Multiplicative bias correction for discrete kernels

Abstract

In this paper, we prove that two multiplicative bias correction techniques (MBC) can be applied for discrete kernels in the context of probability mass function estimation. First, some properties of the MBC discrete kernel estimators (bias, variance and mean integrated squared error) are investigated. Second, the popular cross-validation technique is adapted for bandwidth selection. Finally, a simulation study and a real data application for discrete data illustrate the performance of the MBC estimators based on dirac discrete uniform and triangular discrete kernels.

Appendix

We present a sketch of proofs of Theorems 1 and 2. We provide the proofs when the discrete triangular kernel is used. The proofs using the other kernels can be given similarly.

1.1 Sketch of the proof of Theorem 1

1.1.1 Bias

First, note that $\mathbb {E}\left( \widehat{f}_{DT,h}(x)\right) =\mathbb {E}(f(\mathcal {T}))$, where the random variable $\mathcal {T}\sim DT(a;x,h)$. By using a fourth-order discrete Taylor expansion around $\mathcal {T}=x$ for

$$\begin{aligned} I_{h}(x)=\mathbb {E}(\widehat{f}_{DT,h})=\sum K_{DT,h}(y)f(y)=\mathbb {E}(f(\mathcal {T})), \end{aligned}$$

we have

$$\begin{aligned} I_{h}(x)=f(x)+\sum \limits _{j=1}^{4}\frac{f^{(j)}}{j!} \mathbb {E}(\mathcal {T}-x)^{j}+o(\mathbb {E}(\mathcal {T}-x)^{4}). \end{aligned}$$

By using the property of the discrete triangular random variable and a Taylor expansion around $h=0$,

$$\begin{aligned} \mathbb {E}(\mathcal {T}-x)= & {} 0,\\ \mathbb {E}(\mathcal {T}-x)^{2}= & {} \left\{ \log (a+1)S(a)-2\sum \limits _{k=1}^{a}k^{2}\log (k)\right\} h \\&+\left\{ \frac{\log ^{2}(a+1)}{2}S(a)-\sum \limits _{k=1}^{a}k^{2}\log ^{2} (k)\right\} h^{2}+o(h^{2}),\\ \mathbb {E}(\mathcal {T}-x)^{3}= & {} 0,\\ \mathbb {E}(\mathcal {T}-x)^{4}= & {} \left\{ \log (a+1)R(a)-2\sum \limits _{k=1}^{a}k^{4}\log (k)\right\} h \\&+\left\{ \frac{\log ^{2}(a+1)}{2}R(a)-\sum \limits _{k=1}^{a}k^{4}\log ^{2} (k)\right\} h^{2}+o(h^{2}), \end{aligned}$$

where

$$\begin{aligned} R(a)=\frac{2}{5}a^{5}+a^{4}+\frac{2}{3}a^{3}-\frac{1}{15}a. \end{aligned}$$

The Taylor expansion of $I_{h}(x)$ around $h=0$ is then given by

$$\begin{aligned} I_{h}(x)=f(x)\left\{ 1+\frac{l_{1}(x,f)}{f(x)}h+\frac{l_{2}(x,f)}{f(x)}h^{2} +o(h^{2})\right\} , \end{aligned}$$

where

$$\begin{aligned} l_{1}(x,f)&=\frac{f^{(2)}(x)}{2}\left( \log (a+1)S(a)-2\sum \limits _{k=1}^{a} k^{2}\log (k)\right) \\&\quad + \frac{f^{(4)}(x)}{24}\left( \log (a+1)R(a)-2\sum \limits _{k=1}^{a} k^{4}\log (k)\right) \end{aligned}$$

and

$$\begin{aligned} l_{2}(x,f)&=\frac{f^{(2)}(x)}{2}\left( \frac{\log ^{2}(a+1)}{2}S(a) -\sum \limits _{k=1}^{a}k^{2}\log ^{2}(k)\right) \\&\quad + \frac{f^{(4)}(x)}{24}\left( \frac{\log ^{2}(a+1)}{2}R(a) -\sum \limits _{k=1}^{a}k^{4}\log ^{2}(k)\right) . \end{aligned}$$

Similarly, $I_{h/c}(x)=\mathbb {E}\left( \widehat{f}_{DT,h/c}(x)\right)$ can be approximated by

$$\begin{aligned} I_{h/c}(x)=f(x)\left\{ 1+\frac{1}{c}\frac{l_{1}(x,f)}{f(x)}h +\frac{1}{c^{2}}\frac{l_{2}(x,f)}{f(x)}h^{2}+o(h^{2})\right\} . \end{aligned}$$

Now, we define

$$\begin{aligned} \widehat{f}_{DT,h}(x)=I_{h}(x)+Z \end{aligned}$$

and

$$\begin{aligned} \widehat{f}_{DT,h/c}(x)=I_{h/c}(x)+W. \end{aligned}$$

The estimator $\tilde{f}_{TS,DT}$ can be written as follows:

$$\begin{aligned} \tilde{f}_{TS,DT}= \left\{ I_{h}(x)\right\} ^{\frac{1}{1-c}}\left\{ 1+\frac{Z}{I_{h}(x)}\right\} ^{\frac{1}{1-c}} \left\{ I_{h/c}(x)\right\} ^{-\frac{c}{1-c}}\left\{ 1+\frac{W}{I_{h/c}(x)}\right\} ^{-\frac{c}{1-c}}. \end{aligned}$$

Using the expansion $(1+t)^{\alpha }=1+\alpha t+o(t^{2})$, we then have

$$\begin{aligned} \tilde{f}_{TS,DT}(x)= & {} \left\{ I_{h}(x)\right\} ^{\frac{1}{1-c}}\left\{ I_{h/c}(x)\right\} ^{-\frac{c}{1-c}} +\frac{1}{1-c}Z\left\{ \frac{I_{h}(x)}{I_{h/c}(x)}\right\} ^{-\frac{c}{1-c}}\nonumber \\&-\frac{c}{1-c}W\left\{ \frac{I_{h}(x)}{I_{h/c}(x)}\right\} ^{\frac{1}{1-c}}+O\left\{ (Z+W)^{2}\right\} . \end{aligned}$$

(4)

Based on Assumption 2 and using the same calculations as in Hirukawa (2010) and Terrell and Scott (1980), we can show easily that

$$\begin{aligned} \mathbb {E}\left( \tilde{f}_{TS,DT}(x)\right) =f(x)+\frac{1}{c}\left[ \frac{1}{2}\left\{ \frac{l^{2}_{1}(x,f)}{f(x)}-l_{2}(x,f)\right\} \right] h^{2}+o(h^{2}). \end{aligned}$$

1.1.2 Variance

For the variance, from Eq. (4) we have

$$\begin{aligned} \mathrm{Var}\left( \tilde{f}_{TS,DT}(x)\right)= & {} \mathbb {E}\left( \frac{1}{1-c}Z-\frac{c}{1-c}W\right) ^{2}+o(n^{-1})\\= & {} \mathrm{Var}\left( \frac{1}{1-c}\widehat{f}_{DT,h}(x)-\frac{c}{1-c}\widehat{f}_{DT,h/c}(x)\right) +o(n^{-1})\\= & {} \frac{1}{(1-c)^{2}}\mathrm{Var}\left( \widehat{f}_{DT,h}(x)\right) +\frac{c^{2}}{(1-c)^{2}}\mathrm{Var}\left( \widehat{f}_{DT,h/c}(x)\right) \\&-\frac{2c}{(1-c)^{2}}\mathrm{cov}\left( \widehat{f}_{DT,h}(x),\widehat{f}_{DT,h/c}(x)\right) . \end{aligned}$$

First, note that the terms $\mathrm{Var}\left( \widehat{f}_{DT,h}(x)\right)$ and $\mathrm{Var}\left( \widehat{f}_{DT,h/c}(x)\right)$ are given by [see Kokonendji et al. (2007)]:

$$\begin{aligned} \mathrm{Var}\left( \widehat{f}_{DT,h}(x)\right)= & {} f(x)(1-f(x))K^{2}_{DT,h}(x)+o\left( \frac{1}{n}\right) \\= & {} \frac{f(x)}{n}(1-f(x))\frac{(1+a)^{2h}}{P^{2}(a,h)}+o\left( \frac{1}{n}\right) , \end{aligned}$$

and

$$\begin{aligned} \mathrm{Var}\left( \widehat{f}_{DT,h/c}(x)\right)= & {} f(x)(1-f(x))K^{2}_{DT,h/c}(x)+o\left( \frac{1}{n}\right) \\= & {} \frac{f(x)}{n}(1-f(x))\frac{(1+a)^{2h/c}}{P^{2}(a,h/c)}+o\left( \frac{1}{n}\right) . \end{aligned}$$

Now,

$$\begin{aligned}&\mathrm{cov}\left( \widehat{f}_{DT,h}(x),\widehat{f}_{DT,h/c}(x)\right) \\&\quad = \mathbb {E}\left( \widehat{f}_{DT,h}(x)\widehat{f}_{DT,h/c}(x)\right) -\mathbb {E}\left( \widehat{f}_{DT,h}(x)\right) \mathbb {E}\left( \widehat{f}_{DT,h/c} (x)\right) \\&\quad =\frac{1}{n^{2}}\sum \limits _{i=1}^{n}\sum \limits _{j=1}^{n}\mathbb {E} \left( K_{DT,h}(X_{i})K_{DT,h/c}(X_{j})\right) -\mathbb {E} \left( K_{DT,h}(X_{i})\right) \mathbb {E}\left( K_{DT,h/c}(X_{j})\right) \\&\quad =\frac{1}{n}\mathbb {E}\left( K_{DT,h}(X_{i})K_{DT,h/c}(X_{i})\right) +\frac{(n-1)}{n}\mathbb {E}\left( K_{DT,h}(X_{i})\right) \mathbb {E}\left( K_{DT,h/c} (X_{j})\right) \\&\qquad -\mathbb {E}\left( K_{DT,h}(X_{i})\right) \mathbb {E}\left( K_{DT,h/c}(X_{j})\right) \\&\quad =\frac{1}{n}\mathbb {E}\left( K_{DT,h}(X_{i})K_{DT,h/c}(X_{i})\right) -\frac{1}{n}\mathbb {E}\left( K_{DT,h}(X_{i})\right) \mathbb {E}\left( K_{DT,h/c} (X_{j})\right) \\&\quad =\frac{1}{n}K_{DT,h}(x)K_{DT,h/c}(x)f(x)-\frac{1}{n}K_{DT,h}(x)K_{DT,h/c} (x)f^{2}(x)+o\left( \frac{1}{n}\right) \\&\quad =\frac{1}{n}f(x)(1-f(x))\frac{(1+a)^{h}}{P(a,h)}\frac{(1+a)^{h/c}}{P(a,h/c)} +o\left( \frac{1}{n}\right) . \end{aligned}$$

Therefore, the variance of $\tilde{f}_{TS,DT}(x)$ is given by

$$\begin{aligned} \mathrm{Var}\left( \tilde{f}_{TS,DT}(x)\right) =\frac{f(x)(1-f(x))}{n(1-c)^{2}} \left( \frac{(1+a)^{h}}{P(a,h)}-c\frac{(1+a)^{h/c}}{P(a,h/c)}\right) ^{2}+o \left( \frac{1}{n}\right) , \end{aligned}$$

which corresponds to the results in Theorem 1.

1.2 Sketch of the proof of Theorem 2

1.2.1 Bias

At first, the estimator $\tilde{f}_{JLN,DT}$ can be written as [see Hirukawa (2010)]

$$\begin{aligned} \tilde{f}_{JLN,DT}(x)=f(x)\left\{ 1+\frac{\widehat{f}_{DT}(x)-f(x)}{f(x)} \right\} \left\{ 1+(\psi (x)-1)\right\} , \end{aligned}$$

where $\psi (x)=n^{-1}\sum _{i=1}^{n}K_{DT,h}(X_{i})/\widehat{f}_{DT}(X_{i})$. Then, we have

$$\begin{aligned} \mathbb {E}\left( \tilde{f}_{JLN,DT}(x)\right)= & {} f(x)+f(x)\mathbb {E} \left\{ \frac{\widehat{f}_{DT}(x)-f(x)}{f(x)}\right\} +f(x) \mathbb {E}\left\{ \psi (x)-1\right\} \\&+f(x)\mathbb {E}\left\{ \left( \frac{\widehat{f}_{DT}(x)-f(x)}{f(x)}\right) \left( \psi (x)-1\right) \right\} . \end{aligned}$$

By using Assumption 2 and the properties of DT random variables, the terms $\mathbb {E}\left\{ \frac{\widehat{f}_{DT}(x)-f(x)}{f(x)}\right\}$, $\mathbb {E}\left\{ \psi (x)-1\right\}$ and $\mathbb {E}\left\{ \left( \frac{\widehat{f}_{DT}(x)-f(x)}{f(x)}\right) \left( \psi (x)-1\right) \right\}$ can be approximated following the same procedures as in Hirukawa (2010). Thus, $\mathbb {E}(\widehat{f}_{JLN-DT})$ is approximated by

$$\begin{aligned} \mathbb {E}(\tilde{f}_{JLN-DT}(x))= & {} f(x)-f(x)\left[ \frac{1}{2}\left\{ \log (a+1)S(a)-2\sum \limits _{k=1}^{a} k^{2}\log (k)\right\} q^{(1)}(x)\right] h^{2}\\&-f(x) \left[ \frac{1}{24}\left\{ \log (a+1)R(a)-2\sum \limits _{k=1}^{a}k^{4} \log (k)\right\} q^{(2)}(x)\right] h^{2}+o(h^{2}),\\ \mathbb {E}(\tilde{f}_{JLN-DT}(x))= & {} f(x)-f(x)\left[ \frac{1}{2}\left\{ \log (a+1)S(a)-2\sum \limits _{k=1}^{a} k^{2}\log (k)\right\} q^{(1)}(x)\right. \\&\left. +\frac{1}{24}\left\{ \log (a+1)R(a)-2\sum \limits _{k=1}^{a}k^{4} \log (k)\right\} q^{(2)}(x)\right] h^{2}+o(h^{2}), \end{aligned}$$

where $q(x)=l_{1}(x,f)/f(x)$ with $l_{1}(x,f)$ given in the proof of Theorem 1.

1.2.2 Variance

Note that following Hirukawa (2010) and Jones et al. (1995), we can show that $\tilde{f}_{JLN,DT}(x)$ is equivalent to

$$\begin{aligned} \tilde{f}_{JLN-DT}(x)=f(x)\frac{1}{n}\sum \limits _{i=1}^{n}\frac{K_{DT,h} (X_{i})}{f(X_{i})}. \end{aligned}$$

It follows that

$$\begin{aligned} \mathrm{Var}\left( \widetilde{f}_{JLN-DT}(x)\right)= & {} f^{2}(x)\frac{1}{n}\mathrm{Var}\left\{ \frac{K_{DT,h}(X_{i})}{f(X_{i})}\right\} \\= & {} f^{2}(x)\frac{1}{n}\left\{ \mathbb {E}\left( \frac{K^{2}_{DT,h}(X_{i})}{f^{2}(X_{i})}\right) -\left[ \mathbb {E}\left( \frac{K_{DT,h}(X_{i})}{f(X_{i})}\right) \right] ^{2}\right\} \\= & {} \frac{f^{2}(x)}{n}\left\{ \frac{K^{2}_{DT,h}(x)}{f(x)}-K^{2}_{DT,h}(x)\right\} +o\left( \frac{1}{n}\right) \\= & {} \frac{f(x)}{n}(1-f(x))\frac{(1+a)^{2h}}{P^{2}(a,h)}+o\left( \frac{1}{n}\right) . \end{aligned}$$

Therefore, we obtain the approximation for the variance given in Theorem 2. $\square$