Quasi-concave density estimation

Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. Closely related maximum Renyi entropy estimators that impose weaker concavity restrictions on the fitted density are also considered, notably a minimum Hellinger discrepancy estimator that constrains the reciprocal of the square-root of the density to be concave. A limiting form of these estimators constrains solutions to the class of quasi-concave densities.Comment: Published in at http://dx.doi.org/10.1214/10-AOS814 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

An Alternative Approach to Functional Linear Partial Quantile Regression

We have previously proposed the partial quantile regression (PQR) prediction procedure for functional linear model by using partial quantile covariance techniques and developed the simple partial quantile regression (SIMPQR) algorithm to efficiently extract PQR basis for estimating functional coefficients. However, although the PQR approach is considered as an attractive alternative to projections onto the principal component basis, there are certain limitations to uncovering the corresponding asymptotic properties mainly because of its iterative nature and the non-differentiability of the quantile loss function. In this article, we propose and implement an alternative formulation of partial quantile regression (APQR) for functional linear model by using block relaxation method and finite smoothing techniques. The proposed reformulation leads to insightful results and motivates new theory, demonstrating consistency and establishing convergence rates by applying advanced techniques from empirical process theory. Two simulations and two real data from ADHD-200 sample and ADNI are investigated to show the superiority of our proposed methods

Quantile tomography: using quantiles with multivariate data

The use of quantiles to obtain insights about multivariate data is addressed. It is argued that incisive insights can be obtained by considering directional quantiles, the quantiles of projections. Directional quantile envelopes are proposed as a way to condense this kind of information; it is demonstrated that they are essentially halfspace (Tukey) depth levels sets, coinciding for elliptic distributions (in particular multivariate normal) with density contours. Relevant questions concerning their indexing, the possibility of the reverse retrieval of directional quantile information, invariance with respect to affine transformations, and approximation/asymptotic properties are studied. It is argued that the analysis in terms of directional quantiles and their envelopes offers a straightforward probabilistic interpretation and thus conveys a concrete quantitative meaning; the directional definition can be adapted to elaborate frameworks, like estimation of extreme quantiles and directional quantile regression, the regression of depth contours on covariates. The latter facilitates the construction of multivariate growth charts---the question that motivated all the development

Influence of fall height on high impact polystyrene deformation and characteristics of drop weight test

This study deals with high impact polystyrene (HIPS) which was subjected the drop-weight test. HIPS is a polymer produced by the reaction between butadiene synthetic elastomer and styrene (5-14 %) which contains the crystal polymer in certain amounts and is commonly used in mechanical engineering applications where machine parts are exposed to impact loading. The injection moulded HIPS samples were subjected the penetration test at different fall heights and the results were subsequently evaluated and discussed. It was found out that all fall heights are suitable for HIPS penetration, but the optimal one is 50 J because of the smallest variation range. Higher heights are not needed because of increasing power consumption of the test device. From the results, it is clear, that HIPS is not so highly impact resistant material as for example HDPE, because of that is this material suitable for applications where is not often exposed to too big impacts at high velocities. © The Authors, published by EDP Sciences, 2017

Impact resistance of high-density polyethylene against falling penetrator with different potential energy

This study deals with high-density polyethylene (HDPE) which was subjected the drop-weight test. HDPE is a semicrystalline thermoplastic polymer which is commonly used in many applications and mainly in the automotive industry because of its properties. The injection moulded HDPE samples were subjected the penetration test at different potential energies and the results were subsequently evaluated and discussed. The samples were tested in the range of potential energies from 30 to 230 J. The first sample penetration occurred at the energy 50 J. It was found out that the potential energy 30 J is too small for penetration of this material, which shows the impact resistance of HDPE. © The Authors, published by EDP Sciences, 2017