Nonparametric Estimation of a Hazard Rate Function with Right Truncated Data

Abstract

Left truncation and right truncation coexist in a truncated sample. Earlier researches focused on left truncation. Lagakos et al. (Biometrika 75:515–523, 1988) proposed to transform right truncated data to left truncated data and then apply the methods developed for left truncation. Interpretation of survival quantities, such as the hazard rate function, in reverse-time is not natural. Though it is most interpretable, researchers seldom use the forward-time hazard function. In this book chapter we studied the nonparametric inference for the hazard rate function with right truncated data. Kernel smoothing techniques were used to get smoothed estimates of hazard rates. Three commonly used kernels, uniform, Epanechnikov, and biweight kernels were applied on the AIDS data to illustrate the proposed methods.

Appendix

Asymptotic properties of kernel estimator of intensity were established by Ramlau-Hansen (1983). In this study we exploratively investigate the limiting distribution of $\widehat {\alpha }(t)$. In the following context, “≈” indicates asymptotic equivalence. Note that $(nb)^{1/2}[\widehat {\alpha }(t)-\alpha ^s(t)]$ can be expressed as

$$\displaystyle \begin{aligned} (nb)^{1/2}[\widehat{\alpha}(t)-\alpha^s(t)]=\frac{(nb)^{1/2}}{b}\int_0^\tau K\left(\frac{t-u}{b}\right)\left[\frac{-\widehat{G}(u)}{1-\widehat{G}(u)}d(\widehat{A}^*-A^*)(u)\right] \end{aligned}$$

$$\displaystyle \begin{aligned} -\frac{(nb)^{1/2}}{b}\int_0^\tau K\left(\frac{t-u}{b}\right) \left(\frac{\widehat{G}(u)}{1-\widehat{G}(u-)}-\frac{G(u)}{1-G(u-)}\right)dA^*(u) \end{aligned}$$

For the first term on the right-hand side of the above equation, it can be shown that

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \frac{(nb)^{1/2}}{b}\int_0^\tau K\left(\frac{t-u}{b}\right)\frac{-\widehat{G}(u)}{1-\widehat{G}(u-)}d(\widehat{A}^*-A^*)(u)\\ &\displaystyle \approx&\displaystyle \sqrt{\frac{n}{b}}\int_\tau^0 K\left(\frac{t-u}{b}\right)\frac{G(u)}{1-G(u-)}d(\widehat{A}^*-A^*)(u)\\ &\displaystyle =&\displaystyle \sqrt{\frac{n}{b}}\int_\tau^0 K\left(\frac{t-u}{b}\right)\frac{G(u)}{1-G(u-)} J(u) \frac{d\bar M^*(u)}{\bar{Y}(u)} \end{array} \end{aligned} $$

To investigate the second term on the right-hand side, we first consider the Taylor series expansion,

$$\displaystyle \begin{aligned}\frac{\widehat{G}(u)}{1-\widehat{G}(u^-)}-\frac{G(u)}{1-G(u-)}\approx \frac{d}{dA^*(u)}\left(\frac{G(u)}{1-G(u-)}\right) (\widehat{A}^*-A^*)(u).\end{aligned}$$

Then we will have

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \frac{(nb)^{1/2}}{b}\int_0^\tau K\left(\frac{t-u}{b}\right)\left[-\left(\frac{\widehat{G}(u)}{1-\widehat{G}(u-)}-\frac{G(u)}{1-G(u-)}\right)dA^*(u)\right]\\ {} &\displaystyle \approx&\displaystyle \sqrt{\frac{n}{b}}\int_0^\tau K\left(\frac{t-u}{b}\right)\left[-d\left(\frac{G(u)}{1-G(u-)}\right) (\widehat{A}^*-A^*)(u)\right]\\ {} &\displaystyle =&\displaystyle \sqrt{\frac{n}{b}}\int_0^\tau K\left(\frac{t-u}{b}\right)\left[-d\left(\frac{G(u)}{1-G(u-)}\right) \int_\infty^u J(x)\frac{d\bar M^*(x)}{\bar{Y}(x)}\right]\\ {} &\displaystyle =&\displaystyle \sqrt{\frac{n}{b}}\int_\tau^0\left[-\int_0^u K\left(\frac{t-y}{b}\right) d\left(\frac{G(y)}{1-G(y-)}\right) \right]J(x)\frac{d\bar M^*(x)}{\bar{Y}(x)}. \end{array} \end{aligned} $$

Combining the above results, $(nb)^{1/2}[\widehat {\alpha }_n(t)-\alpha _n^s(t)]$ is asymptotically equal to

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \sqrt{\frac{1}{nb}}\int_\tau^0 \left[ K\left(\frac{t-u}{b}\right)\frac{G(u)}{1-G(u-)}\right.\\ {} &\displaystyle &\displaystyle \left.-\int_0^u K\left(\frac{t-y}{b}\right) d\left(\frac{G(y)}{1-G(y-)}\right) \right]J(u) \frac{d\bar M^*(u)}{\bar{Y}(u)/n}. \end{array} \end{aligned} $$

(22.15)

Through the martingale central limit theorem, when n →∞, b → 0, nb →∞, $(nb)^{1/2}[\widehat {\alpha }(t)-\alpha ^s(t)]$ converges in distribution to a normal random variable with mean zero and the following variance function,

$$\displaystyle \begin{aligned}\frac{1}{b}\int_\tau^0 \left[ K\left(\frac{t-u}{b}\right)\frac{G(u)}{1-G(u-)}-\int_0^u K\left(\frac{t-x}{b}\right) d\left(\frac{G(x)}{1-G(x-)}\right) \right]^2 \frac{\alpha(u)du}{y(u)}.\end{aligned}$$

In addition, it needs to prove that $(nb)^{1/2}[\alpha _n^s(t)-\alpha (t)]$ is asymptotically negligible. Some regularity conditions for establishing such a result can be found in Ramlau-Hansen (1983, §4). We do not investigate this topic here.