Smoothed Isotonic Estimators of a Monotone Baseline Hazard in the Cox Model

We consider the smoothed maximum likelihood estimator and the smoothed Grenander‐type estimator for a monotone baseline hazard rate λ ₀ in the Cox model. We analyze their asymptotic behaviour and show that they are asymptotically normal at rate n ^{m /(2m +1)}, when λ ₀ is m ≥2 times continuously differentiable, and that both estimators are asymptotically equivalent. Finally, we present numerical results on pointwise confidence intervals that illustrate the comparable behaviour of the two methods.     

Testing discontinuities in nonparametric regression

In nonparametric regression, it is often needed to detect whether there are jump discontinuities in the mean function. In this paper, we revisit the difference-based method in [13] and propose to further improve it. To achieve the goal, we first reveal that their method is less efficient due to the inappropriate choice of the response variable in their linear regression model. We then propose a new regression model for estimating the residual variance and the total amount of discontinuities simultaneously. In both theory and simulation, we show that the proposed variance estimator has a smaller mean-squared error compared to the existing estimator, whereas the estimation efficiency for the total amount of discontinuities remains unchanged. Finally, we construct a new test procedure for detection of discontinuities using the proposed method; and via simulation studies, we demonstrate that our new test procedure outperforms the existing one in most settings.     

Consistency of The MMGLE under the piecewise proportional hazards models with interval-censored data

Wong et al. [(2018), ‘Piece-wise Proportional Hazards Models with Interval-censored Data’, Journal of Statistical Computation and Simulation, 88, 140–155] studied the piecewise proportional hazards (PWPH) model with interval-censored (IC) data under the distribution-free set-up. It is well known that the partial likelihood approach is not applicable for IC data, and Wong et al. (2018) showed that the standard generalised likelihood approach does not work either. They proposed the maximum modified generalised likelihood estimator (MMGLE) and the simulation results suggest that the MMGLE is consistent. We establish the consistency and asymptotically normality of the MMGLE.     

Discriminating between distributions using feed-forward neural networks

A relevant problem in many applicatory contexts is to test whether some given observations follow one of two possible probability distributions. The vast literature produced over the years on this topic does not identify a tool which can be easily adopted to any situation but only finds solutions to specific comparisons. Recently, an easy to implement procedure for discrimination between two distributions based on feed-forward neural networks has been proposed giving interesting results. In this work this procedure is further investigated in terms of power, neural network architecture and expected statistical properties of the test statistic for small, moderate and large sample sizes, in a wide range of symmetric and skewed alternatives.     

Asymptotic normality of a conditional hazard function estimate in the single index for quasi-associated data

Abstract

The main goal of this paper is to study the estimation of the conditional hazard function of a scalar response variable Y given a hilbertian random variable X in functional single-index model. We construct an estimator of this nonparametric function and we study its asymptotic properties, under quasi-associated structure. Precisely, we establish the asymptotic normality of the constructed estimator. We carried out simulation experiments to examine the behavior of this asymptotic property over finite sample data.     

Large-sample confidence intervals for risk measures of location–scale families

For a loss distribution belonging to a location–scale family, _Fμ,σ$F_{\mu ,\sigma }$, the risk measures, Value-at-Risk and Expected Shortfall are linear functions of the parameters: μ+τσ$\mu +\tau \sigma$ where τ$\tau$ is the corresponding risk measure of the mean-zero and unit-variance member of the family. For each risk measure, we consider a natural estimator by replacing the unknown parameters μ$\mu$ and σ$\sigma$ by the sample mean and (bias corrected) sample standard deviation, respectively. The large-sample parametric confidence intervals for the risk measures are derived, relying on the asymptotic joint distribution of the sample mean and sample standard deviation. Simulation studies with the Normal, Laplace and Gumbel families illustrate that the derived asymptotic confidence intervals for Value-at-Risk and Expected Shortfall outperform those of Bahadur (1966) and Brazauskas et al. (2008), respectively. The method can also be effectively applied to Log-location-scale families whose supports are positive reals; an illustrative example is given in the area of financial credit risk.     

Nonparametric estimation of the number of components of a superposition of renewal processes

Suppose all events occurring in an unknown number (ν)$(\nu )$ of iid renewal processes, with a common renewal distribution F  , are observed for a fixed time τ$\tau$, where both ν$\nu$ and F   are unknown. The individual processes are not known a priori, but for each event, the process that generated it is identified. For example, in software reliability application, the errors (or bugs) in a piece of software are not known a priori, but whenever the software fails, the error causing the failure is identified. We present a nonparametric method for estimating ν$\nu$ and investigate its properties. Our results show that the proposed estimator performs well in terms of bias and asymptotic normality, while the MLE of ν$\nu$ derived assuming that the common renewal distribution is exponential may be seriously biased if that assumption does not hold.     

Bayesian estimation of a proportional hazards model for double-censored durations

Double-censored data consist of uncensored, left- and right-censored observations and occur in survival time analysis. In this paper, parametric Bayes estimation is investigated for a proportional hazards model with durations subject to double-censoring. We prove consistency and asymptotic normality of the posterior mean with the Bernstein–von Mises theorem. In addition, we estimate asymptotic standard errors. A simulation study shows that the finite-sample performance is similar to that of the maximum likelihood estimator. Finally, the proposed model is applied to rating transition data. The analysis suggests that an upgrade of a rating increases the duration in that class by about 10 days on average.     

Asymptotic normality for the estimator of non parametric regression model under ϕ-mixing errors

In this article, by using the Rosenthal-type inequality and the Bernstein's big-block and small-block procedure, we establish the asymptotic normality for the estimators of non parametric regression model based on ?-mixing errors. The result obtained in the article generalizes some corresponding ones for some dependent random variables.