Abstract
Most studies for NA random variable is under complete sampling setting, which is actually an relatively ideal condition in application. The paper relaxes this condition to the censoring incomplete sampling data and considers the topic for kernel estimation of the density function together with the hazard function based on the Kaplan-Meier estimator. The strong asymptotic properties for the two estimators are firstly established.
Introduction
Definition (Joag-Dev and Proschan, 1983) Random sequences {T i , 1 < i ≤ n} are said to be negatively associated (NA) if for every pair of disjoint subsets B 1 and B 2 from {1, 2, · · · , n}, cov (f 1 (T i ; i ∈ B 1 ), f 2 (T j ; j ∈ B 2 )) ≤ 0, where there exists the covariance for f 1 (·) and f 2 (·) with increasing for every variable (or decreasing for every variable). A sequence of random variables {T i ; i ≥ 1} is said to be NA if every finite subfamily is NA.
Obviously, NA random variables includes independent random variable as a special case, and also describes many other random variable, for example the random sampling without replacement in a finite population. Many researchers have studied the property of NA random variables and published a large number of articles. For example, Su et al. (1997) established a probability inequality and some moment inequalities for the partial sum of a NA sequence, which contributed to prove some properties for strictly stationary NA sequences such as weak invariance principle. The results in Shao (2000) showed that most of the well-known inequalities, such as the Kolmogorov exponential inequality and the Rosenthal maximal inequality, are still hold for NA random variables. Wu Most studies for NA random variable is under complete sampling setting, however, which is actually an relatively ideal condition in application. When study survival data, censoring incomplete situation is often encountered in data sampling. Let (T i , Y i ), i = 1, · · · , n, denote a sequence of nonnegative random variable vector where T i is the true survival time of interest which is right censored by the censoring random variable Y i . It is assumed that T i is independent of Y i , but there are not assumed to be mutually independent for T i 's and Y i 's, which are all NA in our paper. And then the observed data is (X i , δ i ) in the censorship model, where
and I(A) is the indicator of the random event A. For its simplicity, assumed that T i have a common unknown continuous marginal distribution function F (x) = P (T i ≤ x) and denote its survival distribution S T (t) = 1 − F (t). The random censoring times Y i , i = 1, · · · , n, being independent of the random variables T i 's , are assumed to have a common distribution function G(y) = P (Y i ≤ y) with its survival distribution S Y (t) = 1−G(t). Meanwhile, denote L(·) as the distribution of the observed variable X i 's, and write its survival distribution as S X (t) = 1 − L(t). For any distribution function H(·), we define the left and right endpoints of its support as a H and τ H by a H = inf{x : H(x) > 0}, τ H = sup{x : H(x) < 1} in our paper.
The distribution function L(·) can be consistently estimated by the empirical distribution function L n (t), which is defined as follows:
the number of uncensored or censored observations no less than time t.
For drawing nonparametric inference about F (·) based on the censored observations
, introduce a stochastic process on [0, ∞) as follows:
the number of uncensored observations no larger than time t. The well-known nonparametric maximum likelihood estimation F n (·) of F (·) was the Kaplan-Meier (K-M) estimator (Kaplan and Meier, 1958) , which is usually used to estimate F (·) for the incomplete data (X i , δ i ):
where the jump
Define the sub-distribution function
F (0) = 0, hence we have by integration by parts that
and then
The estimation for the hazard function h(·) is also an interesting issue in survival analysis, which is defined as follows when there is further assumed that F (·) has a densityf (·):
Its correspondingly cumulative hazard function is defined as
The above representation of H(·) in term of F * (·) and S X (·) suggests the empirical estimation for H(·) by
and
the empirical distribution functions of F * (·).
we may verify that the estimators F n (·) and H n (·) can be respectively represented as
and
where
are the order statistics of X 1 , X 2 , . . . , X n and δ (i) is the concomitant of X (i) . The K-M estimator F n (x) and the estimator H n (x) have been generally accepted as a substitute for the usual empirical estimator of distribution function F (·) and the cumulative hazard function H(·) in the case of right censoring, respectively, which help to study other estimators such as the kernel density estimator and the kernel hazard estimator in the following.
A kernel estimator for f (·) based on F n (·) can be constructed as
where k(·) is a smooth probability kernel function and {b n , n ≥ 1} is a sequence of bandwidth tending to zero at appropriate rates.
Similarly, we can also construct an kernel estimator for the hazard function h(·) under the NA sampling data, which is defined by
The smoothed estimators f n (·) and h n (·) have attracted the attention of many investiga- For present our main results, definē
The main purpose of this paper is to study the asymptotic properties of kernel smoothing density estimator f n (·) and hazard estimator h n (·) based on censoring NA data. Under certain regularity conditions, we establish the strong asymptotic properties for the two estimators with the convergent rates being O(b
, where {b n , n ≥ 1} will be defined in the main results.
Main results and their proofs
Theorem 1 Under the conditions of Lemma 1, and assume that k(·) is the bounded variation probability kernel density on the finite interval (r, s), where r < 0 < s. Suppose that density
where the sequence {b n ; n ≥ 1} satisfies b
Theorem 2 Under the conditions of Lemma 1, and assume that k(·) is the bounded variation probability kernel density on the finite interval (r, s), where r < 0 < s. The density
2)
where the sequence {b n ; n ≥ 1} satisfies b
Remark 1 Theorem 1 and Theorem 2 are fundamental results in studying censoring NA data, which can be useful in deriving some asymptotic properties for the kernel density estimator f n (·) and the hazard function estimator h n (·), respectively. Take for example, if one can establish the similar main results in Hall (1981) for NA data, then using Theorem 1, the following proposition may hold, which is the next issue we will consider.
Proposition Suppose that the sequence {b n ; n ≥ 1} satisfies b n → 0 and (a) (ln n) 2 (nb n ln ln n) → 0,
then there will be lim sup
where ϕ(f, G) is some functional for f (·) and G(·).
We firstly present two lemmas (Wu and Chen, 2013 ) that help to prove our theorem.
Lemma 1 Let {T n ; n ≥ 1} and {Y n ; n ≥ 1} be two sequences of NA random variables.
Suppose that the sequences {T n ; n ≥ 1} and {Y n ; n ≥ 1} are independent. Then, for any
For positive reals x and t, and δ taking value 0 or 1, write
Lemma 2 Let {T n ; n ≥ 1} and {Y n ; n ≥ 1} be two sequences of NA random variables.
Suppose that the sequences {T n ; n ≥ 1} and {Y n ; n ≥ 1} are independent. Then, for any
and
where sup
Remark 2 Note that by the definition of η(X, t, δ),
Therefore, we can obtain by Lemma 2 that
Meanwhile, one can establish the following result by the fact that {X n ; n ≥ 1} and {(X n , δ n ); n ≥ 1} are all NA random variable sequences according to Joag-Dev and Proschan (1983).
Lemma 3 Under the conditions of Lemma 1, for any
For simplicity and without loss of generality, it can be assumed that a F = 0 in the following proof procedure.
Proof of Theorem 1 According to Remark 2, f n (x) −f n (x) can be expressed as
Thus, we have the following formula (2.10)
Using the partial integration for I 11 , we have
Then, combine the property of empirical process and (2.6), when n is large enough,
and
Again, note that
Integrating by parts for I 12 , we have for 0 < τ < τ L
Since density function f (·) = F ′ (·) and g(·) = G ′ (·) are bounded in the closed interval [0, τ ], where means thatL
is also bounded in the interval [0, τ ], therefore by empirical process
where M is some positive constant number.
Thus, combining equations (2.10), (2.11), (2.12) and (2.13), we have
On the other hand, similar to the discussion of I 12 , This completes the proof by combining (2.9), (2.14), (2.15) and (2.16).
Proof of Theorem 2 Note the strong asymptotic expression from (2.6) that
n (s) L 2 (s) dF * (s) + r 2n (t), and the counterpart to the term I 11 ,
Then following the proof procedure of Theorem 1, we can also establish Theorem 2, and this ends the proof.

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