On estimation of a density function in multiplicative censoring

Abstract

This paper considers non-parametric density estimation in the context of multiplicative censoring. A new estimator for the density function is proposed and consistency of the proposed estimator is investigated. Simulations are drawn to illustrate the results and to show how the estimator behaves for finite samples.

Appendices

Appendix A: Proofs of the main results

Proof of Theorem 1

Under ${\mathbf {K(1)}},$ $\mathbf {H(2)}$ and according to Lemma 7.1 we have

$$\begin{aligned} \lim _{m\rightarrow \infty }E\left( \check{g}_{m}(x)\right) =g(x). \end{aligned}$$

(6.1)

Let

$$\begin{aligned} q^{*}_{n}(z)=\frac{1}{nh_{n}}\sum _{i=1}^{n}K_{2}\left( \frac{z-T_{i}}{h_{n}}\right) , \end{aligned}$$

be an ordinary kernel density estimator function of $q$, that is defiend by (2.1), and based on the sample of $(T_1,\ldots ,T_n)$ where $T_i=-\ln Z_i.$ An interchange of expectation and integration, justified by Fubini’s Theorem and $\mathbf { K(3)},$ show that

$$\begin{aligned} E\left( K_*\left( \frac{(z -W_{1})}{h_{n}}\right) \left| \right. T_{1}\right) =K_{2}\left( \frac{z-T_{1}}{h_{n}}\right) . \end{aligned}$$

(6.2)

Then it follows from (6.2) that

$$\begin{aligned} E\left( q_{n}(z)\right) =E\left( q^{*}_{n}(z)\right) . \end{aligned}$$

Thus $q_{n}$ has the same bias as the ordinary kernel density estimator $q^{*}_{n}$. ${\mathbf {K(1)}}, {{\mathbf {H(1)}}}$ and Lemma 7.1 follow

$$\begin{aligned} \lim _{n\rightarrow \infty }E\left( q^{*}_{n}(z )\right) =q(z), \end{aligned}$$

hence

$$\begin{aligned} \lim _{n\rightarrow \infty }E\left( \tilde{g}_{n}(x)\right)&= \frac{1}{x}\lim _{n\rightarrow \infty }E\left( q^{*}_{n}(-\ln x)\right) \nonumber \\&= \frac{1}{x} q(-\ln x)\nonumber \\&= g(x). \end{aligned}$$

(6.3)

Therefore from (6.1), (6.3) and Assumption ${\mathbf {P}}$, we have

$$\begin{aligned} \lim _{k \rightarrow \infty }E\left( \hat{g}_k(x)\right)&= p\lim _{m\rightarrow \infty }E\left( \check{g}_{m}(x)\right) + (1-p) \lim _{n\rightarrow \infty }E\left( \tilde{g}_{n}(x)\right) \nonumber \\&= p g(x)+(1-p) g(x) \nonumber \\&= g(x) \end{aligned}$$

(6.4)

Assumption ${{\mathbf {H(2)}}}$ and Lemma 7.2 follow that

$$\begin{aligned} \lim _{m \rightarrow \infty }var(\check{g}_{m}(x))=0. \end{aligned}$$

(6.5)

By proof of Theorem 2.1 of Stefanski and Carroll (1990), ${\mathbf {K(2)-K(4)}}$ and ${\mathbf {H(1)}}$, we have

$$\begin{aligned} \lim _{ n\rightarrow \infty }var(\tilde{g}_{n}(x))=0. \end{aligned}$$

(6.6)

It can be easily seen that

$$\begin{aligned} var(\hat{g}_k(x))={\hat{p}}^2 var(\check{g}_m(x))+(1-\hat{p} )^2 var(\tilde{g}_n(x)), \end{aligned}$$

(6.7)

so by combining (6.5), (6.6) we have

$$\begin{aligned} \lim _{k\rightarrow \infty }var(\hat{g}_k(x))=0. \end{aligned}$$

(6.8)

Finally, (6.4) and (6.8) complete the proof. $\square$

Proof of Theorem 2

It is easy to see that

$$\begin{aligned} \sup _{\epsilon <x<\tau -\epsilon }\left| \hat{g}_k(x)-g(x)\right|&\le \hat{p} \sup _{\epsilon <x<\tau -\epsilon }\left| \check{g}_{m}(x)-g(x)\right| \nonumber \\&+ (1-\hat{p}) \sup _{\epsilon <x<\tau -\epsilon }\left| \tilde{g}_{n}(x)-g(x)\right| .\nonumber \\&:= I_{1}+I_{2}. \end{aligned}$$

(6.9)

From G(1), G(2), H(3), K(1) and Lemma 7.3 we have

$$\begin{aligned} I_{1}=O\left( \sqrt{\frac{\ln \frac{1}{h_{m}} }{mh_{m}}}\,\,\right) +O\left( h^{2}_{m}\right) ,\quad a.s. \end{aligned}$$

(6.10)

For $I_{2}$ first we have

$$\begin{aligned} \sup _{\epsilon <x<\tau -\epsilon }\left| \tilde{g}_{n}(x)-g(x)\right|&\le \frac{1}{\epsilon }\sup _{\epsilon <x<\tau -\epsilon }\left| q_{n}(-\ln x)-E(q_{n}(-\ln x))\right| \nonumber \\&+\frac{1}{\epsilon }\sup _{\epsilon <x<\tau -\epsilon }\left| E(q_{n}(-\ln x))-q(-\ln x))\right| \nonumber \\&:= I_{3}+I_{4}. \end{aligned}$$

(6.11)

Let $W_i$ has density function $f^*$ with characteristic function $\Psi _{f^{*}}.$ Since $\Psi _{K_{2}}$ is vanished out of $[-1,1]$, we have

$$\begin{aligned} q_{n}(x)-E\left( q_{n}(x)\right) =\frac{1}{2\pi }\int \limits _{-\frac{1}{h_{n}}}^{\frac{1}{h_{n}}}e^{-itx}\left( \hat{\Psi }_{f^*} (t)-\Psi _{f^{*}}(t)\right) \Psi _{K_{2}}(h_{n}t)\left( 1-it\right) dt,\nonumber \\ \end{aligned}$$

(6.12)

where

$$\begin{aligned} \hat{\Psi }_{f^{*}}(t)=\frac{1}{n}\sum _{j=1}^{n}e^{itW_{j}}, \end{aligned}$$

hence

$$\begin{aligned} I_{3}&\le \frac{1}{2\pi }\sup _{|t|\le \frac{1}{h_{n}}}\left| \hat{\Psi }_{f^*}(t)-\Psi _{f^{*}}(t)\right| \int \limits _{-1}^{+1} \left| \Psi _{K_{2}}(u)\left( \frac{1}{h_{n}}-\frac{i u}{{h}^{2}_{n}}\right) \right| du\\&= \frac{1}{2\pi }\Delta _{n}\int \limits _{-1}^{+1} \left| \Psi _{K_{2}}(u)\left( \frac{1}{h_{n}}-\frac{i u}{{h}^{2}_{n}}\right) \right| du, \end{aligned}$$

where

$$\begin{aligned} \Delta _{n}=\sup _{|t|\le \frac{1}{h_{n}}}\left| \hat{\Psi }_{f^*}(t)-\Psi _{f^{*}}(t)\right| . \end{aligned}$$

Lemma 7.4 implies that

$$\begin{aligned} \Delta _{n}=O\left( \sqrt{\frac{\ln n}{n}}\right) \quad a.s., \end{aligned}$$

hence using Assumption ${\mathbf {K(3)}}$ we have

$$\begin{aligned} I_{3}=O\left( \sqrt{\frac{\ln n}{n h^{4}_{n}}}\right) \quad a.s. \end{aligned}$$

(6.13)

Using $\mathbf G(1)$ and $\mathbf K(1)$,

$$\begin{aligned} I_{4}=O\left( h^{2}_n\right) \quad a.s. \end{aligned}$$

(6.14)

Now (6.11), (6.13) and (6.14) yield

$$\begin{aligned} I_{2}=O\left( \sqrt{\frac{\ln n}{nh^{4}_{n}}}\right) +O\left( h^{2}_{n}\right) . \end{aligned}$$

(6.15)

(6.9), (6.10), and (6.15) complete the proof. $\square$

Appendix B

In this section, we consider the kernel density estimator $f_n$ of a univariate density $f$ introduced by Roenblatt (1956),

$$\begin{aligned} f_{n}(x)=\frac{1}{na_{n}}\sum _{j=1}^{n}K\left( \frac{x-\xi _{j}}{a_{n}}\right) , \end{aligned}$$

(7.1)

where $\xi _{1},\xi _{2},\ldots ,\xi _{n}$ are independent observations from distribution $F$ with density function $f$. $K,$ $a_n$ and $F_n$ are kernel function, a bandwidth and empirical distribution function, respectively. Also, the characteristic function of $f$ and its empirical are denoted by

$$\begin{aligned} \Psi _{f}(t)=\int \limits _{\mathbb {R}}e^{itx}f(x) dx,\quad t \in \mathbb {R} \end{aligned}$$

and

$$\begin{aligned} \Psi _{n}(t)=\int \limits _{\mathbb {R}}e^{itx}dF_{n}(t),\quad t \in \mathbb {R}. \end{aligned}$$

Lemma 7.1

(Corollary 1A, Parzen (1962)). The estimator defined by (7.1) is asymptotically unbiased at all points $x$ at which the probability density function is continuous if the constants $a_{n}$ satisfy $\lim _{n \longrightarrow \infty }a_{n}=0$ and if the function $K(\cdot )$ satisfies

$$\begin{aligned} \sup _{-\infty <y <\infty } |K(y)|<\infty , \end{aligned}$$

(7.2)

$$\begin{aligned} \int \limits _{-\infty }^{\infty }|K(y)|dy <\infty , \end{aligned}$$

(7.3)

$$\begin{aligned} \lim _{y \longrightarrow \infty } |y K(y)|=0, \end{aligned}$$

(7.4)

and

$$\begin{aligned} \int \limits _{-\infty }^{\infty }K(y)dy=1. \end{aligned}$$

(7.5)

$\square$

Lemma 7.2

(Theorem 2A, Parzen (1962)) The estimates defined by (7.1) have variances satisfying

$$\begin{aligned} \lim _{n \longrightarrow \infty }n a_{n} Var (f_{n}(x))=f(x)\int \limits _{-\infty }^{\infty }K^{2}(y)dy \end{aligned}$$

at all points $x$ of continuity of $f(x),$ if $\lim _{n \longrightarrow \infty } a_{n}=0$ and the function $K$ satisfies (7.2) and (7.3). $\square$

Lemma 7.3

(Theorem 1.2., Stute (1982)) Let $K$ be of bounded variation and suppose that $K(x)=0$ outside some finite interval $[r,s)$. If

$$\begin{aligned} \sup _{x,y \in J, x\ne y}\frac{|F(x)-F(y)|}{|x-y|}<M<\infty , \end{aligned}$$

with $J=(a, b),$ then for each $\epsilon >0$ and every bandsequence $(a_{n})$, where $\lim _{n\rightarrow \infty }$ $a_{n}=0$ and    $\lim _{n \rightarrow \infty }\frac{\ln ({a^{-1}_{n}})}{na_{n}}=0,$ we have

$$\begin{aligned} \limsup _{n\longrightarrow \infty }\sqrt{\frac{n a_{n}}{2\ln a^{-1}_{n}}}\sup _{t \in J_{\epsilon }}\left| f_{n}(t)-E(f_{n}(t))\right| =C\quad a.s., \end{aligned}$$

where $J_{\epsilon }=(a+\epsilon , b-\epsilon )$ and $C$ is a constant. $\square$

Lemma 7.4

(Example (1), Csörgő (1985)) Suppose that $P\left( |\xi |>x\right) \le Lx^{-\alpha }$ for all large enough $x,$ where $L$ and $\alpha$ are arbitrary positive constants. Then for any $A>0$

$$\begin{aligned} \limsup _{n\longrightarrow \infty }\sqrt{\frac{n}{\ln n}}\Delta _{n}(n^{A})\le C), \end{aligned}$$

where $C$ is a positive constant and

$$\begin{aligned} \Delta _{n}(T)=\sup _{|t|<T}\left| \Psi _{n}(t)-\Psi _{f}(t)\right| \end{aligned}$$

for any extended number $0<T\le \infty$. $\square$