A quantile based test for comparing cumulative incidence functions of competing risks models

In the present note, we develop a nonparametric testing procedure for testing equality of cumulative incidence functions of competing risks models using quantile functions. Asymptotic properties of the test statistic are discussed. Simulation studies and real data examples illustrate the practical utility of the procedure.     

Uniform convergence of nonparametric regressions in competing risk models with right censoring

We consider, in the presence of covariates, non-independent competing risks that are subject to right censoring. We define a nonparametric estimator of the incident regression function through the generalized product-limit estimator of the conditional censorship distribution function. Under suitable conditions, we establish the almost sure uniform convergence of those estimators with an appropriate rate.     

A central limit theorem for the U-statistic

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 2, pp. 269–271, February, 1989.     

A U-statistic and estimation for the inverse Gaussian distribution

Unbiased estimation of the square of the coefficient of variation in the inverse Gaussian distribution is considered. It is shown that the estimator is indeed a U-statistic. The exact distribution of the estimator is also derived.     

A note on the almost sure limit theorem for U-statistic

Summary In this note we prove an almost sure limit theorem for the products of U-statistics.     

Nonparametric estimation of competing risks models with covariates

In competing risks model, several failure times arise potentially. The smallest failure time and its index only are observed. Without specific assumptions, the joint or even the marginal distribution functions of the underlying failure times are not identifiable (A. Tsiatis, Proc. Natl. Acad. Sci. USA 72 (1975) 20). Nonetheless, if each individual is characterized by a “sufficiently informative” set of covariates, these distributions are identifiable under some conditions of regularity (J.J. Heckman and B. Honoré, Biometrika 76 (1989) 325). In this paper, nonparametric kernel estimators of the joint distribution function of failure times conditional on the covariates are proposed. Their weak and strong consistency are discussed.     

A test of independence in some copula models

The purpose of this paper is to provide limit laws for semiparametric estimators of copulas. Some statistical tests of independence are introduced as a consequence of this methodology. We are primarily concernedwith the case where the parameter lies on the boundary of the admissible domain.      

A lack-of-fit test in Tobit errors-in-variables regression models

The problem of fitting a parametric model in Tobit errors-in-variables regression models is discussed in this paper. The proposed test is based on the supremum of the Khmaladze type transformation of a certain partial sum process of calibrated residuals. This framework covers the usual error-free Tobit model as a special case. The asymptotic null distribution of this transformed process is shown to be the same as that of a time transformed standard Brownian motion. Consistency against some fixed alternatives and asymptotic power under some local nonparametric alternatives of this test are also discussed. Simulation studies are conducted to assess the finite sample performance of the proposed test.     

A specification test of stochastic diffusion models

In this paper, we propose a hypothesis testing approach to checking model mis-specification in continuous-time stochastic diffusion model. The key idea behind the development of our test statistic is rooted in the generalized information equality in the context of martingale estimating equations. We propose a bootstrap resampling method to implement numerically the proposed diagnostic procedure. Through intensive simulation studies, we show that our approach is well performed in the aspects of type I error control, power improvement as well as computational efficiency.     

Availability optimization of systems subject to competing risk

This paper considers a competing risk (degradation and sudden failure) maintenance situation. A maintenance model and a repair cost model are presented. The degradation state of the units is continuously monitored. When either the degradation level reaches a predetermined threshold or a sudden failure occurs before the unit reaches the degradation threshold level, the unit is immediately repaired (renewed) and restored to operation. The subsequent repair times increase with the number of renewals. This process is repeated until a predetermined time is reached for preventive maintenance to be performed. The optimal maintenance schedule that maximizes the unit availability subject to repair cost constraint is determined in terms of the degradation threshold level and the time to perform preventive maintenance.     

A nonparametric lack-of-fit test for heteroscedastic regression models

A simple test is proposed for examining the correctness of a given completely specified response function against unspecified general alternatives in the context of univariate regression. The usual diagnostic tools based on residual plots are useful but heuristic. We introduce a formal statistical test supplementing the graphical analysis. Technically, the test statistic is the maximum length of the sequences of ordered (with respect to the covariate) observations that are consecutively overestimated or underestimated by the candidate regression function. Note that the testing procedure can cope with heteroscedastic errors and no replicates. Recursive formulae allowing one to calculate the exact distribution of the test statistic under the null hypothesis and under a class of alternative hypotheses are given.     

Study of n-fold weibull competing risk model

This paper deals with an n-fold Weibull competing risk model. A characterisation of the WPP plot is given along with estimation of model parameters when modelling a given data set. These are illustrated through two examples. A study of the different possible shapes for the density and failure rate functions is also presented.     

A locally asymptotically powerful test for nonlinear autoregressive models

We propose a locally asymptotically powerful test to simultaneously examine hypotheses relative to the parametric form of the conditional mean and the conditional variance functions in a class of nonlinear semi-parametric time series models without a specified error law. On the basis of a modified version of the Le Cam method of Hwang and Basawa (2001), we establish the local asymptotic normality relative to the model. The main result shows that the test statistic built by substituting consistent estimated residuals and parameters for the theoretical ones is asymptotically normal. Its asymptotic power is computed and the result is illustrated by some simulations. To cite this article: F. Chebana, N. Laïb, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

A robust adaptive-to-model enhancement test for parametric single-index models

This paper is devoted to test the parametric single-index structure of the underlying model when there are outliers in observations. First, a test that is robust against outliers is suggested. The Hampel’s second-order influence function of the test statistic is proved to be bounded. Second, the test fully uses the dimension reduction structure of the hypothetical model and automatically adapts to alternative models when the null hypothesis is false. Thus, the test can greatly overcome the dimensionality problem and is still omnibus against general alternative models. The performance of the test is demonstrated by both Monte Carlo simulation studies and an application to a real dataset.     

A semiparametric test of independence in copula models for censored data

We propose a semiparametric test of independence in copula models for bivariate survival censored data. We give the limit laws of the estimate of the parameter and the proposed test statistic under the null hypothesis of independence.     

Factor risk quantification in annuity models

Calculation of risk contributions of sub-portfolios to total portfolio risk is essential for risk management in insurance companies. Thanks to risk capital allocation methods and linearity of the loss model, sub-portfolio (or position) contributions can be calculated efficiently. However, factor risk contribution theory in non-linear loss models has received little interest. Our concern is the determination of factor risk contributions to total portfolio risk where portfolio risk is a non-linear function of factor risks. We employ different approximations in order to convert the non-linear loss model into a linear one. We illustrate the theory on an annuity portfolio where the main factor risks are interest-rate risk and mortality risk.     

On the inconsistency of Bayesian non-parametric estimators in competing risks/multiple decrement models

In the competing risks/multiple decrement model, the joint distribution is often not identifiable given only the observed time of failure and the cause of failure. The traditional approach is consequently to assume a parametric model. In this paper we shall not do this, but rather assume a Bayesian stance, take a Dirichlet process as a prior distribution, and then calculate the posterior distribution given the data. In this paper we show that in dimensions ? 2, the posterior mean yields an inconsistent estimator of the joint probability law, contrary to the common assumption that the prior law ‘washes out’ with large samples. For single decrement mortality tables however, the non-parametric Bayesian method allows a flexible method for adjusting a standard mortality table to reflect mortality experience, or covariate information.     

A modified residual-based test for serial correlation in linear panel data models

This paper suggests a modified serial correlation test for linear panel data models, which is based on the parameter estimates for an artificial autoregression modeled by differencing and centering residual vectors. Specifically, the differencing operator over the time index and the centering operator over the individual index are, respectively, used to eliminate the potential individual effects and time effects so that the resultant serial correlation test is robust to the two potential effects. Clearly, the test is also robust to the potential correlation between the covariates and the random effects. The test is asymptotically chi-squared distributed under the null hypothesis. Power study shows that the test can detect local alternatives distinct at the parametric rate from the null hypothesis. The finite sample properties of the test are investigated by means of Monte Carlo simulation experiments, and a real data example is analyzed for illustration.     

A generalized penalty function in Sparre Andersen risk models with surplus-dependent premium

In a general Sparre Andersen risk model with surplus-dependent premium income, the generalization of Gerber-Shiu function proposed by Cheung et al. (2010a) is studied. A general expression for such Gerber-Shiu function is derived, and it is shown that its determination reduces to the evaluation of a transition function which is independent of the penalty function. Properties of and explicit expressions for such a transition function are derived when the surplus process is subject to (i) constant premium; (ii) a threshold dividend strategy; or (iii) credit interest. Extension of the approach is discussed for an absolute ruin model with debit interest.     

A nonparametric test for conditional symmetry in nonstationary and absolutely regular dynamical models

In this Note, we reconsider the test for symmetry of the errors distribution in a class of heteroscedastic models proposed by Ngatchou-Wandji (2009). In the new study, the observations, as well as the errors, are not necessarily stationary but are required to be absolutely regular.