Kernel density estimation for heavy-tailed distributions using the champernowne transformation

Comments on: hybrid semiparametric Bayesian networks

This note comments on the article of David Atienza, Pedro Larrañaga and Concha Bielza in which they first review recent contributions to Bayesian networks and then introduce a new hybrid version. It combines parametric and nonparametric density estimates for continuous variables by simultaneously allowing for discrete parents. We discuss straightforward extensions of the linear Gaussian parts and potential smoothing over the outcomes of discrete parents and conclude with some minor comments.

Modeling in Forestry Using Mixture Models Fitted to Grouped and Ungrouped Data

The creation and maintenance of complex forest structures has become an important forestry objective. Complex forest structures, often expressed in multimodal shapes of tree size/diameter (DBH) distributions, are challenging to model. Mixture probability density functions of two- or three-component gamma, log-normal, and Weibull mixture models offer a solution and can additionally provide insights into forest dynamics. Model parameters can be efficiently estimated with the maximum likelihood (ML) approach using iterative methods such as the Newton-Raphson (NR) algorithm. However, the NR algorithm is sensitive to the choice of initial values and does not always converge. As an alternative, we explored the use of the iterative expectation-maximization (EM) algorithm for estimating parameters of the aforementioned mixture models because it always converges to ML estimators. Since forestry data frequently occur both in grouped (classified) and ungrouped (raw) forms, the EM algorithm was applied to explore the goodness-of-fit of the gamma, log-normal, and Weibull mixture distributions in three sample plots that exhibited irregular, multimodal, highly skewed, and heavy-tailed DBH distributions where some size classes were empty. The EM-based goodness-of-fit was further compared against a nonparametric kernel-based density estimation (NK) model and the recently popularized gamma-shaped mixture (GSM) models using the ungrouped data. In this example application, the EM algorithm provided well-fitting two- or three-component mixture models for all three model families. The number of components of the best-fitting models differed among the three sample plots (but not among model families) and the mixture models of the log-normal and gamma families provided a better fit than the Weibull distribution for grouped and ungrouped data. For ungrouped data, both log-normal and gamma mixture distributions outperformed the GSM model and, with the exception of the multimodal diameter distribution, also the NK model. The EM algorithm appears to be a promising tool for modeling complex forest structures.

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

Here, we present the first three-dimensional taphonomic analysis of a carnivore-modified assemblage at the anatomical scale of the appendicular skeleton. A sample of ten carcasses composed of two taxa (zebra and wildebeest) consumed by wild lions in the Tarangire National Park (Tanzania) has been used to determine element-specific lion damage patterns. This study presents a novel software for the 3D spatial documentation of bone surface modifications at the anatomical level. Combined with spatial statistics, the present analysis has been able to conclude that despite variable degrees of competition during carcass consumption, lions generate bilateral patterning consisting of substantial damage of proximal ends of stylopodials and zeugopodials, moderate damage of the distal ends of femora and marginal damage of distal ends of humeri and zeugopodials. Of special interest is, specifically, the patterning of tooth marks on shafts according to element, since these are crucial to determine not only the type of carnivore involved in any given bone assemblage, but also the interaction with other agents (namely, hominins, in the past). Lions leave few tooth marks on mid-shaft sections, mostly concentrated on certain sections and orientations of stylopodials and, to a lesser extent, of the proximal tibia. Redundant occurrence of tooth marks on certain bone sections renders them as crucial to attest lion agency in carcass initial consumption. Indirectly, this can also be used to determine whether hominins ever acquired carcasses at lion kills.

EVT-based estimation of risk capital and convergence of high quantiles

We discuss some issues regarding the accuracy of a quantile-based estimation of risk capital. In this context, extreme value theory (EVT) emerges naturally. The paper sheds some further light on the ongoing discussion concerning the use of a semi-parametric approach like EVT and the use of specific parametric models such as the g-and-h. In particular, we discusses problems and pitfalls evolving from such parametric models when using EVT and highlight the importance of the underlying second-order tail behavior.

EVT-based estimation of risk capital and convergence of high quantiles

We discuss some issues regarding the accuracy of a quantile-based estimation of risk capital. In this context, extreme value theory (EVT) emerges naturally. The paper sheds some further light on the ongoing discussion concerning the use of a semi-parametric approach like EVT and the use of specific parametric models such as the g-and-h. In particular, we discusses problems and pitfalls evolving from such parametric models when using EVT and highlight the importance of the underlying second-order tail behavior.

Classification based on hybridization of parametric and nonparametric classifiers

Parametric methods of classification assume specific parametric models for competing population densities (e.g., Gaussian population densities can lead to linear and quadratic discriminant analysis) and they work well when these model assumptions are valid. Violation in one or more of these parametric model assumptions often leads to a poor classifier. On the other hand, nonparametric classifiers (e.g., nearest-neighbor and kernel-based classifiers) are more flexible and free from parametric model assumptions. But, the statistical instability of these classifiers may lead to poor performance when we have small numbers of training sample observations. Nonparametric methods, however, do not use any parametric structure of population densities. Therefore, even when one has some additional information about population densities, that important information is not used to modify the nonparametric classification rule. This paper makes an attempt to overcome these limitations of parametric and nonparametric approaches and combines their strengths to develop some hybrid classification methods. We use some simulated examples and benchmark data sets to examine the performance of these hybrid discriminant analysis tools. Asymptotic results on their misclassification rates have been derived under appropriate regularity conditions.