This paper establishes the functional convergence of the Extreme Nelson--Aalen and Extreme Kaplan--Meier estimators, which are designed to capture the heavy-tailed behaviour of censored losses. The resulting limit representations can be used to obtain the distributions of pathwise functionals with respect to the so-called tail process. For instance, we may recover the convergence of a censored Hill estimator, and we further investigate two goodness-of-fit statistics for the tail of the loss distribution. Using the the latter limit theorems, we propose two rules for selecting a suitable number of order statistics, both based on test statistics derived from the functional convergence results. The effectiveness of these selection rules is investigated through simulations and an application to a real dataset comprised of French motor insurance claim sizes.
We develop a new goodness fit test for Rayleigh distribution for complete as well as right censored data. We use U-Statistic theory to derive the test statistic. First we develop a test for complete data and then discuss, how right censored observations can be incorporated in the testing procedure. The asymptotic properties of the test statistics in both uncensored and censored cases are studied in detail. Extensive Monte Carlo simulation studies are carried out to validate the performance of the proposed tests. We illustrate the procedures using real data sets. We also provide, a goodness of …
Using fixed point characterization, we develop a new goodness of fit test for uniform distribution. We also discuss how the right censored observations can be incorporated in the proposed test procedure. We study the asymptotic properties of the proposed test statistics. A Monte Carlo simulation is carried out to evaluate the finite sample performance of the tests. We illustrate the test procedures using real data sets.
We suggest several goodness-of-fit methods which are appropriate with Type-II right censored data. Our strategy is to transform the original observations from a censored sample into an approximately i.i.d. sample of normal variates and then perform a standard goodness-of-fit test for normality on the transformed observations. A simulation study with several well known parametric distributions under testing reveals the sampling properties of the methods. We also provide theoretical analysis of the proposed method.
In the present paper, we develop a new goodness fit test for gamma distribution using the fixed point characterization. U-Statistic theory is employed to derive the test statistic. We discuss how the right censored observations are incorporated in the test developed here. The asymptotic properties of the test statistic in both censored and uncensored cases are studied in detail. Extensive Monte Carlo simulation studies are carried out to validate the performance of the proposed tests. We also illustrate the test procedure using several real data sets.
In this paper, a new goodness-of-fit test for a location-scale family based on progressively Type-II censored order statistics is proposed. Using Monte Carlo simulation studies, the present researchers have observed that the proposed test for normality is consistent and quite powerful in comparison with existing goodness-of-fit tests based on progressively Type-II censored data. Also, the new test statistic for a real data set is used and the results show that our new test statistic performs well.
We express the joint Rényi entropy of progressively censored order statistics in terms of an incomplete integral of the hazard function, and provide a simple estimate of the joint Rényi entropy of progressively Type-II censored data. Then we establish a goodness of fit test statistic based on the Rényi Kullback-Leibler information with the progressively Type-II censored data, and compare its performance with the leading test statistic. A Monte Carlo simulation study shows that the proposed test statistic shows better powers than the leading test statistic against the alternatives with monotone…
In this article, the weighted empirical likelihood is applied to a general setting of two-sample semiparametric models, which includes biased sampling models and case-control logistic regression models as special cases. For various types of censored data, such as right censored data, doubly censored data, interval censored data and partly interval-censored data, the weighted empirical likelihood-based semiparametric maximum likelihood estimator $(\tildeθ_n,\tilde{F}_n)$ for the underlying parameter $θ_0$ and distribution $F_0$ is derived, and the strong consistency of $(\tildeθ_n,\tilde{F}_n)$…
In risk analysis, a global fit that appropriately captures the body and the tail of the distribution of losses is essential. Modelling the whole range of the losses using a standard distribution is usually very hard and often impossible due to the specific characteristics of the body and the tail of the loss distribution. A possible solution is to combine two distributions in a splicing model: a light-tailed distribution for the body which covers light and moderate losses, and a heavy-tailed distribution for the tail to capture large losses. We propose a splicing model with a mixed Erlang (ME)…
A tail empirical process for heavy-tailed and right-censored data is introduced and its Gaussian approximation is established. In this context, a (weighted) new Hill-type estimator for positive extreme value index is proposed and its consistency and asymptotic normality are proved by means of the aforementioned process in the framework of second-order conditions of regular variation. In a comparative simulation study, the newly defined estimator is seen to perform better than the already existing ones in terms of both bias and mean squared error. As a real data example, we apply our estimation…
We introduce a kernel-based goodness-of-fit test for censored data, where observations may be missing in random time intervals: a common occurrence in clinical trials and industrial life-testing. The test statistic is straightforward to compute, as is the test threshold, and we establish consistency under the null. Unlike earlier approaches such as the Log-rank test, we make no assumptions as to how the data distribution might differ from the null, and our test has power against a very rich class of alternatives. In experiments, our test outperforms competing approaches for periodic and Weibul…
Recently, the characterization based approach for the construction of goodness of fit tests has become popular. Most of the proposed tests have been designed for complete i.i.d. samples. Here we present the adaptation of the recently proposed exponentiality tests based on equidistribution-type characterizations for the case of randomly censored data. Their asymptotic properties are provided. Besides, we present the results of wide empirical power study including the powers of several recent competitors. This study can be used as a benchmark for future tests proposed for this kind of data.
University College London（伦敦大学学院）；University of Cambridge（剑桥大学）
Survival Analysis and Reliability Theory are concerned with the analysis of time-to-event data, in which observations correspond to waiting times until an event of interest such as death from a particular disease or failure of a component in a mechanical system. This type of data is unique due to the presence of censoring, a type of missing data that occurs when we do not observe the actual time of the event of interest but, instead, we have access to an approximation for it given by random interval in which the observation is known to belong. Most traditional methods are not designed to deal …
We propose a new approach for estimating the parameters of a probability distribution. It consists on combining two new methods of estimation. The first is based on the definition of a new distance measuring the difference between variations of two distributions on a finite number of points from their support and on using this measure for estimation purposes by the method of minimum distance. For the second method, given an empirical discrete distribution, we build up an auxiliary discrete theoretical distribution having the same support of the first and depending on the same parameters of the…
Residuals in normal regression are used to assess a model's goodness-of-fit (GOF) and discover directions for improving the model. However, there is a lack of residuals with a characterized reference distribution for censored regression. In this paper, we propose to diagnose censored regression with normalized randomized survival probabilities (RSP). The key idea of RSP is to replace the survival probability of a censored failure time with a uniform random number between 0 and the survival probability of the censored time. We prove that RSPs always have the uniform distribution on $(0,1)$ unde…
Estimating Distributions with Failure Rate Properties from Noisy Quantile Data
Department of Mechanical and Industrial Engineering, University of Toronto（多伦多大学机械与工业工程系）；Rotman School of Management, University of Toronto（多伦多大学罗特曼管理学院）；The Wharton School, University of Pennsylvania（宾夕法尼亚大学沃顿商学院）
