Birnbaum–Saunders power-exponential kernel density estimation and Bayes local bandwidth selection for nonnegative heavy tailed data

Abstract

In this paper, we study the performance of the Birnbaum–Saunders-power-exponential (BS-PE) kernel and Bayesian local bandwidth selection in the context of kernel density estimation for nonnegative heavy tailed data. Our approach considers the BS-PE kernel estimator and treats locally the bandwidth h as a parameter with prior distribution. The posterior density of h at each point x (point where the density is estimated) is derived in closed form, and the Bayesian bandwidth selector is obtained by using popular loss functions. The performance evaluation of this new procedure is carried out by a simulation study and real data in web-traffic. The proposed method is very quick and very competitive in comparison with the existing global methods, namely biased cross-validation and unbiased cross-validation.

Appendix

1.1 Proof of Proposition 1

We here present the calculations of $\hat{\pi }(h|x,X_{1},X_{2},\ldots ,X_{n})$ and $\hat{h}_{n}(x)$ (with quadratic loss function) using the prior (13) and the BS-PE kernel density estimator (5):

$$\begin{aligned}&\hat{\pi }(h|x,X_{1},X_{2},\ldots ,X_{n}) = \frac{\hat{f}_{h}(x)\pi (h)}{\int \hat{f}_{h}(x)\pi (h)dh}\nonumber \\&\quad { =\frac{\frac{\nu }{n2^{\frac{1}{2\nu }+1}\varGamma \left( \frac{1}{2\nu }\right) \varGamma (\alpha )\beta ^{\alpha }h^{\alpha \nu +\frac{3}{2}}}\sum \nolimits _{i=1}^{n}\left( \frac{1}{\sqrt{xX_{i}}}+\sqrt{\frac{x}{X_{i}^{3}}}\right) \exp \left( \frac{-1}{2h^{\nu }}\left( \frac{X_{i}}{x}+\frac{x}{X_{i}}-2\right) ^{\nu }\right) \exp \left( -\frac{1}{\beta h^{\nu }}\right) }{\frac{\nu }{n2^{\frac{1}{2\nu }+1}\varGamma \left( \frac{1}{2\nu }\right) \varGamma (\alpha )\beta ^{\alpha }} \int \frac{1}{h^{\alpha \nu +\frac{3}{2}}}\sum \nolimits _{i=1}^{n}\left( \frac{1}{\sqrt{xX_{i}}}+\sqrt{\frac{x}{X_{i}^{3}}}\right) \exp \left( \frac{-1}{2h^{\nu }}\left( \frac{X_{i}}{x}+\frac{x}{X_{i}}-2\right) ^{\nu }\right) \exp (-\frac{1}{\beta h^{\nu }}) dh}}\nonumber \\&\quad { =\frac{\sum \nolimits _{i=1}^{n}\left( \frac{1}{\sqrt{xX_{i}}}+\sqrt{\frac{x}{X_{i}^{3}}}\right) \frac{1}{h^{\nu \alpha +\frac{3}{2}}}\exp \left( -\frac{1}{h^{\nu }}\left( \frac{(\frac{X_{i}}{x}+\frac{x}{X_{i}}-2)^{\nu }}{2}+\frac{1}{\beta }\right) \right) }{\int \sum \nolimits _{i=1}^{n}\left( \frac{1}{\sqrt{xX_{i}}}+\sqrt{\frac{x}{X_{i}^{3}}}\right) \frac{1}{h^{\nu \alpha +\frac{3}{2}}}\exp \left( -\frac{1}{h^{\nu }}\left( \frac{(\frac{X_{i}}{x}+\frac{x}{X_{i}}-2)^{\nu }}{2}+\frac{1}{\beta }\right) \right) dh}.} \end{aligned}$$

(18)

Now, we consider the denominator of (18),

$$\begin{aligned} \int \hat{f}_{h}(x)\pi (h)dh= & {} \int \sum \limits _{i=1}^{n}\left( \frac{1}{\sqrt{xX_{i}}}+\sqrt{\frac{x}{X_{i}^{3}}}\right) \frac{1}{h^{\nu \alpha +\frac{3}{2}}}\exp \left( -\frac{1}{h^{\nu }}\left( \frac{\left( \frac{X_{i}}{x}+\frac{x}{X_{i}}-2\right) ^{\nu }}{2}+\frac{1}{\beta }\right) \right) dh\nonumber \\= & {} \sum \limits _{i=1}^{n}\left( \frac{1}{\sqrt{xX_{i}}}+\sqrt{\frac{x}{X_{i}^{3}}}\right) \int \frac{1}{h^{\alpha \nu +\frac{3}{2}}}\exp \left( -\frac{1}{h^{\nu }}\left( \frac{\left( \frac{X_{i}}{x}+\frac{x}{X_{i}}-2\right) ^{\nu }}{2}+\frac{1}{\beta }\right) \right) dh.\nonumber \\ \end{aligned}$$

(19)

Let $u=\frac{1}{h^{\nu }}$. Then we can rewrite the above integral as:

$$\begin{aligned} \int \hat{f}_{h}(x)\pi (h)dh= & {} \frac{1}{\nu }\sum _{i=1}^{n}\left( \frac{1}{\sqrt{xX_{i}}}+\sqrt{\frac{x}{X_{i}^{3}}}\right) \int u^{\alpha +\frac{1}{2\nu }-1}\exp \left( -u\left( \frac{\left( \frac{X_{i}}{x}+\frac{x}{X_{i}}-2\right) ^{\nu }}{2}+\frac{1}{\beta }\right) \right) du \nonumber \\= & {} \frac{1}{\nu }\varGamma \left( \alpha +\frac{1}{2\nu }\right) \sum \limits _{i=1}^{n}\left( \frac{1}{\sqrt{xX_{i}}}+\sqrt{\frac{x}{X_{i}^{3}}}\right) \left[ \frac{\left( \frac{X_{i}}{x}+\frac{x}{X_{i}}-2\right) ^{\nu }}{2}+\frac{1}{\beta }\right] ^{-\alpha -\frac{1}{2\nu }}. \end{aligned}$$

(20)

The local bandwidth $\hat{h}_{n}(x)$ is given by

$$\begin{aligned} \hat{h}_{n}(x)= & {} \int h \hat{\pi }(h|x,X_{1},X_{2},\ldots ,X_{n}) dh \nonumber \\= & {} \int \frac{ h\frac{1}{h^{\alpha \nu +\frac{3}{2}}}\sum \nolimits _{i=1}^{n}\left( \frac{1}{\sqrt{xX_{i}}}+\sqrt{\frac{x}{X_{i}^{3}}}\right) \exp \left( -\frac{1}{h^{\nu }}\left( \frac{\left( \frac{X_{i}}{x}+\frac{x}{X_{i}}-2\right) ^{\nu }}{2}+\frac{1}{\beta }\right) \right) }{\frac{1}{\nu }\varGamma (\alpha +\frac{1}{2\nu })\sum \nolimits _{i=1}^{n}\left( \frac{1}{\sqrt{xX_{i}}}+\sqrt{\frac{x}{X_{i}^{3}}}\right) \left[ \frac{\left( \frac{X_{i}}{x}+\frac{x}{X_{i}}-2\right) ^{\nu }}{2}+\frac{1}{\beta }\right] ^{-\alpha -\frac{1}{2\nu }}}dh. \end{aligned}$$

(21)

By the same reasoning for the numerator of (21), the local bandwidth is then given by:

$$\begin{aligned} \hat{h}_{n}(x)=\frac{\varGamma \left( \alpha -\frac{1}{2\nu }\right) \sum \nolimits _{i=1}^{n}\left( \frac{1}{\sqrt{xX_{i}}}+\sqrt{\frac{x}{X_{i}^{3}}}\right) \left[ \frac{\left( \frac{X_{i}}{x}+\frac{x}{X_{i}}-2\right) ^{\nu }}{2}+\frac{1}{\beta }\right] ^{-\alpha +\frac{1}{2\nu }}}{ \varGamma \left( \alpha +\frac{1}{2\nu }\right) \sum \nolimits _{i=1}^{n}\left( \frac{1}{\sqrt{xX_{i}}}+\sqrt{\frac{x}{X_{i}^{3}}}\right) \left[ \frac{\left( \frac{X_{i}}{x}+\frac{x}{X_{i}}-2\right) ^{\nu }}{2}+\frac{1}{\beta }\right] ^{-\alpha -\frac{1}{2\nu }}}, \end{aligned}$$

which corresponds to Eq. (15). Similarly we can use the same technique above for the case of $\tilde{h}_{n}(x)$ given by (16) which completes the proof.