Adaptive Bayesian bandwidth selection in asymmetric kernel density estimation for nonnegative heavy-tailed data

Abstract

In this paper, we consider an interesting problem on adaptive Birnbaum-Saunders-power-exponential (BS-PE) kernel density estimation for nonnegative heavy-tailed (HT) data. Treating the variable bandwidths h i , i = 1, . . . , n of adaptive BS-PE kernel as parameters, we then propose a conjugate prior and estimate the h i 's by using the popular quadratic and entropy loss functions. Explicit formulas are obtained for the posterior and Bayes estimators. Comparison simulations with global unbiased cross-validation bandwidth selection technique were conducted under four HT distributions. Finally, two applications based on HT real data are presented and analyzed.

A 3D taphonomic model of long bone modification by lions in medium-sized ungulate carcasses

“…In this type of low-density point processes, kernel maps tend to over stretch the real clustering areas, creating the false impression that hot spots are bigger than what they actually are 30 . This is due to the fact, as described in the introduction of the present work, that kernel maps are strongly dependent on selected smoothing bandwidths 18 – 20 , 30 . The array of methods (and limiting factors) that determine these bandwithds are responsible for highly variable map outcomes when using one method or another 18 – 20 , 30 , 42 – 44 .…”

“…This is due to the fact, as described in the introduction of the present work, that kernel maps are strongly dependent on selected smoothing bandwidths 18 – 20 , 30 . The array of methods (and limiting factors) that determine these bandwithds are responsible for highly variable map outcomes when using one method or another 18 – 20 , 30 , 42 – 44 . Until now, in none of the published works where kernel maps have been adopted, have bandwithds been statistically justified prior to their use.…”

“…Some of these methods make assumptions (i.e., inhomogeneous Poisson process, Cox Process…) that must not be skipped if properly used. In addition, heavy-tailed distributions make bandwidth selection very difficult 18 – 20 . This impacts the use of these methods in taphonomic work because very frequently, distribution of BSM on bones follow such distribution types.…”

Ake: An R Package for Discrete and Continuous Associated Kernel Estimations

“…Also, these associated kernels are useful for heavy tailed data p.d.f. estimation and can be added later in the package; see, e.g., Ziane et al (2015). The case of multivariate data needs to be taken in consideration; see Kokonendji and Somé (2015) for p.d.f.…”

“…The coming versions of this package will contain, among others, p.d.f. estimation of heavy tailed data (e.g., Ziane et al, 2015) and the estimation of other functionals. The bandwidth selection remains crucial in associated kernel estimations of p.d.f., p.m.f.…”