Nonparametric estimation of conditional distribution functions with longitudinal data and time-varying parametric models

Abstract

Nonparametric estimation and inferences of conditional distribution functions with longitudinal data have important applications in biomedical studies. We propose in this paper an estimation approach based on time-varying parametric models. Our model assumes that the conditional distribution of the outcome variable at each given time point can be approximated by a parametric model, but the parameters are smooth functions of time. Our estimation is based on a two-step smoothing method, in which we first obtain the raw estimators of the conditional distribution functions at a set of disjoint time points, and then compute the final estimators at any time by smoothing the raw estimators. Asymptotic properties, including the asymptotic biases, variances and mean squared errors, are derived for the local polynomial smoothed estimators. Applicability of our two-step estimation method is demonstrated through a large epidemiological study of childhood growth and blood pressure. Finite sample properties of our procedures are investigated through simulation study.

Appendix: Proof of theoretical results

1.1 A.1 Useful approximation for the equivalent kernels

The following approximations for the equivalent kernel function $W_{q,p+1} (t_j, t; h)$ are used in computing the asymptotic bias and variance of $F_{t, \widehat{\theta }(t|x)}^{(q)} \big [ y(t)|x \big ]$:

$$\begin{aligned}&W_{q,Q+1}(t_{j},t; h)=\frac{q!}{Jh^{q+1}g(t)}K_{q,p+1}\Big (\frac{t_{j}-t}{h} \Big ) \big [1+o_{p}(1) \big ], \quad j=1, \ldots ,J; \nonumber \\\end{aligned}$$

(A.1)

$$\begin{aligned}&\sum _{j=1}^{J}W_{q,p+1}(t_{j},t; h) \,(t_{j}-t)^{k}=q!1_{[k=q]},\quad k=1, \ldots , p; \end{aligned}$$

(A.2)

$$\begin{aligned}&\sum _{j=1}^{J} W_{q,p+1}(t_{j},t; h) \,(t_{j}-t)^{p+1}\nonumber \\&\quad =\,q!h^{p-q+1} B_{p+1} \big ( K_{q,p+1} \big ) \big [1+ o_{p}(1) \big ], \quad k=1, \ldots , p; \end{aligned}$$

(A.3)

$$\begin{aligned}&\sum _{j=1}^{J} W_{q,p+1}^{2}(t_{j},t; h)=\frac{(q!)^2}{Jh^{2q+1}g(t)} V \big (K_{q,p+1} \big ) \big [1+ o_{p}(1) \big ], \end{aligned}$$

(A.4)

where $K_{q,p+1}(t)$, $B_{p+1}(K)$ and V(K) are defined in Theorem 1. Proofs of Eqs. (A.1)–(A.4) are given Fan and Zhang (2000, Appendix A, Lemmas 1 and 2).

1.2 A.2 Proof of Theorem 1

First, note that $\widetilde{F}_{t_j, \theta (t_j|x)} \big [ y(t_{j})|x \big ]=F_{t_j, \widetilde{\theta }(t_j|x)} \big [ y(t_{j})|x \big ]$. Using the Eq. (6), the bias of $\widehat{F}_{t, \theta (t|x)}^{(q)} \big [ y(t) \big | x \big ]$ is

$$\begin{aligned} E \big \{ \widehat{F}_{t, \theta (t|x)}^{(q)} \big [ y(t) \big | x \big ] \big \} - F_{t, \theta (t|x)}^{(q)} \big [ y(t) \big | x \big ] = {\mathcal {W}}_1 + {{\mathcal {W}}}_2, \end{aligned}$$

(A.5)

where

$$\begin{aligned} \left\{ \begin{array}{lll} {{\mathcal {W}}}_1 &{}=&{} \sum \nolimits _{j=1}^{J} W_{q,p+1}(t_{j},t; h) \big \{ E \widetilde{F}_{t_j, \theta (t_j|x)} \big [ y(t_{j})|x \big ] - F_{t_j, \theta (t_j|x)} \big [ y(t_{j})|x \big ] \big \}, \\ {{\mathcal {W}}}_2 &{}=&{} \sum \nolimits _{j=1}^{J} W_{q,p+1}(t_{j},t; h) F_{t_j, \theta (t_j|x)} \big [ y(t_{j})|x \big ] - F_{t, \theta (t|x)}^{(q)} \big [ y(t)|x \big ] . \end{array} \right. \end{aligned}$$

(A.6)

It then follows from (15) and (A.1) that

$$\begin{aligned} {{\mathcal {W}}}_1 = \sum _{j=1}^{J} \frac{q!}{Jh^{q+1}g(t)}K_{q,p+1}\Big (\frac{t_{j}-t}{h} \Big ) \big [1+o_{p}(1) \big ] o_p \big ( n_j^{-1/2} \big ) = o_p \big ( n^{-1/2} \big ), \end{aligned}$$

(A.7)

where the second equality holds because, by Assumption A2, $\lim _{n \rightarrow \infty } (n_j/n)$ is bounded between 0 and 1, and $\sum _{j=1}^{J} \big | q! \big [Jh^{q+1}g(t) \big ]^{-1}K_{q,p+1}\big [(t_j-t)/h \big ] \big |$ is bounded. By the Taylor expansions for $F_{t_j, \theta (t_j)} \big [ y(t_{j})|x \big ]$ and the Eqs. (A.2) and (A.3), we have

$$\begin{aligned} {{\mathcal {W}}}_2= & {} \sum _{j=1}^{J} W_{q,p+1}(t_{j},t; h) \Big \{ \sum _{k=0}^{p+1} \Big [ F_{t, \theta (t|x)}^{(k)} \big [ y(t)|x \big ] \frac{(t_j-t)^k}{k!} \Big ] \nonumber \\&+ o_p \big [(t_j-t)^{p+1} \big ] \Big \} - F_{t, \theta (t|x)}^{(q)} \big [ y(t)|x \big ] \nonumber \\= & {} \frac{q!h^{p-q+1}}{p+1} F_{t, \theta (t|x)}^{(p+1)} \big [ y(t)|x \big ] B_{p+1} \big ( K_{q,p+1} \big ) \big [1+ o_{p}(1) \big ]. \end{aligned}$$

(A.8)

Since, by Assumption A1, and the asymptotic expressions of (A.7) and (A.8), ${{\mathcal {W}}}_2$ is the dominating term over ${{\mathcal {W}}}_1$. The asymptotic expression for the bias of $\widehat{F}_{t, \theta (t|x)}^{(q)} \big [ y(t) \big | x \big ]$ follows from (A.6) to (A.8).

Let $\mu _{F}(t_j)= E\big \{ \widetilde{F}_{t_j, \theta (t_j|x)} \big [ y(t) \big | x \big ] \big \}$. Then, by Eq. (6),

$$\begin{aligned} Var \Big \{ \widehat{F}_{t, \theta (t|x)}^{(q)} \big [ y(t) \big | x \big ] \Big \}= & {} E\Big \{ \sum _{j=1}^{J} W_{q,p+1}(t_{j},t; h) \Big [ \widetilde{F}_{t_j, \theta (t_{j}|x)} \big [ y(t_{j}) \big | x \big ] -\mu _{F}(t_j) \Big ] \Big \}^{2} \nonumber \\= & {} {{\mathcal {W}}}_3 + {{\mathcal {W}}}_4, \end{aligned}$$

(A.9)

where, by (15), (A.4) and Assumption A2 with $c_j=c$,

$$\begin{aligned} {{\mathcal {W}}}_3= & {} \sum _{j=1}^{J} W_{q,p+1}^{2}(t_{j},t; h) E \Big \{ \Big [ \widetilde{F}_{t_j, \theta (t_{j}|x)} \big [ y(t_{j}) \big | x \big ]-\mu _{F}(t_j) \Big ]^2 \Big \} \nonumber \\= & {} \sum _{j=1}^{J}W_{q,p+1}^{2}(t_{j},t; h) Var \Big \{ \widetilde{F}_{t_j, \theta (t_{j}|x)} \big [ y(t_{j}) \big | x \big ] \Big ]\nonumber \\= & {} \frac{(q!)^2}{cnJh^{2q+1}g(t)} V \big (K_{q,p+1} \big ) \nonumber \\&\times F'_{t, \theta (t|x)}\big [y(t) \big |x \big ]^T I^{-1}\big [ \theta (t|x) \big ] F'_{t, \theta (t|x)}\big [y(t) \big |x \big ] \big [1+ o_{p}(1) \big ] \end{aligned}$$

(A.10)

and, by (14), Assumptions A2 and A4, the Eq. (A.1) and $\lim _{n \rightarrow \infty } n_{jk}/n=c_{jk}$, there is a constant $C_1>0$ such that, when n is sufficiently large,

$$\begin{aligned} {{\mathcal {W}}}_4= & {} \sum _{j\ne k}^{J} \Big \{ W_{q,p+1}(t_{j},t; h) W_{q,p+1}(t_{k},t; h) \nonumber \\&\times E \Big \{ \big [ \widetilde{F}_{t_j, \theta (t_{j}|x)} \big [ y(t_{j}) \big | x \big ] -\mu _{F}(t_j) \big ] \big [ \widetilde{F}_{t_k, \theta (t_{k}|x)} \big [ y(t_{k}) \big | x \big ]-\mu _{F}(t_k) \big ] \Big \} \Big \} \nonumber \\\le & {} \sum _{j\ne k}^{J}\Big \{ \Big | W_{q,p+1}(t_{j},t; h) \, W_{q,p+1}(t_{k},t; h) \Big | \nonumber \\&\times \Big | Cov\big \{ \widetilde{F}_{t_j, \theta (t_{j}|x)} \big [ y(t_{j}) \big | x \big ], \widetilde{F}_{t_k, \theta (t_{k}|x)} \big [ y(t_{k}) \big | x \big ] \big \} \Big |\Big \} \nonumber \\\le & {} \frac{C_1}{J^2 h^{2q+2} g^2(t)} \sum _{j \ne k}^{J} \Big | K_{q,p+1}\Big ( \frac{t_j-t}{h} \Big ) \, K_{q,p+1} \Big ( \frac{t_k-t}{h} \Big ) \, \Big [ \frac{\rho _F (t_j, t_k |x)}{r(n_j, n_k, n_{jk})} \Big ] \Big |.\nonumber \\ \end{aligned}$$

(A.11)

The bounded support of $K(\cdot )$ implies that, for any t, $K_{q, p+1}\big [(t_j-t)/h \big ]=K_{q, p+1}\big [(t_k-t)/h \big ]=0$ for any $j \ne k$ such that $\big | t_j -t_k \big | \le ah$ for some constant $a>0$.

We now consider the following three situations:

• (i) If $\big | t_j - t_k \big | \le \delta$, by Assumption A2, $Cov\big \{ \widetilde{F}_{t_j, \theta (t_{j}|x)} \big [ y(t_{j}) \big | x \big ], \widetilde{F}_{t_k, \theta (t_{k}|x)} \big [ y(t_{k}) \big | x \big ] \big \}=0$, so that ${{\mathcal {W}}}_4=0$, and, by (A.9), $Var \big \{ \widehat{F}_{t, \theta (t|x)}^{(q)} \big [ y(t) \big | x \big ] \big \}={{\mathcal {W}}}_3$.

• (ii) If $\big | t_j - t_k \big | > \delta \ge ah$, then $K \big [(t_j-t)/h \big ] K \big [(t_k-t)/h \big ] =0$, so that, by (A.11), ${{\mathcal {W}}}_4=0$, and it still follows from (A.9) that $Var \big \{ \widehat{F}_{t, \theta (t|x)}^{(q)} \big [ y(t) \big | x \big ] \big \}={{\mathcal {W}}}_3$.

• (iii) If $\delta < ah$ and $\delta \le \big | t_j - t_k \big | \le ah$, since $K_{q, p+1}(s)$ and $\rho _F (t_j, t_k)$ are bounded, there is $C_2>0$, so that, by (A.11) and $\sum _{j =1}^{J} \sum _{k: \delta \le |t_k-t_j| \le ah} r^{-1}(n_j, n_k, n_{jk}) = o(Jh)$,

  $$\begin{aligned} {{\mathcal {W}}}_4\le & {} \frac{C_2}{J^2 h^{2q+2} g^2(t)} \Big \{ \sum _{j =1}^{J} \sum _{k: \delta \le |t_k-t_j| \le ah} r^{-1}(n_j, n_k, n_{jk}) \Big \} \big [ 1+ o_p(1) \big ] \nonumber \\= & {} o_p \big [ \big ( nJh^{2q+1} \big )^{-1} \big ]. \end{aligned}$$

  (A.12)

Then, by (A.9) and (A.12), $Var \big \{ \widehat{F}_{t, \theta (t|x)}^{(q)} \big [ y(t) \big | x \big ] \big \}={{\mathcal {W}}}_3 \big [1+ o_p(1) \big ]$. Since, for all three situations (i), (ii) and (iii), $Var \big \{ \widehat{F}_{t, \theta (t|x)}^{(q)} \big [ y(t) \big | x \big ] \big \}={{\mathcal {W}}}_3 \big [1+ o_p(1) \big ]$, the asymptotic expression for the variance of $\widehat{F}_{t, \theta (t|x)}^{(q)} \big [ y(t) \big | x \big ]$ follows from (A.9)-(A.11).