A weighted estimator of conditional hazard rate with left-truncated and dependent data

Abstract

Based on empirical likelihood method, we construct new weighted estimators of conditional density and conditional survival functions when the interest random variable is subject to random left-truncation; further, we define a plug-in weighted estimator of the conditional hazard rate. Under strong mixing assumptions, we derive asymptotic normality of the proposed estimators which permit to built a confidence interval for the conditional hazard rate. The finite sample behavior of the estimators is investigated via simulations too.

Appendix

Lemma 2

(Fan and Yao (2003), Proposition 2.6, p. 72) Let $V_1,\ldots , V_m$ be $\alpha$-mixing and complex-valued random variables measurable with respect to the $\sigma$-algebra $\mathscr {F}^{j_1}_{i_1}, \ldots , \mathscr {F}^{j_m}_{i_m}$, respectively, with $1\le i_1<j_1<\cdots <j_m\le n,~ i_{l+1}-j_l\ge w\ge 1$ and $P(|V_j|\le 1)=1$ for $l, j=1, 2, \ldots , m$. Then $|E(\prod ^m_{j=1}V_j)-\prod ^m_{j=1}EV_j|\le 16(m-1)\alpha (w),$ where $\mathscr {F}^b_a=\sigma \{V_i , a\le i\le b\}$ and $\alpha (w)$ is the mixing coefficient.

Lemma 3

(Hall and Heyde (1980), Corollary A.2, p. 278) Suppose that X and Y are random variables such that $E|X|^p<\infty , E|Y|^q<\infty$, where p, $q>1$, $p^{-1}+q^{-1}<1$. Then

$$\begin{aligned} |EXY-EXEY| \le 8\Vert X\Vert _p\Vert Y\Vert _q\Big \{\sup _{A\in \sigma (X), B\in \sigma (Y)} |P(A\cap B) -P(A)P(B)|\Big \}^{1-p^{-1}-q^{-1}}. \end{aligned}$$

Lemma 4

(Liang et al. (2011), Lemma 5.4) Suppose that $\alpha (k)=O(k^{-\gamma })$ for some $\gamma >3$, and that (A0) holds. Then $\sup _{y\ge a_F}|G_n(y)-G(y)|=O_\mathbf P(n^{-1/2})$.