Asymptotic properties of conditional distribution estimator with truncated, censored and dependent data

Abstract

In this paper we study the strong and weak convergence with rates for the estimators of the conditional distribution function as well as conditional cumulative hazard rate function for a left truncated and right censored model. It is assumed that the lifetime observations with multivariate covariates form a stationary α-mixing sequence. Also, the almost sure representations and asymptotic normality of the estimators are established. The finite sample performance of the estimators is investigated via simulations.

Appendix

In this section, we list some preliminary lemmas, which have been used in Sect. 4. Let {Z _i ,i≥1} be a sequence of α-mixing real random variables with the mixing coefficients {α(k)}.

Lemma 4.1

(Hall and Heyde 1980, Corollary A.2, p. 278)

Suppose that X and Y are random variables such that E|X|^p<∞, E|Y|^q<∞, where p, q>1, p ⁻¹+q ⁻¹<1. Then

Lemma 4.2

(Volkonskii and Rozanov 1959)

Let V ₁,…,V _m be α-mixing random variables measurable with respect to the σ-algebra $\mathcal{F}^{j_{1}}_{i_{1}},\ldots, \mathcal{F}^{j_{m}}_{i_{m}}$, respectively, with 1≤i ₁<j ₁<⋯<j _m ≤n, i _l+1−j _l ≥w≥1 and |V _j |≤1 for l,j=1,2,…,m. Then $|E(\prod^{m}_{j=1}V_{j})-\prod^{m}_{j=1}EV_{j}|\leq 16(m-1)\alpha(w)$, where $\mathcal{F}^{b}_{a}=\sigma\{V_{i}, a\leq i\leq b\}$ and α(w) is the mixing coefficient.

Lemma 4.3

(Liebscher 2001, Proposition 5.1)

Assume that EZ _i =0 and |Z _i |≤S<∞ a.s. (i=1,2,…,n). Set $D_{N}=\max_{1\leq j\leq 2N}\mathrm{ Var}(\sum_{i=1}^{j}Z_{i})$. Then, for n,N∈ℕ,0<N≤n/2, ϵ>0, $P(|\sum_{i=1}^{n}Z_{i}|>\epsilon )\leq 4\exp{\{ -\frac{\epsilon ^{2}}{16}(nN^{-1}D_{N}+\frac{1}{3}\epsilon SN)^{-1}\}}+32\frac{S}{\epsilon }n\alpha (N)$.

Lemma 4.4

(Liebscher 1996, Lemma 2.3)

Assume α(k)≤C ₁ k ^−r, for some r>1, C ₁>0. Let sup_{1≤i,j≤n,i≠j} |Cov(Z _i ,Z _j )|:=R ^∗(n)<∞ be satisfied. Moreover, let R _m (n)<∞ for some m, 2r/(r−1)<m≤∞, where R _m (n)=sup_1≤i≤n (E|Z _i |^m)^1/m, for 1≤m<∞, and R _∞(n)=sup_1≤i≤n  ess sup _w∈Ω|Z _i |. Then $\mathrm{Var} (\sum^{n}_{i=1}Z_{i} )\leq n \{C_{2}(r,m)(R_{m}(n))^{2m/(r(m-2))}(R^{*}(n))^{1-m/(r(m-2))}+R^{2}_{2}(n) \}$ holds with $C_{2}(r,m):=\frac{20r-40r/m}{r-1-2r/m}C_{1}^{1/r}$.