A Berry-Esseen type bound for the kernel density estimator based on a weakly dependent and randomly left truncated data

Journal of Inequalities and Applications, Jan 2017

In many applications, the available data come from a sampling scheme that causes loss of information in terms of left truncation. In some cases, in addition to left truncation, the data are weakly dependent. In this paper we are interested in deriving the asymptotic normality as well as a Berry-Esseen type bound for the kernel density estimator of left truncated and weakly dependent data.

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A Berry-Esseen type bound for the kernel density estimator based on a weakly dependent and randomly left truncated data

Asghari and Fakoor Journal of Inequalities and Applications A Berry-Esseen type bound for the kernel density estimator based on a weakly dependent and randomly left truncated data Petros Asghari Vahid Fakoor In many applications, the available data come from a sampling scheme that causes loss of information in terms of left truncation. In some cases, in addition to left truncation, the data are weakly dependent. In this paper we are interested in deriving the asymptotic normality as well as a Berry-Esseen type bound for the kernel density estimator of left truncated and weakly dependent data. left-truncation; weakly dependent; asymptotic normality; Berry-Esseen; α-mixing 1 Introduction α(m) = sup P(A ∩ B) – P(A)P(B) ; A ∈ Fk, B ∈ Fk∞+m , k≥ 2 Preliminaries and notation Suppose that Yi’s and Ti’s for i = , . . . , N are positive random variables with distributions F and G, respectively. Let the joint distribution function of (Y, T) be H∗(y, t) = P(Y ≤ y, T ≤ t) =  y α –∞ G(t ∧ u) dF(u), G(u) dF(u), so the marginal density function of Y is f ∗(y) = α G(y)f (y). A kernel estimator for f is given by fn(y) = nhn i= Gn(y) =  ≤ y < ∞, fˆn(y) = nhn i= For details as regards αn, see []. Using αn, we present a more applicable estimator of f , which is denoted fˆn and is defined as 3 Results Definition  The kernel function K , is a second order kernel function if –∞∞ K (t) dt = , –∞∞ tK (t) dt =  and –∞∞ tK (t) dt > . Assumptions √nhn K Let k = [ p+nq ], km = (m – )(p + q) +  and lm = (m – )(p + q) + p + , in which m = , , . . . , k. Now we have the following decomposition: in which Wni = Jn + Jn + Jn , Jn = Jn = Jn = jnk+, jnk+ = jnm = jnm = in which λn := knq + hn–δ/(+δ)u(q) + qhn, p λn := n (phn + ). Theorem  If the assumptions of Theorem  and A are satisfied, then for y ≥ aF and for large enough n we have nhn fˆn(y) – E fn(y) in which an is defined in (.). in which –δ + hn(p + ) + hn+δ u(q) + λn α(q) / Remark  In many applications, f and G are unknown and should be estimated, so σ (y) is not applicable in these cases. Here we present an estimator for it that is denoted by σˆn(y) and is defined as follows: σˆn(y) = αnfˆn(y)  Gn(y) – K (t) dt. in which cn := max in which an is defined in (.) and cn is defined in (.). 4 Proofs In order to start the proofs of the main theorems, we shall state some lemmas that are used in the proving procedure of the main theorems. For the sake of simplicity let C, C and C , be positive appropriate constants which may take different values at different places. E(XY ) – E(X)E(Y ) ≤ X r Y s P(A ∩ B) – P(A)P(B) σn(y) – σ (y) = O(bn), in which σn(y) = Var Jn + Jn + Jn = Var Jn + Var Jn + Var Jn Var Jn = Var m= i=km Var(Wni) + Cov Jn, Jn + Cov Jn, Jn + Cov Jn , Jn , Cov(Wns, Wnt) E K  Yi – y = n K (t) αf (y + thn) dt – hn G(y + thn) K (t)f (y + thn) dt so it can be concluded that K (t) αf (y + thn) dt + hn G(y + thn) K (t)f (y + thn) dt ≤i<j≤k =: I + II + III . Cov jni, jnj +  Cov(Wni, Wnj) m= km≤i<j≤km+p– ≤i<j≤k s=ki t=kj ki+p– kj+p– ki+p– kj+p– nhn ≤i<j≤k s=ki t=kj now using the notation u(n) := following result: ki+p– kj+p– ki+p– kj+p– G(Y) +δ ≤i<j≤k s=ki t=kj G(Y) +δ ≤i<j≤k s=ki t=kj δ j∞=n (α(j)) δ+ , which is defined before, and A we get the Under Assumption A we can write E K III ≤  Cov(Wni, Wnj) ≤ nhn m= km≤i<j≤km+p– + E K Yih–n y G(αYi) × f ∗(u|u)f ∗(u) du du + K (t)f (y + thn) dt K uh–n y K uh–n y K (s)K (t) ds dt + K (s)K (t) ds dt + K (t)f (y + thn) dt Now, using (.), (.), (.), and (.), we have Var Jn = O knp + hn–δ/+δu(q) + phn = O(), Var Jn = Var =: I + II + III . m= lm≤i<j≤lm+q– Var(Wni) +  Cov jni, jnj ≤i<j≤k Cov(Wni, Wnj) III = O(qhn). Now, using (.) and (.), we have Var Jn = O =: I + II , Var Jn = Var Var(Wni) +  Cov(Wni, Wnj) k(p+q)+≤i<j≤n = O So we can write Var Jn n – k(p + q) + phn Gathering all that is obtained above, ≤i<j≤k ≤i<j≤k k(p+q)+≤i<j≤n Cov jni, jnj +  Cov jni, jnj +  m= km≤i<j≤km+p– m= lm≤i<j≤lm+p– Cov(Wni, Wnj) Cov(Wni, Wnj) Cov(Wni, Wnj) + Cov Jn, Jn + Cov Jn, Jn + Cov Jn , Jn Cov jni, jnj + Cov(Wni, Wnj) m= lm≤i<j≤lm+p– Cov(Wni, Wnj) + Cov Jn, Jn Cov jni, jnj + Cov(Wni, Wnj) and by letting we have ≤i<j≤k ≤i<j≤k k(p+q)+≤i<j≤n + Cov Jn, Jn + Cov Jn , Jn , (.) ≤ On the other hand using (.), (.), and (.), we have Cov Jn, Jn ≤ Var Jn Var Jn Cov Jn, Jn Cov Jn , Jn So for An we can write =  E K  Y – y hn hn G(Y) K (t)f (y + thn) dt = O(hn). From (.) we get the following result: and the proof is completed. Before starting the next lemma, we note that If we let in which Sn = Sn = S = Ynm = Ynm = Ynm = i=lm   P Sn > λn  = O λn  ,   P Sn > λn  = O λn  . Proof With the aid of Lemma  we can write E Sn  = σn(y) E Jn – E Jn √nhn[fn(y) – Efn(y)] σn(y) Sn = Sn + Sn + Sn , So the proof is completed. E Sn  –  = E Sn  – E Sn (...truncated)


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Petros Asghari, Vahid Fakoor. A Berry-Esseen type bound for the kernel density estimator based on a weakly dependent and randomly left truncated data, Journal of Inequalities and Applications, 2017, pp. 1, 2017, DOI: 10.1186/s13660-016-1272-0