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

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.

Introduction

is a population with large, deterministic and finite size N with elements . In sampling from this population we only observe those pairs for which . Suppose that there is at least one pair with this condition. The sample is denoted by . This model is called random left-truncated model (RLTM). We assume that is a stationary α-mixing sequence of random variables and is an independent and identically distributed (i.i.d.) sequence of random variables. The definition of a strong mixing sequence is presented in Definition 1.

Definition 1

Let be a sequence of random variables. The mixing coefficient of this sequence is where denotes the σ-algebra generated by for . This sequence is said to be strong mixing or α-mixing if the mixing coefficient converges to zero as .

Studying the various aspects of left-truncated data is of high interest due to their applicability in much research. One of these applications is in survival analysis. It is well known that in medical research on some specific diseases such as AIDS and dementia, the sampling scheme results in data samples that are left truncated. This model also arises in astronomy [1].

Strong mixing sequences of random variables are widely occurring in practice. One application is in the analysis of time series and in renewal theory. A stationary ARMA-sequence fulfils the strong mixing condition with an exponential rate of mixing coefficient. The concept of strong mixing sequences was first introduced by Rosenblatt [2] where a central limit theorem is presented for a sequence of random variables that satisfies the mixing condition.

The Berry-Esseen inequality or theorem was stated independently by Berry [3] and Esseen [4]. This theorem specifies the rate at which the scaled mean of a random sample converges to the normal distribution for all sample spaces. Parzen [5] derived a Berry-Esseen inequality for the kernel density estimator of an i.i.d. sequence of random variables. Several works were done for left-truncated observations. We can refer to [6] where the distribution of left-truncated data was estimated and asymptotic properties of the estimator were derived. More work was done by Stute [7]. Prakasa Rao [8] presented a Berry-Esseen theorem for the density estimator of a sample that forms a stationary Markov process. Liang and Un̈a-Álvarez [9] have derived a Berry-Esseen inequality for mixing data that are right censored. Yang and Hu [10] presented Berry-Esseen type bounds for kernel density estimator based on a φ-mixing sequence of random variables. Asghari et al. [11, 12] presented a Berry-Esseen type inequality for the kernel density estimator, respectively, for a left-truncated model and for length-biased data.

This paper is organized as follows. In Section 2, needed notations are introduced and some preliminaries are listed. In Section 3, the Berry-Esseen type theorem for the estimator of the density function of the data is presented. In Section 4, the theorems and corollaries of Section 3 are proved.

Preliminaries and notation

Suppose that ’s and ’s for are positive random variables with distributions F and G, respectively. Let the joint distribution function of be in which .

If the marginal distribution function of is denoted by , we have so the marginal density function of Y is A kernel estimator for f is given by In many applications, the distribution function of the truncation random variable G is unknown. So is not applicable in these cases and we need to use an estimator of G. Before starting the estimation details, for any distribution function L on , let and .

Woodroof [6] pointed out that F and G can be estimated only if , and . This integrability condition can be replaced by the stronger condition . Using this assumption, here we use the non-parametric maximum likelihood estimator for G that is presented by Lynden-Bell [13] and is denoted by , in which and .

Using the definition of that is mentioned in the estimation procedure of G and also using the empirical estimators of and , which are denoted by and , we have It can be seen that is actually the empirical estimator of , . This fact gives the following estimator of α: For details as regards , see [14]. Using , we present a more applicable estimator of f, which is denoted and is defined as Note that in (2.2) the sum is taken over i’s for which .

Results

Before presenting the main theorems, we need to state some assumptions. Suppose that and . Woodroof [6] stated that the uniform convergence rate of to G is true for for . Thus, we have to assume that . Let be a compact set such that . As mentioned in the Introduction, is a stationary α-mixing sequence of random variables with mixing coefficient , and is an i.i.d. sequence of random variables.

Definition 2

The kernel function K, is a second order kernel function if , and .

Assumptions

for some in which .

For the conditional density of given (denoted by ), we have for and in a neighborhood of in which M is a positive constant.

K is a positive bounded kernel function such that for and .

K is a second order kernel function.

f is twice continuously differentiable.

Let and be positive integers such that , there exists a constant C such that for n large enough . Also , as .

is a sequence of i.i.d. random variable with common continuous distribution function G, and independent of .

The kernel function is differentiable and Hölder continuous with exponent .

for in which .

The joint density of , , exists and we have for some constant C.

There exists and for the bandwidth we have and which η is such that .

Discussion of the assumptions. A1, A2, and A4 are common in the literature. For example Zhou and Liang [15] used A2 for deconvolution estimator of multivariate density of α-mixing process. A3(i)-(ii) are commonly used in non-parametric estimation. A3(iii) is specially needed for a Taylor expansion. H1-H4 are needed to use Theorem 4.1 of [16] in Theorem 4 here.

Let , so by letting , we can write

Let , and , in which . Now we have the following decomposition: in which From now on, we let , .

Theorem 1

If Assumptions A1-A3(i) and A4 are satisfied and f and G are continuous in a neighborhood of y for , then for large enough n we have in which and

Theorem 2

If the assumptions of Theorem  1 and A5 are satisfied, then for and for large enough n we have in which is defined in (3.3).

Theorem 3

If the assumptions of Theorem  2 are satisfied, G has bounded first derivative in a neighborhood of y and f has bounded derivative of order 2 in a neighborhood of y for , then for large enough n we have in which and and are defined in (3.4).

Remark 1

In many applications, f and G are unknown and should be estimated, so is not applicable in these cases. Here we present an estimator for it that is denoted by and is defined as follows: Using this estimator instead of in Theorem 3, costs a change in the rate of convergence. This change is discussed in the following corollaries.

Corollary 1

Let Assumptions A3, A5 and H1-H4 be satisfied, then for in which

Theorem 4

Let Assumptions A1-A5 and H1-H4 be satisfied. For and for large enough n we have in which is defined in (3.5) and is defined in (3.6).

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, and , be positive appropriate constants which may take different values at different places.

Lemma 1

[17]

Let X and Y be random variables such that and in which r and s are constants such that and . Then we have

Lemma 2

Suppose that Assumptions A1-A3(i) and A4 are satisfied. If f and G are continuous in a neighborhood of y for then as . Furthermore, if f and G have bounded first derivatives in a neighborhood of y for , for such y’s we have in which

Proof

Using the decomposition that is defined in (3.2) we can write As assumed in the lemma, f and G are continuous in a neighborhood of y so they are bounded in this neighborhood. Now under Assumption A3(i) we have so it can be concluded that Lemma 1 for arbitrarily and also the continuity of f in a neighborhood of y gives now using the notation , which is defined before, and A1 we get the following result: Under Assumption A2 we can write Now, using (4.4), (4.5), (4.6), and (4.2), we have By the same argument as is used for and and , it can be concluded that Now, using (4.8) and (4.9), we have Similarly and So we can write Gathering all that is obtained above, and by letting we have On the other hand using (4.7), (4.10), and (4.13), we have So for we can write On the other hand from (4.3), it can easily be concluded that as . Now under Assumptions A1 and A4 , so . If f and G have bounded first derivatives in a neighborhood of y, we can write From (4.14) we get the following result: and the proof is completed. □

Before starting the next lemma, we note that If we let , it can be observed that in which

Lemma 3

Suppose that Assumptions A1-A3(i) and A4 are satisfied and f and G are continuous in a neighborhood of y for . Then for such y’s we have

With the aid of Lemma 2 we can write The same argument shows that , so we have and So the proof is completed. □

In the following let in which , , are independent random variables with the same distribution as , . φ and are, respectively, the characteristic functions of and . Also let and .

Lemma 4

Under the assumptions of Lemma  3, for we have the following:

It can easily be seen that , and Using (4.25) and Lemma 2, we can write On the other hand, from Lemma 2 we know that , so substituting this in (4.26), gives the result,  □

Lemma 5

[18]

Let be a stationary sequence with mixing coefficient and suppose that , , and there exist and such that and also . In this case, for any , there exists a constant C, for which we have

Lemma 6

Under the assumptions of Lemma  3 for we have

Using [19], Theorem 5.7, for we can write On the other hand, using Lemma 5 there exists such that for any Let , for and and , so we have From Lemma 4, , so the proof is completed. □

Lemma 7

[20]

Let be a stationary sequence with mixing coefficient . Suppose that p and q are positive integers. Let in which . If such that , there exists a constant such that

Lemma 8

Under the assumptions of Lemma  3 for we have

By letting in [19], Theorem 5.3, p.147, for any we have Now by letting in Lemma 7, there exists a constant for which we have Now using (4.32) and (4.33) we have so On the other hand applying Lemma 6 gives so By choosing we get the following result: and the lemma is proved. □

Lemma 9

[21]

Let X and Y be random variables. For any we have

Proof of Theorem 1

Using (4.21) and Lemma 9, for any and we can write By choosing and and using Lemma 3, we have On the other hand using Lemmas 8, 4, and 6 we have So the proof is completed. □

Proof of Theorem 2

According to Lemma 9 for any we can write and From Lemma 5.2 of [16] we have and from [22] we have So we can write Now by choosing and using Theorem 1 we get the result  □

Proof of Theorem 3

By the triangular inequality and using Lemma 1 for we have Here we used the fact that the event does not happen for the selected a.

From the inequality , it can be concluded that Under Assumptions A3(ii) and A3(iii), use of the Taylor expansion yields So from (4.48), (4.49), (4.50), Theorem 2, and Lemma 2, we have  □

Proof of Corollary 1

Using the triangular inequality it can be seen that Under Assumptions A3, A5, H1-H4, Theorem 4.1 of [16] we obtain From (4.45) and (4.52) we have Using (4.53) and (4.52) in (4.51) proves the corollary. □

Proof of Theorem 4

Using the triangular inequality we can write By Assumptions A1-A3(i), A4 and A5, Theorem 3 results in the following: in which is defined in Theorem 3.

Under Assumptions A3, A5, H1-H4, Corollary 1 results in the following: Substituting (4.55) and (4.56) in (4.54) proves the theorem. □

Conclusions

In this paper we obtained Berry-Esseen type bounds for the kernel density estimator based on left-truncated and strongly mixing data. Here it is concluded that in RLTM, which is also dealing with weak dependency, we can get asymptotic normality but comparing the results with [11] we see that the rates get much more complicated and also slower.