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On the convergence rates of kernel estimator and hazard estimator for widely dependent samples
Journal of Inequalities and Applications volumeÂ 2018, ArticleÂ number:Â 71 (2018)
Abstract
In this paper, we establish a Bernsteintype inequality for widely orthant dependent random variables, and obtain the rates of strong convergence for kernel estimators of density and hazard functions, under some suitable conditions.
1 Introduction
Let \(\{X_{n}, n\geq1\}\) be a sequence of random variables with an unknown marginal probability density function \(f(x)\) and distribution function \(F(x)\). Assume that \(K(x)\) is a known kernel function, the kernel estimate of \(f(x)\) and the empirical distribution function of \(F(x)\) are given by
where \(\{h_{n}, n\geq1\}\) is a sequence of positive bandwidths tending to zero as \(n\to\infty\), and \(I(\cdot)\) is the indicator of the event specified in the parentheses. Denote the hazard rate of distribution \(F(x)\) by \(\lambda (t)=f(t)/(1F(t))\), and it can be estimated by
Works devoted to the estimation of probability density and hazard rate functions include the following. Izenman and Tran ([1], 1990) discussed the uniform consistency and sharp rates of convergence under strong mixing and absolute regularity conditions. Cai ([2], 1998) established the asymptotic normality and the uniform consistency with rates of the kernel estimators for density and hazard functions under a censored dependent model. Liebscher ([3], 2002) derived the rates of uniform strong convergence for density and hazard rate estimators for right censoring based on a stationary strong mixing sequence. Liang et al. ([4], 2005) obtained the optimal convergence rates of the nonlinear wavelet estimators of the hazard rate function when the survival times form a stationary strong mixing sequence. Bouezmarni et al. ([5], 2011) proposed new estimators based on the gamma kernels for density and hazard rate functions which are free of bias, and achieved the optimal rate of convergence in terms of integrated mean squared error, and so on.
On the other hand, different from strong mixing and negatively associated random variables, widely orthant dependence random variables (defined below) were introduced by Wang and Cheng ([6], 2011). Chen et al. ([7], 2016) proved a new type of Nagaevâ€™s inequality and a refined inequality of widely dependent random variables, and as applications, investigated elementary renewal theorems and weighted elementary renewal theorem. Now, let us recall the following definition of widely orthant dependence.
Definition 1.1
For random sequence \(\{X_{n}, n\geq1 \}\), if there exists a finite real sequence \(\{g_{U}(n), n\geq1\}\) satisfying, for each \(n\geq1\) and for all \(x_{i}\in(\infty, \infty)\), \(1\leq i \leq n\),
and there also exists a finite real sequence \(\{g_{L}(n), n\geq1\}\) satisfying, for each \(n\geq1\) and for all \(x_{i}\in(\infty, \infty)\), \(1\leq i \leq n\),
then a random sequence \(\{X_{n}, n\geq1\}\) is called widely orthant dependent (WOD) with dominating coefficients \(g(n)=\max\{ g_{U}(n), g_{L}(n)\}\).
Now, we will give two real examples of WOD sequences. The first example of WOD that satisfies the conditions of the main results is given as follows (see ExampleÂ 1.1) by the framework of Farlieâ€“Gumbelâ€“Morgenstern (FGM) dependence (see Cambanis [8], 1991). The second example (see ExampleÂ 1.2) does not satisfy the conditions of the paper, but it is useful as an example in the article.
Example 1.1
A sequence \(\{X_{n}, n\geq1\}\) of random variables on \((\Omega, B, P)\) is called FGM if for any \(n\in N\) and \((x_{1}, \ldots, x_{n})\in R_{n}\),
the constants \(a(\cdot, \cdot)\) are admissible if the \(2^{n}\) inequalities \(1+\sum_{1\leq j< k\leq n}a(j,k)\varepsilon_{j}\varepsilon _{k}\geq0\) for all \(\varepsilon_{j}=M_{j}\) or \(1m_{j}\) hold, where \(M_{j}\) and \(m_{j}\) are the supremum and the infimum of the set \(\{\{ F_{i}(x), x\in R\} \setminus\{0,1\}\}\). If for some integer i the marginal \(F_{i}(\cdot)\) is absolutely continuous, then \(M_{j}=1\) and \(m_{j}=0\), hence \(\varepsilon_{i}=\pm1\). Next, by Hashorva and HÃ¼sler ([9], 1999), we have \(\sum_{1\leq j< k\leq n}a(j,k)=O(n)\), \(n\in N\). Hence, from (1.4) and (1.5), we can take \(g_{L}(n)=O (1+\sum_{1\leq j< k\leq n}a(j,k) )=O(n)\), then \(P\{X_{1}\leq x_{1}, \ldots, X_{n}\leq x_{n}\}\leq g_{L}(n)\prod_{i=1}^{n} F_{i}(x_{i})\). This implies that the FGM sequence is a WOD sequence, and the conditions of the main results and lemmas are satisfied.
Example 1.2
Assume that the random vectors \((\xi_{n}, \eta_{n}) \), \(n=1,2,\ldots \)â€‰, are independent and for each integer \(n\geq1\), the random variables \(\xi_{n}\) and \(\eta_{n}\) are dependent according to the Farlieâ€“Gumbelâ€“Morgenstern copula with the parameter \(a_{n}\in[1,1]\). Suppose that the distributions of \(\xi_{n}\) and \(\eta_{n}\), \(n=1,2, \ldots \)â€‰, are absolutely continuous, denoted by \(F_{\xi_{n}}\) and \(F_{\eta_{n}}\), \(n=1,2, \ldots \)â€‰, respectively. By Sklarâ€™s theorem (see Chap.Â 2 of Nelsen RB ([10], 2006)), for each integer \(n\geq1\) and any \(x_{n}, y_{n}\in(\infty,+\infty)\), we can construct the cumulative distribution function of \((\xi_{n}, \eta_{n})\) as follows:
and
Therefore, for each \(n\geq1\), we have
Then \(\{(\xi_{n}, \eta_{n})\}\) is a sequence of independent bivariate WOD random variables. Thus, for all \(x_{j}, y_{j}\in R\),
and
From this, and by DefinitionÂ 1.1, it is easy to see that the sequence \(\{\xi_{1}, \eta_{1}, \ldots, \xi_{n}, \eta_{n}, \ldots\} \) is WOD with \(g_{L}(n)=g_{U}(n)=2^{n}\), but the condition of LemmaÂ 2.5 is not satisfied.
We can further refer to some large sample properties of nonparameter estimate based on WOD samples. For instance, Wang et al. ([11], 2013) studied the strong consistency of estimator of fixed design regression model for WOD samples. Shi and Wu ([12], 2014) discussed the strong consistency of kernel density estimator for identically distributed WOD samples. Li et al. ([13], 2015) studied the pointwise strong consistency for a kind of recursive kernel estimator based WOD samples.
In this paper, we attempt to establish a Bernsteintype inequality and to derive the rates of strong convergence for the estimators of density and hazard rate functions for WOD samples. Throughout the paper, all limits are taken as n tends to âˆž, and \(c, C, C_{1}, C_{2}, \ldots \) denote positive constants whose values may change from one place to another, unless specified otherwise.
2 Assumptions and some auxiliary results
For the sake of simplicity, some assumptions on kernel function \(K(\cdot)\) and density function \(f(x)\) are listed below.

(A1)
\(K(u)\in L_{1}\), \(\int_{\infty}^{+\infty} K(u)\,du =1\), \(\sup_{x\in R}(1+x)K(x)\leq c<\infty\).

(A2)
\(\int_{\infty}^{+\infty} u^{r}K(u)\,du =0\), \(r=1,2,\ldots, s1\), \(\int_{\infty}^{+\infty} u^{s} K(u)\,du =M\neq0\), where M is a positive constant, \(s\geq2\) is some positive integer.

(A3)
\(f(x)\in C_{2,\alpha}\), where Î± is a positive constant, \(C_{2,\alpha}\) implies that \(f(x)\) is 2nd differentiable, \(f^{\prime\prime}(x)\) is a continuous function, and \(f^{\prime \prime}(x)\leq\alpha\).
The following proposition for WOD random sequence comes from Corollary 3 in Shen ([3], 2013), which will be used in the following.
Lemma 2.1

(i)
Let a random sequence \(\{X_{n}, n\geq1 \}\) be WOD with dominating coefficients \(g(n)\). If \(\{ G_{n}(\cdot), n\geq1\}\) is a nondecreasing (or nonincreasing) function sequence, then the random sequence \(\{ G_{n}(X_{n}), n\geq1\}\) is still WOD with the same dominating coefficients \(g(n)\).

(ii)
Let a random sequence \(\{X_{n}, n\geq1 \}\) be WOD, then for each \(n\geq1\) and any \(s > 0\),
$$E\exp\Biggl\{ s\sum_{i=1}^{n}X_{i} \Biggr\} \leq g(n)\prod_{i=1}^{n}E\exp \{sX_{i}\}. $$
Remark 2.1
Condition (A1) is a reasonable condition, we can refer to condition (2) in Cai ([14], 1993) and condition (II) in Theorem S of Lin ([15], 1987). And by (A1), we can get the following lemma.
Lemma 2.2
(see Lemma 3, [14], 1993, or [15], 1987)
Let \(K(x)\) be satisfied (A1), and \(f(\cdot) \in{L_{1}}\), then

(i)
for the continuous point of \(f(x)\),
$$\begin{gathered} \lim_{h_{n} \to0} h_{n}^{1} \int_{R} K\biggl(\frac{x  u}{h_{n}}\biggr)f(u)\,du = f(x),\\ \lim _{h_{n} \to0} h_{n}^{1} \int_{R} K^{2}\biggl(\frac{x  u}{h_{n}}\biggr) f(u) \,du = f(x) \int_{R} {K^{2}} (u)\,du;\end{gathered} $$ 
(ii)
for all \(x,y \in R\), \(x \ne y\),
$$\lim_{h_{n} \to0} {h_{n}^{  1}} \int_{R} {K \biggl(\frac{{x  u}}{h_{n}} \biggr)} K \biggl( \frac{{y  u}}{h_{n}} \biggr)\,du = 0. $$
Lemma 2.3
By Lemma 3.4 of Li ([16], 2017), we obtain that
Now, we will establish a Bernsteintype inequality for a WOD random sequence as follows.
Lemma 2.4
Let a random sequence \(\{X_{n}, n\geq1 \}\) be WOD with dominating coefficients \(g(n)\), and \(EX_{i}=0\), \(X_{i}\leq d_{i}\) for \(i=1,\ldots, n\), where \(\{d_{i}, 1\leq i\leq n\}\) is a sequence of positive constants. For \(t>0\), if \(t\cdot\max_{1\leq i \leq n} {d_{i}}\leq1\), then for any \(\varepsilon>0\),
Proof
By \(1\leq i\leq n\), \(tX_{i}\leq1 \) a.s. and noting that \(1+x\leq e^{x}\) for \(x\in R\), we have that
For any \(\varepsilon>0\), using Markovâ€™s inequality and LemmaÂ 2.1(ii), we can get
By LemmaÂ 2.1(i), we see that the random sequence \(\{ X_{n},n\geq1\}\) is still WOD with dominating coefficients \(g(n)\), then
Combining (2.1) and (2.2), we complete the proof of LemmaÂ 2.4.â€ƒâ–¡
Corollary 2.1
Let a random sequence \(\{X_{n}, n\geq1 \}\) be WOD with dominating coefficients \(g(n)\), and \(EX_{i}=0\), \(X_{i}\leq d\), a.s. for \(i=1,\ldots , n\), where d is a positive constant. Set \(\sigma_{n}^{2}=n^{1}\sum_{i=1}^{n}EX_{i}^{2}\), then for any \(\varepsilon>0\),
Proof
By LemmaÂ 2.4, taking \(t=\varepsilon/(2\sigma _{n}^{2}+d\varepsilon)\), we can get CorollaryÂ 2.1 immediately.â€ƒâ–¡
Remark 2.2
By comparison, the conditions of LemmaÂ 2.4 are weaker than those of Theorem 4 in Shen ([17], 2013). So, LemmaÂ 2.4 is a generalization of Theorem 4 in Shen ([17], 2013), and it extends the Bernsteintype inequality in Wang ([18], 2015, Lemma 2.2) from END to WOD sequence.
Lemma 2.5
Let a random sequence \(\{ X_{n}, n\geq1 \}\) be WOD with dominating coefficients \(g(n)\). \(F(x)\) is a continuous distribution function. If there exists a positive constant \(\{ {{\tau_{n}}} \}\) such that \({\tau_{n}} \to0\) and \(n{\tau_{n}^{2}}/[\log(n g^{3}(n))]\to\infty\), then
In particular, taking \(\tau_{n} = n^{1/2} [\log(n g^{3}(n))(\log\log n)^{\delta}]^{1/2}\) for some \(\delta>0\), we have
Proof
The proof is based on a modification of the proof of Lemma 3.5 in Li ([16], 2017). We only outline the difference. Let \(x_{n,k}\) satisfy \(F(x_{n,k})=k/n\), \(k=1,2,\ldots,n1\), and \(\xi_{ik} = I(X_{i}\leq x_{n,k})  EI(X_{i}\leq x_{n,k})\), then
By LemmaÂ 2.1, it is easy to see that a random sequence \(\{\xi _{ik}, i\geq1\}\) is WOD with \(E\xi_{ik}=0\), \(\xi_{ik} \le2\) for fixed k. Thus, by LemmaÂ 2.4, and by choosing \(t=\varepsilon \tau_{n}/4\) for n large enough, we have
Therefore, by the Borelâ€“Cantelli lemma, we obtain the result of LemmaÂ 2.5.â€ƒâ–¡
3 Main results and proofs
Theorem 3.1
Let a random sequence \(\{ X_{n}, n\geq1 \}\) be WOD with dominating coefficients \(g(n)\), and let (A1)â€“(A3) hold true. Let \(K(\cdot)\) be a bounded monotonic density function, and \(n h_{n}^{6}/[\log(n g^{2}(n))(\log \log n)^{\delta}]\to0 \) for some \(\delta> 0\). Then, for \(f(x)\in C_{2,\alpha}\),
Weakening the condition of density kernel \(K(\cdot)\) from bounded monotonic density function to bounded variation function, we can get the result as follows.
Corollary 3.1
Let a random sequence \(\{ X_{n}, n\geq1 \}\) be WOD with dominating coefficients \(g(n)\), and let (A1)â€“(A3) hold true. If the density kernel \(K(\cdot)\) is a Borel measurable and bounded variation function and \(nh_{n}^{6}/[\log(n g^{2}(n))(\log\log n)^{\delta}]\to0 \) for some \(\delta> 0\), then, for \(f(x)\in C_{2,\alpha}\),
Theorem 3.2
If, in addition to the assumptions of TheoremÂ 3.1 and LemmaÂ 2.5, the distribution function \(F({x_{0}}) < 1\), then for \(x \le{x_{0}}\),
Corollary 3.2
If, in addition to the assumptions of CorollaryÂ 3.1 and LemmaÂ 2.5, the distribution function \(F({x_{0}}) < 1\), then for \(x \le{x_{0}}\),
Remark 3.1
From TheoremsÂ 3.1 and 3.2, CorollariesÂ 3.1 and 3.2, the rates of strong convergence can nearly reach \(O(n^{2/5})\) by choosing bandwidth \(h_{n}=O(n^{1/5})\). So the rates in here reach the same order of convergence as those in Li ([16], 2017) and Li and Yang ([19], 2005). Note that negative association implies WOD, and END also implies WOD, but the converse does not hold. Then the results of this paper are the generalization of those in Li ([16], 2017) and Li and Yang ([19], 2005).
Proof of TheoremÂ 3.1
Writing
where
By LemmaÂ 2.1, we see that a random sequence \(\{ Y_{j}, 1\leq j\leq n \}\) is still WOD with dominating coefficients \(g(n)\). And by \(K(x)\) is bounded, we can get that \(EY_{j}=0\), \(Y_{j} \le C_{3}\),
Set \(\theta_{n} = \{ nh_{n}^{2}/ [\log(n g^{2}(n))(\log\log n)^{\delta}] \} ^{1/2}\), by CorollaryÂ 2.1, for any \(\varepsilon>0\),
Thus, by the Borelâ€“Cantelli lemma, we have
Hence, using (3.1) and (3.2), we get
Note that
and
Then, by LemmaÂ 2.3, we have
It follows by using (3.3)â€“(3.5) that
This completes the proof of TheoremÂ 3.1.â€ƒâ–¡
Proof of CorollaryÂ 3.1
By \(K(x)\) is a bounded variation function, we can write that \(K(x)=K_{1}(x)K_{2}(x)\), where \(K_{1}(x)\) and \(K_{2}(x)\) are two monotone increasing functions. Then
where
Then the following proof of CorollaryÂ 3.1 is the same as the proof of TheoremÂ 3.1, here it is omitted.â€ƒâ–¡
Proof of TheoremÂ 3.2
Set \(S(x)=1F(x)\), \(S_{n}(x)=1F_{n}(x)\). By the hazard rate estimator (1.2), we get
Note that \(0 < S(x_{0}) \le S(x) \le1\) for \(x \le x_{0}\), and \(\sup_{x} f(x) \le C_{5}\). It follows by TheoremÂ 3.1 and LemmaÂ 2.5 that
and
On the other hand, for n large enough, as \(x \le x_{0}\), we have
Hence, from (3.7), (3.8), and (3.9), we have
Then we obtain TheoremÂ 3.2.â€ƒâ–¡
Proof of CorollaryÂ 3.2
The proof is analogous to the one of TheoremÂ 3.2 by CorollaryÂ 3.1, and here it is also omitted.â€ƒâ–¡
4 Conclusion
In Sect.Â 2, we give a Bernsteintype inequality for WOD random sequence which extends the Bernsteintype inequality based END sequence. In Sect.Â 3, using the Bernsteintype inequality, we obtain the rates of strong convergence for the estimators of density and hazard rate functions. By choosing the bandwidth \(h_{n}=O(n^{1/6})\), the rates of strong convergence can nearly reach \(O(n^{1/3})\).
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Acknowledgements
The authors thank the referee and the editor for their very valuable comments on an earlier version of this paper. Liâ€™s work was supported by the National Natural Science Foundation of China (NSFC) (11461057), the National Science Foundation of Jiangxi (20161BAB201003). Zhouâ€™s work was supported by the State Key Program of National Natural Science Foundation of China (71331006), the State Key Program in the Major Research Plan of National Natural Science Foundation of China (91546202), National Center for Mathematics and Interdisciplinary Sciences (NCMIS), Key Laboratory of RCSDS, AMSS, CAS (2008DP173182) and Innovative Research Team of Shanghai University of Finance and Economics (IRTSHUFE13122402).
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Li, Y., Zhou, Y. & Liu, C. On the convergence rates of kernel estimator and hazard estimator for widely dependent samples. J Inequal Appl 2018, 71 (2018). https://doi.org/10.1186/s1366001816591
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DOI: https://doi.org/10.1186/s1366001816591