Characterization of bathtub distributions via percentile residual life functions

In reliability theory and survival analysis, many set of data are generated by distributions with bathtub shaped hazard rate functions. Launer (1993) established several relations between the behaviour of the hazard rate function and the percentile residual life function. In particular, necessary conditions were given for a special type of bathtub distributions in terms of percentile residual life functions. The purpose of this paper is to complete the study initiated by Launer (1993) and to characterize (necessary and sufficient conditions) all types of bathtub distributions.Percentile residual life, Bathtub hazard rate, Aging notions,

The percentile residual life up to time t0: ordering and aging properties

Motivated by practical issues, a new stochastic order for random variables is introduced by comparing all their percentile residual life functions until a certain instant. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are studied, and also an application in Reliability Theory is described. Finally, we present some characterization results of the decreasing percentile residual life up to time t0 aging notion.Aging notion, Hazard rate, Mean residual life, Percentile residual life, Reliability, Stochastic ordering

Comparing quantile residual life functions by confidence bands

A quantile residual life function is the quantile of the remaining life of a surviving subject, as it varies with time. In this article we present a nonparametric method for constructing confidence bands for the difference of two quantile residual life functions. These bands provide evidence for two random variables ordering with respect to a quantile residual life order introduced in Franco-Pereira et al. (2010). A simulation study has been carried out in order to evaluate and illustrate the performance and the consistency of this new methodology. We also present applications to real data examples.Quantile residual life, Confidence bands

The decreasing percentile residual life aging notion

Earlier researchers have studied some aspects of the classes of distribution functions with decreasing ?-percentile residual life (DPRL(?)), 0Reliability theory, Hazard rate, Stochastic orders, Aging notions, Nonparametric estimation, Strongly uniform consistency

New stochastic comparisons based on tail value at risk measures

In this article we provide a new criterion for the comparison of claims, when we have conditional claims arising in stop loss contracts or contracts with franchise deductible. These stochastic comparisons are made on the basis of the Tail Value at Risk (also known as conditional tail expectation), just for a fixed level and beyond. In particular, we explain the interest of comparing these quantities, study some preservation properties and, in addition, we provide sufficient conditions for its study. Finally we illustrate its usefulness with some examples.The research of Félix Belzunce is funded by the Ministerio de Economía, Industria y Competitividad (Spain) under grant MTM2016-79943-P (AEI/FEDER, UE). Alba M. Franco-Pereira acknowledges support received from the Ministerio de Economía y Competitividad (Spain) under grant MTM2017-89422-P and has received financial support from the Xunta de Galicia (Centro Singular de Investigación de Galicia accreditation 2016-2019) and the European Union (European Regional Development Fund - ERDF). She also acknowledges funding from Banco Santander and Complutense University of Madrid (project PR26/16-5B-1). Julio Mulero wants to acknowledge the support received from the Conselleria d’Educació, Investigació, Cultura i Esport (Generalitat de la Comunitat Valenciana) under grant GV/2017/015 and the Ministerio de Economía, Industria y Competitividad (Spain) under grant MTM2016-79943-P (AEI/FEDER, UE)

The percentile residual life up to time t(o): Ordering and aging properties

Motivated by practical issues, a new stochastic order for random variables is introduced by comparing all their percentile residual life functions until a certain instant. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are studied and also an application in reliability theory is described. Finally, we present some characterization results of the decreasing percentile residual life up to time to aging notion

Non Parametric ROC Summary Statistics

Receiver operating characteristic (ROC) curves are useful statistical tools for medical diagnostic testing. It has been proved its capability to assess diagnostic marker’s ability to distinguish between healthy and diseased subjects and to compare different diagnostic markers. In this paper we introduce non parametric ROC summary statistics to assess a ROC curve across the entire range of F P F s ∈ (0, 1) as well as over a restricted range of F P F s and compare them with some existing ones through a simulation study and through some real data examples. We also show their capability to compare two diagnostic markers

The epigraph and the hypograph indexes as useful tools for clustering multivariate functional data

The proliferation of data generation has spurred advancements in functional data analysis. With the ability to analyze multiple variables simultaneously, the demand for working with multivariate functional data has increased. This study proposes a novel formulation of the epigraph and hypograph indexes, as well as their generalized expressions, specifically tailored for the multivariate functional context. These definitions take into account the interrelations between components. Furthermore, the proposed indexes are employed to cluster multivariate functional data. In the clustering process, the indexes are applied to both the data and their first and second derivatives. This generates a reduced-dimension dataset from the original multivariate functional data, enabling the application of well-established multivariate clustering techniques that have been extensively studied in the literature. This methodology has been tested through simulated and real datasets, performing comparative analyses against state-of-the-art to assess its performance.Comment: 32 page

Testing the equality of a large number of means of functional data

Given k independent samples of functional data, this paper deals with the problem of testing for the equality of their mean functions. In contrast to the classical setting, where k is kept fixed and the sample size from each population increases without bound, here k is assumed to be large and the size of each sample is either bounded or small in comparison to k. A new test is proposed. The asymptotic distribution of the test statistic is stated under the null hypothesis of equality of the k mean functions as well as under alternatives, which allows us to study the consistency of the test. Specifically, it is shown that the test statistic is asymptotically free distributed under the null hypothesis. The finite sample performance of the test based on the asymptotic null distribution is studied via simulation. Although we start by assuming that the data are functions, the proposed test can also be applied to finite dimensional data. The practical behavior of the test for one dimensional data is numerically studied and compared with other tests

Inference on the symmetry point-based optimal cut-off point and associated sensitivity and specificity with application to SARS-CoV-2 antibody data

Acknowledgments. This work was supported by grants PID2019-104681RB-I00. Data courtesy of Dr Konstantina Kontopoulou.In the presence of a continuous response test/biomarker, it is often necessary to identify a cut-off point value to aid binary classification between diseased and non-diseased subjects. The symmetry-point approach which maximizes simultaneously both types of correct classification is one way to determine an optimal cut-off point. In this article, we study methods for constructing confidence intervals independently for the symmetry point and its corresponding sensitivity, as well as respective joint nonparametric confidence regions. We illustrate using data on the generation of antibodies elicited two weeks post-injection after the second dose of the Pfizer/BioNTech vaccine in adult healthcare workers