Prediction model-based kernel density estimation when group membership is subject to missing

Abstract

The density function is a fundamental concept in data analysis. When a population consists of heterogeneous subjects, it is often of great interest to estimate the density functions of the subpopulations. Nonparametric methods such as kernel smoothing estimates may be applied to each subpopulation to estimate the density functions if there are no missing values. In situations where the membership for a subpopulation is missing, kernel smoothing estimates using only subjects with membership available are valid only under missing complete at random (MCAR). In this paper, we propose new kernel smoothing methods for density function estimates by applying prediction models of the membership under the missing at random (MAR) assumption. The asymptotic properties of the new estimates are developed, and simulation studies and a real study in mental health are used to illustrate the performance of the new estimates.

Appendix

In this appendix, we give proofs for Theorem 1–4.

Proof of Theorem 1

We first show the asymptotic distribution of $\widetilde{f}_{{\text {MS}}}(t;h)$ in Theorem 1 (a). Let $u_{i}=D_{i}R_{i}+d_{i}(1-R_{i})$ and $f_{h}(t)=E\left[ K_{h}(t-T_{i})\mid D_{i}=1\right]$, as defined in (3.3), based on (3.1), we have

$$\begin{aligned}&\sqrt{n}\left[ \widetilde{f}_{MS}(t;h)-f_{h}(t)\right] =\sqrt{n}\left[ \frac{\frac{1}{n}\sum {}_{i=1}^{n}{u}_{i}K_{h}(t-T_{i})}{\frac{1}{n}\sum {}_{i=1}^{n}{u}_{i}}-f_{h}(t)\right] \nonumber \\&=\sqrt{n}\left[ \frac{\frac{1}{n}\sum {}_{i=1}^{n}{u}_{i}K_{h}(t-T_{i} )}{\frac{1}{n}\sum {}_{i=1}^{n}{u}_{i}}-\frac{\frac{1}{n}\sum {}_{i=1}^{n} {u}_{i}K_{h}(t-T_{i})}{p}+\frac{\frac{1}{n}\sum {}_{i=1}^{n}{u}_{i} K_{h}(t-T_{i})}{p}-f_{h}(t)\right] \nonumber \\&=\frac{1}{p}\frac{1}{\sqrt{n}}\left[ \frac{\sum {}_{i=1}^{n}{u}_{i} K_{h}(t-T_{i})}{\sum {}_{i=1}^{n}{u}_{i}}\sum \limits _{i=1}^{n}\left( p-{u} _{i}\right) +\sum \limits _{i=1}^{n}\left[ {u}_{i}K_{h}(t-T_{i})-pf_{h}(t)\right] \right] . \end{aligned}$$

(7.1)

For any given h,  at point t,  as $n\rightarrow \infty$ ,  by the Weak Law of Large Numbers (WLLN), we have

$$\begin{aligned} \frac{\sum {}_{i=1}^{n}{u}_{i}K_{h}(t-T_{i})}{\sum {}_{i=1}^{n}{u}_{i} }\rightarrow f_{h}(t). \end{aligned}$$

(7.2)

By the Central Limit Theory (CLT), we have

$$\begin{aligned} \frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}\left( p-{u}_{i}\right) \rightarrow N(0,Var({u}_{i})) \end{aligned}$$

(7.3)

and

$$\begin{aligned} \frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}\left[ {u}_{i}K_{h}(t-T_{i})-pf_{h} (t)\right] \rightarrow N(0,Var({u}_{i}K_{h}(t-T_{i}))). \end{aligned}$$

(7.4)

By applying Slutsky’s theorem and with the consideration of the correlation between (7.3) and (7.4), we have

$$\begin{aligned} \sqrt{n}\left[ \widetilde{f}_{{\text {MS}}}(t;h)-f_{h}(t)\right] \rightarrow N(0,\sigma _{1}^{2}), \end{aligned}$$

where $\sigma _{1}^{2}=\frac{1}{p^{2}}Var\left( {u}_{i}K_{h}(t-T_{i})-{u} _{i}f_{h}(t)\right)$. The asymptotic distribution of $\widetilde{f} _{{\text {MS}}}(t;h)$ in Theorem 1(a) has been proved.

Replacing $u_{i}=D_{i}R_{i}+d_{i}(1-R_{i})\$ by $u_{i}=d_{i}$, and applying a similar argument for the proof of Theorem 1 (a), we can prove Theorem 1 (b). $\square$

Proof of Theorem 2

Let f(t) be the density function for the diseased population, and let $z_{i}=\left( T_{i}-t\right) /h,$ then

$$\begin{aligned} Bias\left[ \widetilde{f}_{{\text {MS}}}(t;h)\right]&=E\left[ \widetilde{f} _{{\text {MS}}}(t;h)-f(t)\right] \nonumber \\&=E\left[ \widetilde{f}_{{\text {MS}}}(t;h)-f_{h}(t)+f_{h}(t)-f(t)\right] \nonumber \\&=E\left[ \widetilde{f}_{{\text {MS}}}(t;h)-f_{h}(t)\right] +E\left[ f_{h} (t)-f(t)\right] . \end{aligned}$$

(7.5)

Based on Theorem 1, we have

$$\begin{aligned} E\left[ \widetilde{f}_{{\text {MS}}}(t;h)-f_{h}(t)\right] =0\text { as } n\rightarrow \infty . \end{aligned}$$

Since both $f_{h}(t)$ and f(t) are defined for the diseased population, we have

$$\begin{aligned} f_{h}(t)-f(t)&=\int K_{h}(T_{i}-t)f(T_{i})dT_{i}-f(t)\nonumber \\&=\int \frac{1}{h}K(z_{i})f(t+hz_{i})hdz_{i}-f(t)\nonumber \\&=\int K(z_{i})\left[ f(t)+f^{\prime }(t)hz_{i}+\frac{1}{2}f^{\prime \prime }(t)h^{2}z_{i}^{2}+o(h^{2})\right] dz_{i}-f(t)\nonumber \\&=\frac{1}{2}h^{2}\mu _{2}(K)f^{\prime \prime }(t)+o(h^{2}). \end{aligned}$$

(7.6)

Combining (7.5) and (7.6), we have $Bias\left[ \widetilde{f}_{{\text {MS}}}(t;h)\right] =\frac{1}{2}h^{2}\mu _{2}(K)f^{\prime \prime } (t)+o(h^{2}).$

Next, we derive the variance of $\widetilde{f}_{MS}(t;h).$ Let $w(t)=E\big ( \pi _{i}d_{i}+d_{i}^{2}(1-\pi _{i})\mid T_{i}=t\big )$, $z_{i}=\left( T_{i}-t\right) /h$ and g(t) be the population density function of T. Based on Theorem 1, the asymptotic variance for $\widetilde{f} _{{\text {MS}}}(t)$ is

$$\begin{aligned} Var(\widetilde{f}_{{\text {MS}}}(t))&=\frac{1}{np^{2}}Var\left[ \left( K_{h} (t-T_{i})-f_{h}(t)\right) \left( D_{i}R_{i}+d_{i}(1-R_{i})\right) \right] \\&=\frac{1}{np^{2}}E\left[ \left( K_{h}(t-T_{i})-f_{h}(t)\right) \left( D_{i}R_{i}+d_{i}(1-R_{i})\right) \right] ^{2}\\&=\frac{1}{np^{2}}E\left[ \left( K_{h}(t-T_{i})-f_{h}(t)\right) ^{2}\left( D_{i}R_{i}+d_{i}^{2}(1-R_{i})\right) \right] \\&=\frac{1}{np^{2}}E\left\{ E\left[ \left( K_{h}(t-T_{i})-f_{h}(t)\right) ^{2}\left( D_{i}R_{i}+d_{i}^{2}(1-R_{i})\right) \mid T_{i}=t,x_{i}=x\right] \right\} \\&=\frac{1}{np^{2}}E\left[ \left( K_{h}(t-T_{i})-f_{h}(t)\right) ^{2}\left( \pi _{i}d_{i}+d_{i}^{2}(1-\pi _{i})\right) \right] \\&=\frac{1}{np^{2}}E\left\{ E\left[ \left( K_{h}(t-T_{i})-f_{h}(t)\right) ^{2}\left( \pi _{i}d_{i}+d_{i}^{2}(1-\pi _{i})\right) \mid T_{i}=t\right] \right\} \\&=\frac{1}{np^{2}}E\left[ \left( K_{h}(t-T_{i})-f_{h}(t)\right) ^{2}w(T_{i})\right] ,\\&=\frac{1}{np^{2}}\int \left( \frac{1}{h^{2}}K^{2}(z_{i})+o\left( \frac{1}{h^{2} }\right) \right) w(t+hz_{i})g(t+hz_{i})hdz_{i}\\&=\frac{g(t)w(t)}{nhp^{2}}\int K^{2}(z_{i})dz_{i}+o\left( \frac{1}{nh}\right) \\&=\frac{g(t)w(t)}{nhp^{2}}\int K^{2}(z_{i})dz_{i}+o\left( \frac{1}{nh}\right) . \end{aligned}$$

Hence, the asymptotic variance of $\widetilde{f}_{{\text {MS}}}(t)$ is

$$\begin{aligned} \frac{g(t)R(K)}{nhp^{2}}E\left[ d_{i}^{2}+\pi _{i}(d_{i}-d_{i}^{2})\mid T_{i}=t\right] +o\left( \frac{1}{nh}\right) ,\text { with }R(K)=\int K^{2}(t)dt. \end{aligned}$$

Let $w(t)=E\left( d_{i}^{2}\mid T_{i}=t\right) .$ Based on Theorem 1, with a similar argument, the asymptotic variance for $\widetilde{f}_{BG}(t)$ can be derived as below:

$$\begin{aligned}&\frac{1}{np^{2}}Var\left[ \left( K_{h}(t-T_{i})-f_{h}(t)\right) d_{i}\right] \\&\quad =\frac{1}{np^{2}}E\left[ \left( K_{h}(t-T_{i})-f_{h}(t)\right) ^{2} d_{i}^{2}\right] \\&\quad =\frac{1}{np^{2}}E\left[ \left( K_{h}(t-T_{i})-f_{h}(t)\right) ^{2}w(T_{i})\right] \\&\quad =\frac{g(t)R(K)}{np^{2}}E\left[ d_{i}^{2}\mid T_{i}=t\right] +o\left( \frac{1}{nh}\right) . \end{aligned}$$

$\square$

Proof of Theorem 3

We first show the asymptotic distribution of in Theorem 3 (a). Suppose we have a prediction model (3.8) and the parameters are estimated from (3.9). Let $\widehat{{\beta }}$ be the estimate of $\beta$. Based on the Taylor expansion of (3.9) at $\beta$, we have

$$\begin{aligned} \widehat{{\beta }}-{\beta }=\frac{1}{n}\mathbf {I}^{-1}\sum \limits _{i=1}^{n}\Psi i+o\left( \frac{1}{n}\right) , \end{aligned}$$

where $\mathbf {I=-}E[\frac{\partial \Psi _{i}}{\partial {\beta }^{T}}].$ If $\widehat{{\beta }}$ are estimated from the score equation, $\mathbf {I}$ is the Fisher information matrix.

Since

the asymptotic distribution of $\sqrt{n}\left[ \widetilde{f}_{{\text {MS}}} (t)-f_{h}(t)\right]$ is already given in Theorem 1(a). We will focus on deriving the asymptotic distribution of $\sqrt{n}\left[ \widehat{f}_{{\text {MS}}}(t)-\widetilde{f}_{{\text {MS}}}(t)\right]$.

Let $\widehat{u}_{i}=D_{i}R_{i}+\widehat{d}_{i}(1-R_{i})=D_{i}R_{i} +g(x_{i};\widehat{{\beta }})(1-R_{i}).$ Based on (3.10) and (3.1), by applying WLLE, we have

The second term

$$\begin{aligned} II= & {} \frac{1}{\sqrt{n}p}\sum \limits _{i=1}^{n}K_{h}(t-T_{i})\left( \widehat{u}_{i}-{u}_{i}\right) \\= & {} \frac{1}{\sqrt{n}p}\sum \limits _{i=1}^{n}K_{h}(t-T_{i})\left[ \frac{\partial u_{i}}{\partial {\beta }^{T}}(\widehat{{\beta }}-\mathbf {\beta })+o(\frac{1}{n})\right] \\= & {} \frac{1}{n}\left[ \sum \limits _{i=1}^{n}K_{h}(t-T_{i})\frac{\partial u_{i} }{\partial {\beta }^{T}}\right] \frac{\sqrt{n}}{p}(\widehat{{\beta }}-{\beta })+o(1)\\= & {} E\left( K_{h}(t-T_{i})\frac{\partial u_{i}}{\partial {\beta }^{T} }\right) \frac{\mathbf {I}^{-1}}{\sqrt{n}p}\sum \limits _{i=1}^{n}\Psi i+o(1), \end{aligned}$$

where $\mathbf {I}=-E[\frac{\partial \Psi _{i}}{\partial {\beta }^{T}}]$.

For the first term, by Slutsky’s theorem,

$$\begin{aligned} \frac{1}{\frac{1}{n}\Sigma _{i=1}^{n}\widehat{u}_{i}}-\frac{1}{\frac{1}{n}\Sigma _{i=1}^{n}u_{i}}=-\frac{1}{p^{2}}E\left[ \frac{\partial u_{i} }{\partial {\beta }^{T}}({\beta })\right] (\widehat{{\beta }}-{\beta })+o(1). \end{aligned}$$

Thus,

$$\begin{aligned} I&=\frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}\frac{1}{n}\Sigma _{i=1}^{n} \widehat{u}_{i}K_{h}(t-T_{i})\left[ \frac{1}{\frac{1}{n}\Sigma _{i=1} ^{n}\widehat{u}_{i}}-\frac{1}{\frac{1}{n}\Sigma _{i=1}^{n}u_{i}}\right] \\&=\frac{1}{\sqrt{n}}\sum \limits _{i=1}^{n}\left[ (u_{i}+\frac{\partial u_{i} }{\partial {\beta }^{T}}(\widehat{{\beta }}-{\beta } ))K_{h}(t-T_{i})\right] \\&\quad \times \left( -\frac{1}{p^{2}}E\left[ \frac{\partial u_{i}}{\partial {\beta }^{T}}({\beta })\right] (\widehat{{\beta }}-{\beta })\right) +o(1)\\&=-\frac{f_{h}(t)}{p}E\left[ \frac{\partial u_{i}}{\partial \mathbf {\beta }^{T}}({\beta })\right] \sqrt{n}(\widehat{{\beta }} -{\beta })+o(1)\\&=-\frac{f_{h}(t)}{p}E\left[ \frac{\partial u_{i}}{\partial \mathbf {\beta }^{T}}({\beta })\right] \frac{1}{\sqrt{n}}\mathbf {I}^{-1}\sum \limits _{i=1}^{n}\Psi i+o(1). \end{aligned}$$

It follows that

where $\sigma _{3}^{2}=\frac{1}{p^{2}}Var\big ( {u}_{i}K_{h}(t-T_{i} )-f_{h}(t){u}_{i}+\big ( E\left( K_{h}(t-T_{i})\frac{\partial u_{i} }{\partial {\beta }^{T}}\right) -f_{h}(t)E\big ( \frac{\partial u_{i} }{\partial {\beta }^{T}}\big ) \big ) \mathbf {I}^{-1}\Psi i\big )$. Let $\mathbf {c}=E\left[ K_{h}(t-T_{i})(1-R_{i})\frac{\partial g_{i} }{\partial {\beta }^{T}}({\beta })\right]$ and $\mathbf {d} =E\left[ \!(1-R_{i})\frac{\partial g_{i}}{\partial {\beta }^{T} }({\beta })\!\right] .$ Since $\frac{\partial u_{i}}{\partial {\beta }^{T}}=(1-R_{i})\frac{\partial g_{i}}{\partial {\beta }^{T} },$ we have

$$\begin{aligned} \sigma _{3}^{2}=\frac{1}{p^{2}}Var\left\{ u_{i}\left[ K_{h}(t-T_{i} )-f_{h}(t)\right] +\left( \mathbf {c}-f_{h}(t)\mathbf {d}\right) \mathbf {e}^{-1}\Psi _{i}\right\} . \end{aligned}$$

Theorem 3(b) can be proved similarly by replacing $u_{i}$ by $d_{i}$ and $\widehat{u}_{i}$ by $\widehat{d}_{i}$ in the above arguments. $\square$

Proof of Theorem 4

Based on Theorem 3, the asymptotic bias for both and is $f_{h}(t)-f(t).$ Thus, the bias follows from the proof of Theorem 2.

The proof for the asymptotic variance also follows similarly to that of Theorem 2:

Let $w(t)=E\left[ \pi _{i}d_{i}+d_{i}^{2}(1-\pi _{i})\mid T_{i}=t\right] ,$ $w_{1}(t)=E\big \{ u_{i}\left( \mathbf {c}-f_{h}(t)\mathbf {d}\right) \mathbf {e}^{-1}\Psi _{i}\mid T_{i}=t\big \}$, and $w_{2}(T_{i})=E\left[ (\left( \mathbf {c}-f_{h}(t)\mathbf {d}\right) \mathbf {e}^{-1}\Psi _{i} )^{2}\right] .$ Based on Theorem 3, the asymptotic variance for can be derived as below:

$$\begin{aligned}&\frac{1}{np^{2}}Var\left[ u_{i}\left( K_{h}(t-T_{i})-f_{h}(t)\right) +\left( \mathbf {c}-f_{h}(t)\mathbf {d}\right) \mathbf {e}^{-1}\Psi _{i}\right] \\&=\frac{1}{np^{2}}E\left[ \left( K_{h}(t-T_{i})-f_{h}(t)\right) \left( D_{i}R_{i}+d_{i}(1-R_{i})\right) +\left( \mathbf {c}-f_{h}(t)\mathbf {d} \right) \mathbf {e}^{-1}\Psi _{i}\right] ^{2}\\&=\frac{1}{np^{2}}E\left[ \left( K_{h}(t-T_{i})-f_{h}(t)\right) ^{2}\left( D_{i}R_{i}+d_{i}^{2}(1-R_{i})\right) \right. \\&\quad \left. +2\left( K_{h}(t-T_{i})-f_{h}(t)\right) \left( D_{i}R_{i} +d_{i}(1-R_{i})\right) \left( \mathbf {c}-f_{h}(t)\mathbf {d}\right) \mathbf {e}^{-1}\Psi _{i}\right. \\&\quad \left. +(\left( \mathbf {c}-f_{h}(t)\mathbf {d}\right) \mathbf {e}^{-1}\Psi _{i})^{2}\right] \\&=\frac{1}{np^{2}}E\left[ \left( K_{h}(t-T_{i})-f_{h}(t)\right) ^{2}w(T_{i})+2\left( K_{h}(t-T_{i})-f_{h}(t)\right) w_{1}(T_{i})+w_{2} (T_{i})\right] \\&=\frac{1}{np^{2}}\int \left( \frac{1}{h^{2}}K^{2}(z_{i})\right) w(t+hz_{i})g(t+hz_{i})hdz_{i}+o\left( \frac{1}{nh}\right) \\&=\frac{g(t)w(t)}{nhp^{2}}\int K^{2}(z_{i})dz_{i}+o\left( \frac{1}{nh}\right) . \end{aligned}$$

Hence, the asymptotic variance of is

$$\begin{aligned} \frac{g(t)R(K)}{nhp^{2}}E\left[ d_{i}^{2}+\pi _{i}(d_{i}-d_{i}^{2})\mid T_{i}=t\right] +o\left( \frac{1}{nh}\right) ,\text { where }R(K)=\int K^{2}(t)dt. \end{aligned}$$

Similarly, we can prove the asymptotic variance of in (3.16). $\square$