In this paper, we derive the exact null distribution of Banerjee’s test [2] for assessing usual stochastic ordering in the two-sample problem. Critical values of the test are provided for some significance levels. Additionally, we obtain the power function under proportional hazards model alternatives.
The problem of nonparametric estimation of the ratio of two densities is a classical topic in discriminant analysis, machine learning and statistical classification. The aim of the paper is to develop the theory of efficient nonparametric estimation of the ratio under the mean integrated squared error (MISE) criterion. For the first time in the literature, a sharp lower bound for minimax MISE is developed for the oracle who knows data, an underlying Sobolev functional class of the ratios, and the reference (denominator) density. Then a data driven estimator is proposed that matches performance of the oracle. It is shown that the problem of ratio estimation may be dramatically more complicated than estimation of a density, and its complexity is captured by the coefficient of difficulty which is a special functional of the two densities defining the ratio.
Defining the notion of a multivariate quantile has been an open problem for more than half a century, motivating a plethora of possible solutions. Of these, the approach of Chakraborty and Chaudhuri [8] and Koltchinskii [26] leading to M-quantiles, is very appealing for its mathematical elegance, combining elements of convex analysis and probability theory. The key idea is the description of a convex function (the K-function) whose gradient (the K-transform) is in one-to-one correspondence between all of ℝ^d and the unit ball in ℝ^d . By analogy with the d=1 case where the K-transform is a cumulative distribution function like object (an M-distribution), the fact that its inverse is guaranteed to exist lends itself naturally to providing the basis for the definition of a quantile function for all d≥ 1 . The resulting M-quantiles have seen applications in a variety of fields, with outlier detection featuring prominently. We show that for odd d≥ 3 it is not the first derivative, but a poly-Laplacian of the K-function that is proportional to the density function. (For even values of d no relationship is apparent.) Two-dimensional examples from non-standard distributions reinforce this point, and illustrate a feature of the K-transform whereby high density regions of the support are mapped to larger volumes, thereby producing a magnification effect that moves inliers closer to the boundary than outliers. In conclusion, the K-transform should not be viewed as a distribution function in higher dimensions, and its inverse does not lead to a ‘‘useful’’ quantile function, in the sense that its level set contours do not plausibly delineate outlying regions.
This paper proposes a novel measure, the quantile-based reversed aging intensity (QRAI) function, which serves as a quantile-based analogue of the classical reversed aging intensity function traditionally defined within the distribution function framework. The QRAI function offers valuable characterizations for specific parametric lifetime models and provides new insights into aging properties when compared with existing aging classes. The paper also explores stochastic comparisons of random variables using the QRAI measure and establishes its relationship with well-known stochastic orderings. We have obtained the QRAI function for the series and parallel systems, and studied how component and system level QRAI functions are connected. Two non-parametric estimators of the QRAI function are proposed, and their performance is assessed through simulation studies under selected parametric models. Additionally, a real data application is presented to demonstrate the practical usefulness of the proposed estimators.
In many areas such as toxicology, chemistry and environmental sciences, data observed usually include left-censoring, that is, data below some practical limit of detection do not get observed. In this paper, we first provide a review of different non-parametric estimators for the cumulative distribution function under such a left-censoring situation. One of them is based on the chain rule and the other one is based on counting processes. We then propose a new estimator for the cumulative distribution function based on a non-parametric likelihood approach using reversed hazard rate. We then take on a classical non-parametric likelihood approach to derive the same estimator. We also propose another estimator for the cumulative distribution function using the relationship between the cumulative distribution function and the reversed hazard rate function. We then carry out an empirical comparison of all these estimators and make some comparative comments. Finally, we conclude with an application to real data.
