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 ∈ Fk, 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 –∞∞ tK (t) dt > .
Assumptions
√nhn 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 uh–n y K uh–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) + phn
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)