 9 October 2009 Available online 17 October 2009 MSC: primary 62G08 secondary 62G10

abstract In this paper, we establish the pointwise and uniform moderate deviations limit results for the deconvolution kernel density estimator in the errors-in-variables model, when the measurement error possesses an ordinary smooth distribution. The results are similar to the moderate deviations theorems for the classical kernel density estimators, but a factor related to the ordinary smooth order is needed to account for the measurement errors. © 2009 Elsevier B.V. All rights reserved.

1. Introduction In many practical situations, the random variable X cannot be observed directly, instead, a surrogate Y can be obtained which is related to X in the manner Y = X + ε , where ε is the measurement error with a known density function, X and ε are independent. This is the so-called errors-in-variables or the convolution model. The errors-in-variables modeling is widely used in microfluorimetry, nutrition study, electrophoresis, biostatistics, and some other areas. The investigations on its theories and applications have received much attention in the past decades. A typical example of the errors-in-variables model is given in Fan (1991) about the AIDS study, where Y may be the time from some starting point to the time that symptoms appear, ε may be the time from the start point to the time that infection occurs, and X is the time from the occurrence of infection to the time of symptoms. A comprehensive introduction to the errors-in-variables model, as well as many other real data examples, can be found in Fuller (1987), Carroll et al. (1995) and the references therein. The problem of interest in errors-in-variables model is to estimate the density function of X . Because X is not observable and its density is unknown, the kernel type estimator cannot be applied here. The most commonly used estimator for the density fX of X is the deconvolution kernel density estimator, which is based on the additive structure of X and ε . It is well known that this additivity structure, and the independence of X and ε imply that φY (t ) = φX (t )φε (t ) for t ∈ R, where φ· (t ) denotes the characteristic function of a random variable which is specified by the subscript. Since the density function of ε , denoted by fε , is known hence φε is known, and Y is observable hence φY can be estimated, so the inverse Fourier transformation on φˆ Y (t )/φε (t ) can provide us an estimator for fX , where φˆ Y (t ) is an estimate of the characteristic function of Y . Using a kernel function K with bandwidth hn , Stefanski and Carroll (1991) consider the following so-called deconvolution kernel density estimator of X : fˆn (x) =

1 2π

Z

∞

exp(−itx) −∞

φK (thn )φˆ n (t ) dt , φε (t )

i=

√ −1

(1.1)

where φK (t ) is the characteristic function of some kernel function K , φˆ n (t ) is the empirical characteristic function of the Pn sample Y1 , . . . , Yn defined by φˆ n (t ) = n−1 i=1 exp(−itYi ), and hn has the same meaning as the bandwidth in the classical E-mail address: [email protected]. 0167-7152/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2009.10.003

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W. Song / Statistics and Probability Letters 80 (2010) 169–176

kernel density estimators. Let gn (y) =

1 2π

Z

∞

exp(−ity) −∞

φK (t ) dt , φε (t /hn )

then (1.1) has the following form which resembles the classical kernel density estimator, fˆn (x) =

n 1X 1

n i=1 hn

 gn

x − Yi



hn

.

gn is significantly different from the classical kernel functions in that gn might be a complex function, and it depends on the bandwidth hn , hence on the sample size. The large sample behavior of fˆn (x) strongly depends on the smoothness of the distribution of the measurement error ε . Using the terms in Fan and Truong (1993), a distribution is called ordinary smooth if the tails of its characteristic function decay to 0 at an algebraic rate; it is called supersmooth if its characteristic function has tails approaching 0 exponentially fast. It is known that the local and global rates of convergence of the sequences of deconvolution kernel density estimators are slower than that of the classical kernel density estimators. Moreover, the convergence rate are much slower in the supersmooth cases than in the ordinary smooth cases. Related works on the consistency and the asymptotic normality of the deconvolution kernel density estimator fˆn (x) can be found in Carroll and Hall (1988), Devroye (1989), Fan (1991), and Fan and Liu (1997) and the references therein. But the literature seems scant in the discussion of the moderate and large deviation results for the deconvolution kernel density estimators, while there are many such works on the classical kernel estimators, see Louani (1998) and Gao (2003) for more details. The current paper will try to fill this void partly. In particular, we will focus on the moderate deviation limit theorems when the measurement error has an ordinary smooth distribution. Further research will be conducted in the future for the large deviation results for the ordinary smooth measurement models, and the moderate and large deviation results for the supersmooth cases. Another interesting topic will be the moderate and large deviation results for the nonparametric prediction in measurement error regression models based on the work of Carroll et al. (2009). Since a parametric rate n−1/2 can be achieved for the nonparametric prediction, so the deviation results should be similar to the nonparametric prediction in the classical regression models. The paper is organized as follows. Section 2 will present the necessary assumptions for the theory, the pointwise and uniform moderate deviation results will be also introduced; the proofs of the main results will be provided in Section 3. 2. Assumptions and main results The ordinary smooth condition on ε often takes the following form (C1) t β φε (t ) → c , t β+1 φε0 (t ) → −β c , as t → +∞, with some constant c 6= 0 and β ≥ 0. Moreover, φε (t ) 6= 0, for all t, where β is called the smooth order. For example, the Gamma density function α p xp−1 exp(−α x)I (x > 0)/0 (p) is ordinary smooth with β = p, and the double exponential density function exp(−|x|)/2 is ordinary smooth with β = 2. The following condition is needed to guarantee the boundedness of the density function Y , which is similar to condition (A1) in Gao (2003). (C2) The density function of ε is continuous and fε (t ) → 0 as t → ∞. Obviously, the above mentioned Gamma and double exponential distributions satisfy (C2). As for the kernel function, we shall assume (C3) φ second bounded integrable derivatives. R K∞(t ) is a symmetric function, having R∞ (C4) −∞ [|φK (t )| + |φK0 (t )|]|t |β dt < ∞, −∞ |t |2β |φK (t )|2 dt < ∞. The following condition will be imposed on the density function of X . (C5) The second derivative of the unknown fX exists and is continuous. For the bandwidth hn , we shall assume (C6) hn , bn are sequence of positive numbers such that, nhn → ∞,

bn nhn

→ 0,

nhn log hn b2n

→ 0.

Similar to Giné and Guillou (2001) and Gao (2003), the following condition is needed when using Giné and Guillou (2001)’s exponential inequality to prove the uniform moderate deviation result, (C7) gn (y) is a bounded, square integrable function in the linear span (the set of finite linear combinations) of functions q ≥ 0 satisfying the following property: the subgraph of q, {(s, u) : q(s) ≥ u}, can be represented as a finite number of Boolean operations among sets of the form {(s, u) : w(s, u) ≥ ψ(u)}, where w is a polynomial on R2 and ψ is an arbitrary real function.

W. Song / Statistics and Probability Letters 80 (2010) 169–176

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√ √ As an example, if ε has a double exponential distribution fε (x) = ( 2σ02 ) exp(− 2|x|/σ0 ) with the characteristic function √ φε (t ) = (1 + σ02 t 2 /2)−1 , and K is further chosen to be the Gaussian kernel K (x) = ( 2π )−1 exp(−x2 /2), then one can show that y2 1 exp − gn (y) = √ 2 2π



σ02

 2 ( y − 1 ) . 2

 1−

2hn

Easy to see that gn (y) is a bounded real function of bounded variation, hence gn (y) satisfies (C7). Under above conditions, we can show the following pointwise moderate deviation limit result for the estimator fˆn (x). Theorem 2.1. Suppose (C1)–(C6) hold. Then for any x ∈ R, any closed set F ⊂ R, any open set G ⊂ R, we have lim sup

nhn b2n

n→∞

lim inf

nhn b2n

n→∞

β+1

nhn

log P

! [fˆn (x) − E fˆn (x)] ∈ F

bn β+1

nhn

log P

≤ − inf Ix (λ), λ∈F

! [fˆn (x) − E fˆn (x)] ∈ G ≥ − inf Ix (λ), λ∈G

bn

where

λ2 , 2G(x)

Ix ( t ) =

G(x) =

fY (x)

Z

2π |c |2

∞

|t |2β |φK (t )|2 dt .

−∞

and fY is the density function of Y . Furthermore, we also have the following uniform moderate deviation limit theorem for the estimator fˆn (x). Theorem 2.2. If assumptions (C1)–(C7) hold, then for any λ > 0, lim

nhn b2n

n→∞

β+1

log P

nhn

bn

!



ˆ

fn − E fˆn

> λ = −I (λ),

∞

where k · k∞ denotes the supreme norm of a function and

λ2 , I (λ) = 2kfY k∞ L

L=

1 2π |c |2

Z

∞

|t |2β |φK (t )|2 dt .

−∞

If we further assume (C8) φK (t ) = 1 + O(|t |2 ) as t → 0, β+3

(C9) nhn /bn → 0 as n → ∞, (C10) fX (x) and its second derivative is bounded, then the following corollary holds. Corollary 2.1. Suppose (C1)–(C10) hold, then for any x ∈ R, any closed set F ⊂ R, any open set G ⊂ R, lim sup

nhn b2n

n→∞

lim inf n→∞

nhn b2n

β+1

log P

nhn

! [fˆn (x) − fX (x)] ∈ F

bn β+1

log P

nhn

bn

≤ − inf Ix (λ), λ∈F

! [fˆn (x) − fX (x)] ∈ G ≥ − inf Ix (λ), λ∈G

and for any λ > 0, lim

n→∞

nhn b2n

β+1

log P

nhn

bn



ˆ

fn − fX (x)

! ∞

> λ = −I (λ),

where Ix (λ) and I (λ) are the same as in Theorems 2.1 and 2.2, respectively. Condition (C8) is also used in Fan (1991). A typical example of K satisfying (C8) is the standard normal density function. Condition (C9) guarantees the bias of fˆn (x) converges to 0 sufficiently fast, while the condition (C10) ensures this convergence is uniform in x ∈ R.

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W. Song / Statistics and Probability Letters 80 (2010) 169–176

3. The proofs of main results Proof of Theorem 2.1. From Fan (1991), conditions (C1), (C3)–(C5) imply

|hβn gn (y)| ≤ B(y) = ˆ min(C2 , C1 /y),

(3.1)

where C1 , C2 are two constants, and hβn gn (y) → C (y) = ˆ

∞

Z

1 2π c

exp(−ity)t β φK (t )dt

(3.2)

−∞ β

as n → ∞. For convenience, let Zni = hn gn ((x − Yi )/hn ) . Note that assumption (C1) implies that the density function fY (·) of Y is bounded. Therefore, for any k ≥ 3, we have

 k  Z ∞ β β x−y fY (y)dy = hn h gn (y) k fY (x − yh)dy E |Zni | = h g n n n hn −∞ −∞ Z ∞ Z ∞ β k h gn (y) dy ≤ hn kfY (y)k∞ |B(y)|k dy. ≤ hn kfY (y)k∞ · n ∞

Z

k

−∞

−∞

To proceed, we need an upper bound for the above integration. Let d = max{C1 , C2 }. Then

Z

∞

|B(y)|k dy = 2dk

∞

Z

−∞

[1 ∧ 1/yk ]dy = 2dk 0

1

Z

dy + 2dk

∞

Z

0

1



1 dy = 2dk 1 + yk k−1 1



≤ 3dk .

Therefore, E |Zn1 − EZn1 |k ≤ 3 · 2k dk hkfY (y)k∞ and

k ∞ ∞ X X 1 tbn 1 2k dk t k bkn E (Zn − EZn ) ≤ 3hn kfY (y)k∞ k! nh k! nk hk n

k=3

k=3

≤ 3hn kfY (y)k∞



2tdbn

3

nhn

 exp

2tdbn



nhn

.

By assumption (C6), we have

k  3 ! ∞ X 2tdbn 1 tbn E (Zn − EZn ) = O hn . k! nhn nhn k=3 Since Y1 , . . . , Yn are i.i.d., so

  Φn,x (t ) = E exp[tbn hβn (fˆn (x) − E fˆn (x))] #!   "  n n tbn tbn X (Zni − EZni ) = E exp (Zn1 − EZn1 ) = E exp nhn i=1

"

tbn

= 1+

nhn

nhn

E (Zn1 − EZn1 ) +

t 2 b2n 2n2 h2n

E (Zn1 − EZn1 ) + O 2



2tdbn nhn

!#n

3 hn

.

By (3.1) and (3.2), we can obtain 1 2 h− n E (Zn1 − EZn1 ) →

fY (x) 2π |c |2

∞

Z

|t |2β |φK (t )|2 dt = ˆ G(x).

−∞

Therefore, we have

" Φn,x (t ) = 1 +

t 2 b2n 2n2 hn

G(x)(1 + o(1)) + O



2tdbn nhn

!#n

3 hn

.

Finally, by letting n → ∞,

Φx (t ) = lim

n→∞

nhn b2n

log Φn,x (t ) =

t2 2

G(x),

and the dual of Φx (t ) is given by Ix (t )= ˆ sup[t λ − Φx (t )] = t ∈R

λ2 . 2G(x)

Note that Φx (t ) is differentiable with respect to t ∈ R, from Gärtner–Ellis Theorem, we have the desired result.



W. Song / Statistics and Probability Letters 80 (2010) 169–176

173

To show the uniform moderate deviation result, we need an exponential inequality for the empirical process from Giné and Guillou (2001). For the sake of completeness, we state the exponential inequality below. Lemma 3.1. Let F be a measurable uniformly bounded VC class of functions, and let σ 2 and U be any numbers such that σ 2 ≥ supf ∈F Var (f ), U ≥ supf ∈F kf k∞ , and 0 < σ ≤ U /2. Then, there exist constants C and L depending only on the characteristic of the class F , such that the inequality

n

X

P [f (ξi ) − Ef (ξi )]

i=1

!

  t log 1 + ≤ L exp −

>t

LU

∞



tU

√ √ L( nσ + U log(U /σ ))2

holds for all t ≥C

U log

U

σ

r

√

nσ

+

U

log

! .

σ

` More about the VC (Vapnik–Chervonenkis) class of functions, please see Pollard (1984). We also need the following lemmas which are similar to the lemmas 2.4 and 2.5 in Gao (2003). Lemma 3.2. Let assumptions (C1)–(C7) hold. For any 0 < δ < 1, let In,k , k = 1, 2, . . . , ln , be ln intervals with length δ hn , such 1 −1 2 that {In,k , k = 1, . . . , ln } is a covering of [−h− n , hn ] and ln ≤ 2/δ hn + 1. Let xn,k ∈ In,k , 1 ≤ k ≤ ln , n ≥ 1. Then for any ε > 0, nhn

lim lim sup

δ→0

b2n

n→∞

β+1

!

nhn

log P

sup sup |fˆn,k (x) − E fˆn,k (x)| ≥ ε

bn

= −∞,

1≤k≤ln x∈In,k

where fˆn,k (x) = fˆn (x) − fˆn (xn,k ). β

Proof. By (C7), the class of functions F = {hn gn ((x − ·)/hn ), x ∈ R, hn 6= 0} is a bounded measurable VC class of functions, so the following classes of functions

 F n ,k =

hβn gn



x−· hn



− hβn gn



x n ,k − ·



hn

 ; x ∈ I n ,k ,

k = 1, . . . , ln ; n ≥ 1

are measurable VC classes of functions. Moreover, there is a common VC characteristic for all these classes that does not depend on k and n. Note that

Z 

hβn gn



x−y hn



− hβn gn



2

hn

Z 



hβn gn (z ) − hβn gn

= hn

xn,k − y

xn,k − x hn

fY (y)dy

2 +z

fY (x − hz )dz ≤ 2kfY k∞ ηhn .

β

Now let us take Un = 2khn gn (y)k∞ , F = Fn,k , and σn2 = 2kfY k∞ ηhn . Then for n large enough, we have σn ≤ Un /2. Too √ √ 1/2 β see this point, note that, from (3.1), Un = 2khn gn (y)k∞ = O(1), and σn = O(hn ). We also have nσn ≥ Un log(Un /σn ) because of hn → 0 and nhn / log(1/hn ) → ∞. Moreover, from (C6), one has Un log(Un /σn ) +

lim sup

√

nσn

√

√

log(Un /σn )

≤ lim sup

bn

n→∞

2 nσn

√

log(Un /σn )

bn

n→∞

= 0.

Consequently, there exists a positive integer number n0 such that for any n ≥ n0 ,



bn ≥ C Un log(Un /σn ) +

√

√

√

nσn

p



log(Un /σn )

√

and nσn + Un log(Un /σn ) ≤ 2 2ηnhn kfY k∞ . Now applying Giné and Guillou (2001)’s lemma, for n ≥ n0 , one has β+1

P

nhn

bn

! sup sup |fˆn,k (x) − E fˆn,k (x)| ≥ ε

1≤k≤ln x∈In,k

  bn ε ≤ Lln exp − log 1 + 2LUn

bn ε U n 4LnhkfY k∞ η



.

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W. Song / Statistics and Probability Letters 80 (2010) 169–176

Therefore, nhn

lim lim sup

δ→0

b2n

n→∞

nhn

≤ lim sup

b2n

n→∞

≤−

ε

β+1

!

nhn

log P

sup sup |fˆn,k (x) − E fˆn,k (x)| ≥ ε

bn

1≤k≤ln x∈In,k

nhn ε

log Lln − lim

2LUn bn

n→∞

2

Un bn ε

 log 1 +



4nhLηkfY k∞

.

8L2 ηkfY k∞

Letting η → 0 implies the desired result.



Lemma 3.3. Assume (C1)–(C7) hold. Then for any ε > 0, β+1

nhn

lim sup

log P

b2n

n→∞

nhn

sup

bn

−1 −1 x6∈[−hn ,hn ]

ˆ fn (x) − E (fˆn (x)) ≥ ε

! = −∞.

Proof. Set



hβn gn

Fn =



x−·



hn

 1 −1 ; x 6∈ [−h− , h ] . n n

Then Fn , n ≥ 1 are measurable VC classes of functions with VC characteristic (A, v) that does not depend on n. Note that (C1) implies fY (y) → 0 as y → ∞. So, for any η ∈ (0, ε), there exists an n0 > 0 such that for any n ≥ n0 , h2nβ

sup −1 −1 x6∈[−hn ,hn ]

Z

gn2



x−y



hn

fY (y)dy ≤ ηhn .

β

Now let Un = khn gn (x)k∞ , F = Fn , and σn2 = ηhn . Then for n large enough, σn ≤ Un /2, (C6), one has, Un log(Un /σn ) +

lim sup

√

nσn

√

√

log(Un /σn )

≤ lim sup

bn

n→∞

2 nσn

√

nσn ≥ Un

√

log(Un /σn )

bn

n→∞

= 0.

Consequently, there exists a positive integer number n0 such that for any n ≥ n0 ,



bn ≥ C Un log(Un /σn ) + and

√

nσn + Un

√

√

nσn



log(Un /σn )

p

√

log(Un /σn ) ≤ 2 ηnhn . Now applying Giné and Guillou’s lemma, for n ≥ n0 , one has

β+1

P

!

nhn

bn

   bn ε Un bn ε ≤ L exp − log 1 + . LUn 4Lnhη

|fˆn (x) − E fˆn (x)| ≥ ε

sup −1 −1 x6∈[−hn ,hn ]

Therefore, lim sup

β+1

nhn

n→∞

log P

b2n

≤ lim

nhn

n→∞

b2n

!

nhn

|fˆn (x) − E fˆn (x)| ≥ ε

sup

bn

−1 −1 x6∈[−hn ,hn ]

nhn ε

log L − lim

n→∞

LUn bn

Un bn ε

 log 1 +

Letting η → 0 implies the desired result.

4nhLη

 ≤−

ε2 . 4L2 η



Proof of Theorem 2.2. The lower bound is easy. In fact, for any x ∈ R, from Theorem 2.1 we have lim inf n→∞

nhn b2n

≥ lim inf n→∞

β+1

log P nhn b2n

nhn

bn

! ˆ ˆ kf n − E f n k∞ ≥ λ β+1

nhn

log P

bn

! ˆ ˆ kfn (x) − E fn (x)k ≥ λ ≥ −Ix (λ).

Hence lim inf n→∞

nhn b2n

β+1

log P

nhn

bn

! ˆ ˆ kfn − E fn k∞ ≥ λ ≥ −I (λ).

√

log(Un /σn ). From

W. Song / Statistics and Probability Letters 80 (2010) 169–176

175

To show the opposite inequality, note that kfˆn − E (fˆn )k∞ equals to

( max

|fˆn (x) − E fˆn (x)|,

sup −1 −1 x∈[−hn ,hn ]

) ˆ ˆ |fn (x) − E fn (x)| ,

sup −1 −1 x6∈[−hn ,hn ]

and supx∈[−h−1 ,h−1 ] |fˆn (x) − E fˆn (x)| is bounded above by n

n

max { sup |fˆn,k (x) − E fˆn,k (x)| + |fˆn (zn,k ) − E fˆn (zn,k )|}.

1≤k≤ln x∈I n,k

By Lemmas 3.2 and 3.3 and using inequality: max{log a, log b} ≤ log(a + b) ≤ log 2 + max{log a, log b}, a ≥ 0, b ≥ 0, we have that for any 0 < ε < λ, lim sup

nhn b2n

n→∞

≤ lim sup n→∞

β+1

log P nhn b2n

≤ lim lim sup δ→0

nhn

n→∞

! kfˆn − E fˆn k∞ > λ

bn

β+1

log P nhn b2n

!

nhn

bn

|fˆn (x) − E fˆn (x)| > λ

sup −1 −1 x∈[−hn ,hn ]

β+1

" log P

nhn

bn

! max sup |fˆn,k (x) − E fˆn,k (x)| ≥ ε

1≤k≤ln x∈I n,k

β+1

!#

nhn

+P

max |fˆn (zn,k ) − E fˆn (zn,k )| > λ − ε

bn

1≤k≤ln

≤ lim lim sup δ→0

n→∞

nhn b2n

β+1

nhn

log P

bn

! max |fˆn,k (zn,k ) − E fˆn,k (zn,k )| > λ − ε

1≤k≤ln

.

By Chebyshev’s inequality, β+1

nhn

P

bn

! max |fˆn,k (zn,k ) − E fˆn,k (zn,k )| > λ − ε

1≤k≤ln

 ≤ ln max



exp −

1≤k≤ln

 ≤ ln exp −

b2n nhn

b2n

  (λ − ε)t Φzn,k (t ) nhn "

(λ − ε)t

1+

b2n

n2 h n

max Φx (t )(1 + o(1)) + O x∈R



2tdbn

!#n

3

nhn

hn

.

Therefore, lim sup n→∞

nhn b2n

β+1

log P

nhn

bn

!

  kfˆn − E fˆn k∞ > λ ≤ − (λ − ε)t − max Φx (t ) . x∈R

The desired result follows from the fact that the last inequality holds for arbitrary 0 < ε < λ and any t > 0.



Proof of Corollary 2.1. To show the first two claims one only has to investigate the term Φn,x (t ). Now Φn,x (t ) = β

E exp(tbn hn [fˆn (x) − fX (x)]), it can be rewritten as

Φn,x (t ) = E exp(tbn hβn [fˆn (x) − E fˆn (x)]) · exp(tbn hβn [E fˆn (x) − fX (x)]). So, it suffices to show that nhn b2n

· tbn hβn [E fˆn (x) − fX (x)] → 0

as n → ∞. The conditions (C8) and (C10) imply E fˆn (x) − fX (x) = O(h2n ) uniformly in x ∈ R, so the desired result follows from (C9). The third claim follows from similar argument, hence omitted here for brevity.

(3.3)



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