Denote the integer lattice points in the N-dimensional Euclidean space by for . Let be a strictly stationary random field with common density on the real line R. Throughout this paper, let , , denote ( ) for and , and . The limit process denotes
for some constant .
For a set of sites , denotes the σ-field generated by the random variables . denotes the cardinality of S, and denotes the Euclidean distance between S and , that is . We will use the following mixing coefficient
where C is some positive constant, as , and is a symmetric positive function nondecreasing in each variable.
If , then is called strongly mixing. In Carbon et al.  , it is assumed that h satisfies either
where . Conditions (2) and (3) are also used by Neaderhouser  and Takahata  , respectively and are weaker than the strong mixing condition.
In recent years, there is a growing interest in statistical problem for random fields, because spatial data are modeled as finite observations of random fields. For asymptotic properties of kernel density estimators for spatial random fields, one can refer to Tran  , Hallin et al.   , Cheng et al.  , El Machkouri   , Wang and Woodroofe  , among others. For spatial regression models, see, Biau and Cadre  , Lu and Chen  , Hallin et al.  , Gao et al.  , Carbon et al.  , Dabo-Niang and Yao  .
The purpose of this paper is going to investigate the convergence rate of asymptotic normality of frequency polygon estimation of density function for mixing random fields. The frequency polygon has the advantage to be conceptually and computationally simple. Furthermore, Scott  showed that the rate of convergence of frequency polygon is superior to the histogram for smooth densities, and similar to those of kernel estimators. In recent years, frequency polygon estimator is given increasing attention. For example, key references that can be found for non-spatial random variables are Scott  , Beirlant et al.  , Carbon et al.  , Yang  , Xin et al.  , etc. For spatial random fields, the references on frequency polygon are Carbon  , Carbon et al  , Bensad and Dabo-Niang  and El Machkouri  . For continuous indexed random fields, Bensad and Dabo-Niang  derived the integrated mean squared error of frequency polygon and the optimal uniform strong rate of convergence. For discretely indexed random fields, Carbon  obtained the optimal bin width based on asymptotically minimize integrated error and the rate of uniform convergence, Carbon  derived the asymptotic normality of frequency polygon under the mixing conditions that the function h in (1.1) satisfies (2) or (3), El Machkouri  established the asymptotic normality of frequency polygon for strongly mixing coefficients (that is, ). However, the convergence rate of asymptotic normality of frequency polygon has not been discussed in these literature. In this paper, we will prove a Berry-Esseen bound of frequency polygon and the convergence rate of asymptotic normality under weaker mixing conditions, which include strongly mixing condition.
This paper is organized as follows: Next section presents the main results. Section 3 gives some lemmas, which will be used later. Section 4 provides the proofs of theorems. Throughout this paper, the letter C will be used to denote positive constants whose values are unimportant and may vary, but not dependent on .
2. Main Results
Suppose that we observe on a rectangular region . Consider a partition of the real line into equal intervals of length , where is the bin width and . For , consider the two adjacent histogram bins and . Denote the number of observations falling in these intervals respectively by and . Then the values of the histogram in these previous bins are given by
Thus the frequency polygon estimation of the density function is defined as follows
We know that the curve estimated by the frequency polygon is a non-smooth curve, but it tends to be a smooth density curve as the interval length of interpolation gradually tends to zero. So we always assume that tends to zero as . In addition, we need the following basic assumptions.
Assumption (A1) The density with bounded derivative. For all and some constant ,
where is the conditional density of given .
Assumption (A2) The random field satisfies (1) with for some .
Under Assumption (A2), we can take such that , then . Carefully checking the proof of Theorem 3.1 in Carbon et al  , we find that the conditions (2) and (3) are not used, in fact, it only uses the positive constant . Therefore, by Theorem 3.1 in Carbon et al.  , we obtain the following result on asymptotic variance.
Proposition 1 Suppose that Assumption (A1) and (A2) are satisfied. Then, for , we have
It should be reminded that, as in Remark 3 in El Machkouri (2013), it should be instead of for the asymptotic variance .
Let , and denote the distribution function of . Now we give our main results as follow.
Theorem 1. Suppose that Assumption (A1) and (A2) hold. Assume that there exist integers and such that
where and . Then, for such that and as , we have
where and .
Remark 1. In the theorem above, it does not need to assume that because from (8).
Theorem 1 provides a general result for Berry-Esseen bound of frequency polygon estimation. Some specific bounds can be obtained by choosing different , p and q.
Theorem 2. Suppose that Assumption (A1) and (A2) hold. Let for some . Denote that , and for some .
1) If and
2) or if (2) is satisfied and
3) or if (3) is satisfied and
then, for such that and as , we have
Carbon  proved that the optimal bin width for asymptotical mean square error
where , when . For the optimal bin width, it is ease to get the following result by Theorem 2.
Corollary 1. Suppose that Assumption (A1) and (A2) hold and . Let . 1) If
for some , then, for such that ,
2) If tends to zero exponentially fast as u tends to infinity, then, for such that ,
Remark 2. The asymptotic normality of frequency polygon under the strongly mixing conditions established by Carbon  and El Machkouri  . As far as we know, however, the convergence rate of asymptotic normality has not been studied. Our conclusions make an effort in this respect.
In the later proof, we need to estimate the upper bounds of covariance and variance of dependent variables. The following two lemmas give the upper bounds of covariance and variance respectively.
Lemma 1. Roussas and Ioannides  suppose that and are - measurable and -measurable random variables, respectively. If a.s. and a.s., then
Lemma 2. Gao et al.  let assumption (A1) and (A2) be satisfied. Suppose that the integer vectors , and satisfy for . Then there exists a positive constant C, which is no depending on , and , such that
Lemma 3. Lemma 3.7 in Yang  suppose that and are two random variable sequences, is a positive constant sequence, and . If
then for any ,
Proof of Theorem 1 We will use the methodology of using “small” and “big” blocks which is similar to that of Carbon et al.  . For , define
and . Then
Now we divide into the sum of large blocks and the sum of small blocks. According to the block size method, we assume and satisfy (8). Assume for some integer vector , we have . If it is not this case, there will be a remainder term in the splitting block, but it will not change the proof much. For , let
an so on. Note that
For each integer , define
and . Then
Enumerate the random variables in an arbitrary manner and refer to them as . Note that . Using Theorem 4 in Rio  or Lemma 4.5 in Carbon et al.   , there exists , independent random variables, independent of with the same law verifying
Let and . Thus
By Lemma 3, it is sufficient to show that
Obviously, from (27)
it follows (29). Now consider that
By Lemma 2,
Define . By Lemma 1,
Combining (34)-(36), we have
similarly, for . Thus, we obtain (30) from (33).
Finely, to show that (31). Clearly,
Define . Recalling (36), we have
and by Lemma 2
Combining (38)-(40) yields that and , so that from . Hence
Let . Note that for . From (40), we have
yields (31) by Berry-Esseen theorem. Complete the proof.
Proof of Theorem 2 In Theorem 1, take and where and for and . Notes that . Then
First consider the case (1), that is that and the condition (10) holds. At this time, we have
The condition (10) implies that . Combining this with (45) and (46), we can get that
From (43),(44), (47) and (48), it is ease to know that
It follows the desired result (13). For the case (2) and the case (3), the proving methods are similar to the method used to prove the case (1). Complete the proof.
The frequency polygon estimation has the advantage of simple calculation. It can save calculation cost in the face of large data, so it is a valuable and worth studying method. In the existing literature, the asymptotic normality of the frequency polygon estimation has been studied, but its convergence rate has not been established. This paper proves a Berry-Esseen bound of the frequency polygon and derives the convergence rate of asymptotic normality under weaker mixing conditions. In particularly, for the optimal bin width , it is showed that the convergence rate of asymptotic normality reaches to when mixing coefficient tends to zero exponentially fast. These conclusions show that the asymptotic normality of the frequency polygon estimator also has a good convergence rate under the dependent samples. Therefore, when the sample size is large, the normal distribution can be used to give a better confidence interval estimation.
This research was supported by the Natural Science Foundation of China (11461009) and the Scientific Research Project of the Guangxi Colleges and Universities (KY2015YB345).
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