左截断右删失下乘积限过程的振动模的强极限定理

本文在左截断右删失模型下获得了乘积限过程和累积失效率过程的振动模和Ｌｉｐｓｃｈｉｔｚ－１２模的强一致收敛的精确速度．作为定理的应用，推导了各种核密度估计和失效率估计的强一致收敛的精确速度．     

Strong limit theorems for oscillation moduli of PL-process and cumulative hazard process under truncation and censorship with applications

In this paper we obtain exact rates of uniform convergence for oscillation moduli and Lipschitz-1/2 moduli of PL-process and cumulative hazard process when the data are subject to left truncation and right censorship. Based on these results, the exact rates of uniform convergence for various types of density and hazard function estimators are derived. Research supported by the Postdoctoral Programme Foundation and the National Natural Science Foundation of China     

Sato Processes in Default Modelling

Abstract

In reduced form default models, the instantaneous default intensity is the classical modelling object. Survival probabilities are then given by the Laplace transform of the cumulative hazard defined as the integrated intensity process. Instead, recent literature tends to specify the cumulative hazard process directly. Within this framework we present a new model class where cumulative hazards are described by self-similar additive processes, also known as Sato processes. Furthermore, we analyse specifications obtained via a simple deterministic time change of a homogeneous Lévy process. While the processes in these two classes share the same average behaviour over time, the associated intensities exhibit very different properties. Concrete specifications are calibrated to data on all the single names included in the iTraxx Europe index. The performances are compared with those of the classical Cox–Ingersoll–Ross intensity and a recently proposed class of intensity models based on Ornstein–Uhlenbeck-type processes. It is shown that the time-inhomogeneous Lévy models achieve comparable calibration errors with fewer parameters and with more stable parameter estimates over time. However, the calibration performance of the Sato processes and the time-change specifications are practically indistinguishable.     

Some properties of bivariate empirical hazard processes under random censoring

In Campbell (1982, IMS Lecture Notes—Monograph Series Vol. 2, pp. 243–256, IMS, Hayward, CA) and Campbell and Földes (1982, Proceedings, Internat. Colloq. Nonparametric Statist. Inform., 1980, North-Holland, New York) some asymptotic properties of bivariate empirical hazard processes under random censoring are given. Taking the representation of the empirical hazard process for bivariate randomly censored samples in Campbell, op. cit., as a starting point and restricting attention to strong properties, we obtain a speed of strong convergence for the weighted bivariate empirical hazard processes as well as a speed of strong uniform convergence for bivariate hazard rate estimators. Our approach is based on a local fluctuation inequality for the bivariate hazard process and differs from the martingale methods quite often used in the univariate case.     

随机截断数据下乘积限过程的振动模

文中研究了随机截断数据下的剩积限过程的振动行为，证明了其振动模的收敛速度与完全样本下经验过程振动模的收敛速度相一致。     

ON LOCAL CONTINUITY MODULI FOR GAUSSIAN PROCESSES

ＬＵＣＨＵＡＮＲＯＮＧ（陆传荣）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨａｎｇｚｈｏｕＵｎｉｖｅｒｓｉｔｙ，Ｈａｎｇｚｈｏｕ３１００２８，Ｃｈｉｎａ）（ＴｈｉｓｗｏｒｋｉｓｓｕｐｐｏｒｔｅｄｂｙｔｈｅＮａｔｉｏｎａｌＮａｔｕｒａｌＳｃｉｅ...     

Additive–multiplicative hazards model with current status data

The additive–multiplicative hazards (AMH) regression model specifies an additive and multiplicative form on the hazard function for the counting process associated with a multidimensional covariate process, which contains the Cox proportional hazards model and the additive hazards model as its special cases. In this paper, we study the AMH model with current status data, where the cumulative hazard hazard function is assumed to be nonparametric and is estimated using B-splines with monotonicity constraint on the functional, while a simultaneous sieve maximum likelihood estimation is proposed to estimate regression parameters. The proposed estimator for the parameter vector is shown to be asymptotically normal and semiparametric efficient. The B-splines estimator of the functional of the cumulative hazard function is shown to achieve the optimal nonparametric rate of convergence. A simulation study is conducted to examine the finite sample performance of the proposed estimators and algorithm, and a real data example is presented for illustration.     

A new method for proving weak convergence results applied to nonparametric estimators in survival analysis

Using the limit theorem for stochastic integral obtained by Jakubowski et al. (Probab. Theory Related Fields 81 (1989) 111–137), we introduce in this paper a new method for proving weak convergence results of empirical processes by a martingale method which allows discontinuities for the underlying distribution. This is applied to Nelson–Aalen and Kaplan–Meier processes. We also prove that the same conclusion can be drawn for Hjort's nonparametric Bayes estimators of the cumulative distribution function and cumulative hazard rate.     

Locally adaptive hazard smoothing

Summary We consider nonparametric estimation of hazard functions and their derivatives under random censorship, based on kernel smoothing of the Nelson (1972) estimator. One critically important ingredient for smoothing methods is the choice of an appropriate bandwidth. Since local variance of these estimates depends on the point where the hazard function is estimated and the bandwidth determines the trade-off between local variance and local bias, data-based local bandwidth choice is proposed. A general principle for obtaining asymptotically efficient data-based local bandwiths, is obtained by means of weak convergence of a local bandwidth process to a Gaussian limit process. Several specific asymptotically efficient bandwidth estimators are discussed. We propose in particular an, asymptotically efficient method derived from direct pilot estimators of the hazard function and of the local mean squared error. This bandwidth choice method has practical advantages and is also of interest in the uncensored case as well as for density estimation.Research supported by UC Davis Faculty Research Grant and by Air Force grant AFOSR-89-0386Research supported by Air Force grant AFOSR-89-0386     

A NOTE ON OSCILLATION MODULUS OF PL-PROCESS AND ITS APPLICATIONS UNDER RANDOM CENSORSHIP

The strong limit results of oscillation modulus of PL-process are established in this paper when the density function is not continuous function for censored data. The rates of convergence of oscillation modulus of PL-process are sharp under week condition. These results can be used to derive laws of the iterated logarithm of random bandwidth kernel estimator and nearest neighborhood estimator of density under continuous conditions of density function being not assumed.     

On some conditional characteristics of hazard rate processes induced by external shocks

Stochastic failure models for systems under randomly variable environment (dynamic environment) are often described using hazard rate process. In this paper, we consider hazard rate processes induced by external shocks affecting a system that follow the nonhomogeneous Poisson process. The sample paths of these processes monotonically increase. However, the failure rate of a system can have completely different shapes and follow, e.g., the upside-down bathtub pattern. We describe and study various ‘conditional properties’ of the models that help to analyze and interpret the shape of the failure rate and other relevant characteristics.     

On a first-passage problem for a cumulative process with exponential decay

A first-passage problem for a cumulative process is investigated. The cumulative process is assumed to be generated by a Poisson process, and the amplitude generated by an event is assumed to decay exponentially. An integral equation for the probability density of the first-passage time until the total amplitude exceeds a pre-specified threshold level is derived. The Laplace transform of the probability density of the first-passage time is obtained explicity when each amplitude generated by an event is distributed exponentially. The mean first-passage times are given in a closed form and plotted versus the threshold level.     

A Berry-Esséen Bound for the Product-Limit Estimator under Left-Truncation and Right-Censoring

For left-truncated and right-censored data, the product-limit estimator F?_xis a well-known nonparametric estimator for the distribution function F_x of the target variable X such as the survival time. Since F?_xas a very complicated product form we establish first the Berry-Esseen bound for the cumulative hazard estimator of F_x The cumulative hazard estimator can be represented as a U-statistic. By using the result in Helmers and van Zwet [6], we derive the Berry-esséen bound for this U-statistic. Then Berry-Esseen bounds for the distribution of the cumulative hazard estimator and the normal distribution and the distribution of the product-limit estimator and the normal distribution are obtained.     

Hazard Rate Estimation in Nonparametric Regression with Censored Data

Consider a regression model in which the responses are subject to random right censoring. In this model, Beran studied the nonparametric estimation of the conditional cumulative hazard function and the corresponding cumulative distribution function. The main idea is to use smoothing in the covariates. Here we study asymptotic properties of the corresponding hazard function estimator obtained by convolution smoothing of Beran's cumulative hazard estimator. We establish asymptotic expressions for the bias and the variance of the estimator, which together with an asymptotic representation lead to a weak convergence result. Also, the uniform strong consistency of the estimator is obtained.     

A strong limit theorem for the oscillation modulus of the uniform empirical quantile process

Stute (1982) and Mason, Shorack and Wellner (1983) have recently completed a thorough study of the limiting behavior of the oscillation of the uniform empirical process. In this paper, the corresponding oscillation behavior of the uniform empirical quantile process is investigated. It is shown to be closely related to the limiting behavior of the maximum k-spacing of n independent Uniform (0, 1) random variables, where k can possibly be a function of n. Results of this type are directly applicable to the study of the strong consistency properties of various types of density estimators.     

具有Vasicek利率的风险过程的破产概率的分解(英文)

In this paper, it is assumed that an insurer with a jump-diffusion risk process would invest its surplus in a bond market, and the interest structure of the bond market is assumed to follow the Vasicek interest model. This paper focuses on the studying of the ruin problems in the above compounded process. In this compounded risk model, ruin may be caused by a claim or oscillation. We decompose the ruin probability for the compounded risk process into two probabilities： the probability that ruin caused by a claim and the probability that ruin caused by oscillation. Integro-differential equations for these ruin probabilities are derived. When the claim sizes are exponentially distributed, the above-mentioned integro-differential equations can be reduced into a three-order partial differential equation.     

Maximum likelihood estimation in the proportional hazards cure model

The proportional hazards cure model generalizes Cox’s proportional hazards model which allows that a proportion of study subjects may never experience the event of interest. Here nonparametric maximum likelihood approach is proposed to estimating the cumulative hazard and the regression parameters. The asymptotic properties of the resulting estimators are established using the modern empirical process theory. And the estimators for the regression parameters are shown to be semiparametric efficient.     

Model checking for additive hazards model with multivariate survival data

Multivariate failure time data often arise in biomedical studies due to natural or artificial clustering. With appropriate adjustment for the underlying correlation, the marginal additive hazards model characterizes the hazard difference via a linear link function between the hazard and covariates. We propose a class of graphical and numerical methods to assess the overall fitting adequacy of the marginal additive hazards model. The test statistics are based on the supremum of the stochastic processes derived from the cumulative sum of the martingale-based residuals over time and/or covariates. The distribution of the stochastic process can be approximated through a simulation technique. The proposed tests examine how unusual the observed stochastic process is, compared to a large number of realizations from the approximated process. This class of tests is very general and suitable for various purposes of model fitting evaluation. Simulation studies are conducted to examine the finite sample performance, and the model-checking methods are illustrated with data from an otitis media study.     

关于分数Wiener过程的不可微连续

在本文中，我们推广了Csoego＂和Shao(1994)的结果，给出了阶α（0＜α＜1）的分数Wiener过程的不可微连续模。