Statistical inference based on the nonparametric maximum likelihood estimator under double-truncation

Abstract

Doubly truncated data consist of samples whose observed values fall between the right- and left- truncation limits. With such samples, the distribution function of interest is estimated using the nonparametric maximum likelihood estimator (NPMLE) that is obtained through a self-consistency algorithm. Owing to the complicated asymptotic distribution of the NPMLE, the bootstrap method has been suggested for statistical inference. This paper proposes a closed-form estimator for the asymptotic covariance function of the NPMLE, which is computationally attractive alternative to bootstrapping. Furthermore, we develop various statistical inference procedures, such as confidence interval, goodness-of-fit tests, and confidence bands to demonstrate the usefulness of the proposed covariance estimator. Simulations are performed to compare the proposed method with both the bootstrap and jackknife methods. The methods are illustrated using the childhood cancer dataset.

Appendices

Appendix A: Bootstrap and jackknife algorithms

Simple bootstrap algorithm (Moreira and Uña-Álvarez 2010):

• Step 1: For each $b=1,\;\ldots ,\;B$, draw bootstrap resamples $\{\;(U_{jb}^*,\;T_{jb}^*,\;V_{jb}^*):j=1,\;\ldots ,\;n\;\}$ from $\{\;(U_j ,\;T_j ,\;V_j ):j=1,\;\ldots ,\;n\;\}$, and then compute the NPMLE $\hat{{F}}_b^*(t)$ from them.

• Step 2: Compute the bootstrap variance estimator

  $$\begin{aligned} \hat{{V}}_{\mathrm{Boot}} \{\hat{{F}}(t)\}=\frac{1}{B-1}\sum _{b=1}^B {\{\hat{{F}}_b^*(t)-\bar{{F}}^{*}(t)\}^{2}} , \end{aligned}$$

  where $\bar{{F}}^{*}(t)=\frac{1}{B} \sum _{b=1}^B {\hat{{F}}_b^*(t)}$, and take the $(\alpha /2)\times 100$% and $(1-\alpha /2)\times 100$% points of $\{\;\hat{{F}}_b^*(t):\;b=1,\;\ldots ,\;B\;\}$ for the$(1-\alpha )\times 100$% confidence interval.

Jackknife algorithm:

• Step 1: For each $i=1,\;\ldots ,\;n$, delete the $i$ th sample from $\{\;(U_j ,\;T_j ,\;V_j ):j=1,\;\ldots ,\;n\;\}$, and then compute the NPMLE $\hat{{F}}_{(-i)}(t)$ from the remaining $n-$1 samples.

• Step 2: Compute the jackknife variance estimator

  $$\begin{aligned} \hat{{V}}_{\mathrm{Jack}} \{\hat{{F}}(t)\}=\frac{n-1}{n}\sum _{i=1}^n {\{\hat{{F}}_{(-i)} (t)-\bar{{F}}_{(\cdot )} (t)\}^{2}} , \end{aligned}$$

  where $\bar{{F}}_{(\cdot )} (t)=\frac{1}{n}\sum _{i=1}^n {\hat{{F}}_{(-i)} (t)}$, and the log-transformed $(1-\alpha )\times 100$% confidence interval

  $$\begin{aligned} (\;\hat{{F}}(t)\exp [\;-z_{\alpha /2} \hat{{V}}_{\mathrm{Jack}}^{\mathrm{1/2}} \{\hat{{F}}(t)\}/\hat{{F}}(t)\;],\;\;\hat{{F}}(t)\exp [\;z_{\alpha /2} \hat{{V}}_{\mathrm{Jack}}^{\mathrm{1/2}} \{\hat{{F}}(t)\}/\hat{{F}}(t)\;]\;). \end{aligned}$$

Appendix B: Asymptotic theory

1.1 Appendix B1. Weak convergence of $\sqrt{n}(\;\hat{{F}}(t)-F(t)\;)$

Although not stated explicitly, we assume that the identifiability conditions (Shen 2010, p. 836) are satisfied. Consider the log-likelihood function

$$\begin{aligned} \ell _n (F)/n=\sum _{i=1}^n {(\log f_j -\log F_j )} /n. \end{aligned}$$

For any $h\in Q$, where $Q$ is the set of all uniformly bounded functions, let $H(t)=\int _0^t {h(s)dF(s)}$ and $\hat{{H}}(t)=\int _0^t {h(s)d\hat{{F}}(s)}$ where $h$ satisfies the constraint $\hat{{H}}(\infty )=1$. Suppose that $\hat{{F}}$ is the maximizer of $\ell _n (F)$. Then for any $h\in Q$ and $\varepsilon \ge 0$, we have $\ell _n (\hat{{F}}+\varepsilon \hat{{H}})\le \ell _n (\hat{{F}})$. Hence, the score function $\partial \ell _n (F+\varepsilon H)/\partial \varepsilon |_{\varepsilon =0}$ is equal to

$$\begin{aligned} \Psi _n (F)[h]\equiv \frac{1}{n}\sum _{i=1}^n {\left[ {h(T_i )-\frac{\int {\mathbf{I}(U_i \le s\le V_i )h(s)dF(s} )}{\int {\mathbf{I}(U_i \le s\le V_i )dF(s} )}} \right] } , \end{aligned}$$

for any $h\in Q$. The expectation is defined as

$$\begin{aligned} \Psi (F)[h]\equiv E\left[ {h(T^{*})-\frac{\int {\mathbf{I}(U^{*}\le s\le V^{*})h(s)dF(s)} }{\int {\mathbf{I}(U^{*}\le s\le V^{*})dF(s)} }} \right] . \end{aligned}$$

Consider $\Psi _n (F)[h]$ as a random function defined on $Q$. Accordingly, consider a random map $\Theta \rightarrow l^{\infty }(Q)$, defined by $F\mapsto \Psi _n (F)[\cdot ]$. Then, the equation $\Psi _n (F)[\cdot ]=0$ is considered the estimating function that takes its value on $l^{\infty }(Q)$. It follows that the NPMLE is the Z-estimator that satisfies $\Psi _n (\hat{{F}})[\cdot ]=0$ (van der Vaart and Wellner 1996, p. 309). In the following, we assume that certain regularity conditions for the asymptotic theory for the Z-estimator hold, which include the asymptotic approximation condition, the Fréchet differentiability of the map, and the invertibility of the derivative map.

Then, one can write

$$\begin{aligned} 0=n^{1/2}\Psi _n (\hat{{F}})[h]=n^{1/2}\Psi _n (F)[h]+n^{1/2}\dot{\Psi }_F (\hat{{F}}-F)[h]+o_P (1), \end{aligned}$$

(5)

where $\dot{\Psi }_F (\hat{{F}}-F)[h]$ is the derivative of $\Psi _n (F)[h]$ at $F$ with direction $\hat{{F}}-F$. It follows from the form of $\Psi (F)[\cdot ]$ that

$$\begin{aligned} \dot{\Psi }_F (\hat{{F}}-F)[h]=\frac{d}{dt}\Psi \{\;\hat{{F}}+t(\hat{{F}}-F)\;\}[h]|_{t=0} =-\int {\sigma _F (x)[h]d(\hat{{F}}-F)(x)} . \end{aligned}$$

(6)

It follows from Eqs. (5) and (6) that the NPMLE satisfies the asymptotic linear expression

$$\begin{aligned}&\sqrt{n}\int {\sigma _F (x)[h]d(\hat{{F}}-F)(x)} \nonumber \\&\quad =\frac{1}{\sqrt{n}}\sum _{i=1}^n {\left[ {h(T_i )-\frac{\int {I(U_i \le s\le V_i )h(s)dF(s} )}{\int {I(U_i \le s\le V_i )dF(s} )}} \right] } +o_P (1), \end{aligned}$$

(7)

where the right-side converges weakly to a mean zero Gaussian process with the covariance structure

$$\begin{aligned}&E\left[ {h(T^{*})-\frac{\int {\mathbf{I}(U^{*}\le s\le V^{*})h(s)dF(s} )}{\int {\mathbf{I}(U^{*}\le s\le V^{*})dF(s} )}} \right] \left[ {{h}'(T^*)-\frac{\int {\mathbf{I}(U^{*}\le s\le V^{*}){h}'(s)dF(s} )}{\int {\mathbf{I}(U^{*}\le s\le V^{*})dF(s} )}} \right] \\&\quad =\int {\sigma _F (x)[h]{h}'(x)dF(x)} , \end{aligned}$$

for bounded functions $h$ and ${h}'$. The desired weak convergence of $\sqrt{n}(\;\hat{{F}}(t)-F(t)\;)$ is obtained by setting $h=\sigma _F^{-1} (w_t )$ in Eq. (7).

1.2 Appendix B2: Proof of $\sum _{j=1}^n {w_s (T_j )\hat{{\sigma }}_F^{-1} (w_t )(T_j )\hat{{f}}_j } =\mathbf{W}_s^\mathrm{T} \left\{ {\frac{i_n (\mathbf{f})}{n}} \right\} ^{-1}\mathbf{W}_t$

It follows that

$$\begin{aligned} \hat{{\sigma }}_F (T_j )[h]=\frac{1}{n}\sum _{i=1}^n {J_{ij} \left\{ {\frac{h_j }{\hat{{F}}_i }-\frac{1}{\hat{{F}}_i^2 }\sum _{k=1}^n {J_{ik} h_k \hat{{f}}_k } } \right\} }=\frac{1}{n}\left[ {\frac{h_j \hat{{f}}_j}{\hat{{f}}_j^2 }-\sum _{i=1}^n {\sum _{k=1}^n {\frac{J_{ij} J_{ik} }{\hat{{F}}_i^2 }h_k \hat{{f}}_k } } } \right] .\nonumber \\ \end{aligned}$$

(8)

Note that

$$\begin{aligned} J^{\mathrm{T}}\hbox {diag}\left( {\frac{1}{\mathbf{F}^{2}}} \right) J=\left[ {{\begin{array}{c@{\quad }c@{\quad }c} {\sum _{i=1}^n {\frac{J_{i1} J_{i1} }{F_i^2 }} }&{} \cdots &{} {\sum _{i=1}^n {\frac{J_{i1} J_{in} }{F_i^2 }} } \\ \vdots &{} \ddots &{} \vdots \\ {\sum _{i=1}^n {\frac{J_{in} J_{i1} }{F_i^2 }} }&{} \cdots &{} {\sum _{i=1}^n {\frac{J_{in} J_{in} }{F_i^2 }} } \\ \end{array} }} \right] . \end{aligned}$$

Hence, Eq. (8) with $h={h}'$ and $\sigma _F (x)[{h}']=w_t (x)=\mathbf{I}(x\le t)$ yield

$$\begin{aligned} \left[ {{\begin{array}{c} {w_t (T_1 )} \\ \vdots \\ {w_t (T_n )} \\ \end{array} }} \right]&= \frac{1}{n}\left[ {\left. {\left\{ {\hbox {diag}\left( {\frac{1}{{\hat{\mathbf{f}}}^{2}}} \right) -J^{\mathrm{T}}\hbox {diag}\left( {\frac{1}{{\hat{\mathbf{F}}}^{2}}} \right) J} \right\} } \right| _{\hat{{f}}_n =1-\mathbf{1}_{n-1}^\mathrm{T} {\hat{\mathbf{f}}}} } \right] \left[ {{\begin{array}{c} {h_1 \hat{{f}}_1 } \\ \vdots \\ {h_n \hat{{f}}_n } \\ \end{array} }} \right] \\&= \frac{1}{n}\left[ {\left. {\left\{ {\hbox {diag}\left( {\frac{1}{{\hat{\mathbf{f}}}^{2}}} \right) -J^{\mathrm{T}}\hbox {diag}\left( {\frac{1}{{\hat{\mathbf{F}}}^{2}}} \right) J} \right\} } \right| _{\hat{{f}}_n =1-\mathbf{1}_{n-1}^\mathrm{T} {\hat{\mathbf{f}}}} } \right] D^{\mathrm{T}}\left[ {{\begin{array}{c} {h_1 \hat{{f}}_1 } \\ \vdots \\ {h_{n-1} \hat{{f}}_{n-1} } \\ \end{array} }} \right] , \end{aligned}$$

where the last equation uses the constraint $\sum _{j=1}^n {h_j \hat{{f}}_j } =0$. Multiplying $D$ for both sides, and taking the inverse of the information matrix,

$$\begin{aligned} \left[ {{\begin{array}{c} {\hat{{\sigma }}_F^{-1} (w_t )(T_1 )\hat{{f}}_1 } \\ \vdots \\ {\hat{{\sigma }}_F^{-1} (w_t )(T_{n-1} )\hat{{f}}_{n-1} } \\ \end{array} }} \right] =\left\{ {\frac{i_n ({\hat{\mathbf{f}}})}{n}} \right\} ^{-1}\left[ {{\begin{array}{c} {w_t (T_1 )-w_t (T_n )} \\ \vdots \\ {w_t (T_1 )-w_t (T_n )} \\ \end{array} }} \right] . \end{aligned}$$

It follows that

$$\begin{aligned}&\sum _{j=1}^n {w_s (T_j )\hat{{\sigma }}_F^{-1} (w_t )(T_j )\hat{{f}}_j } =\sum _{j=1}^{n-1} {\{\;w_s (T_j )-w_s (T_n )\;\}\hat{{\sigma }}_F^{-1} (w_t )(T_j )\hat{{f}}_j } \\&\quad =\left[ {{\begin{array}{lll} {w_s (T_1 )-w_s (T_n )}&{} \cdots &{} {w_s (T_{n-1} )-w_s (T_n )\;} \\ \end{array} }} \right] \left\{ {\frac{i_n ({\hat{\mathbf{f}}})}{n}} \right\} ^{-1}\left[ {{\begin{array}{c} {w_t (T_1 )-w_t (T_n )} \\ \vdots \\ {w_t (T_1 )-w_t (T_n )} \\ \end{array} }} \right] \\&\quad =\mathbf{W}_s^\mathrm{T} \left\{ {\frac{i_n ({\hat{\mathbf{f}}})}{n}} \right\} ^{-1}\mathbf{W}_t . \end{aligned}$$