Nonparametric AMOC Changepoint Tests for Stochastically Ordered Alternatives

The problem considered is that of testing on the basis of a finite sequence of independent observations if all the observations have the same distribution versus the alternative that there is a unique change in the distribution and i.i.d. observations after the change are stochastically larger. The distributions before and after the possible change are continuous but not fully specified. We suggest a family of nonparametric tests based on ranks. Asymptotic approximations for the significance level of the test are obtained analytically. Monte Carlo experiments show that the rate of convergence of our asymptotics is fast.     

Change-Point Estimation in Long Memory Nonparametric Models with Applications

We consider the estimation of a change point or discontinuity in a regression function for random design model with long memory errors. We provide several change-point estimators and investigate the consistency of the estimators. Using the fractional ARIMA process as an example of long memory process, we report a small Monte Carlo experiment to compare the performance of the estimators in finite samples. We finish by applying the method to a climatological data example.     

Nonparametric Principal Components Regression

Principal components regression (PCR) is used in resolving the multicollinearity problem but specification bias occurs due to the selection only of the important principal components to be included resulting in the deterioration of predictive ability of the model. We propose the PCR in a nonparametric framework to address the multicollinearity problem while minimizing the specification bias that affects predictive ability of the model. The simulation study illustrated that nonparametric PCR addresses the multicollinearity problem while retaining higher predictive ability relative to parametric principal components regression model.     

Nonparametric estimation of a switching regression model

High-dimensional data often exhibit multi-collinearity, leading to unstable regression coefficients. To address sample selection bias and problems associated with high dimensionality, principal components were extracted and used as predictors in a switching regression model. Since principal component regression often results to decline in predictive ability due to the selection of few principal components, we formulate the model with nonparametric function of principal components in lieu of individual predictors. Simulation studies indicated better predictive ability for nonparametric principal component switching regression over the parametric counterpart while mitigating the adverse effects of multi-collinearity and high dimensionality.     

Nonparametric estimation of response coefficients

In this paper we propose a nonparametric kernel method of estimating response coefficients in the stochastic regressors model. The method is straightforward, and the estimator is easy to calculate. The asymptotic normality of the proposed estimator is established, and an illustrative example is presented.     

Nonparametric regression for threshold data

Consider a detector which records the times at which the endogenous variable of a nonparametric regression model exceeds a certain threshold. If the error distribution is known, the regression function can still be identified from these threshold data. The author constructs estimators for the regression function that are transformations of kernel estimators. She determines the bandwidth that minimizes the asymptotic mean average squared error. Her investigation was motivated by recent work on stochastic resonance in neuroscience and signal detection theory, where it was observed that detection of a subthreshold signal is enhanced by the addition of noise. The author compares her model with several others that have been proposed in the recent past.     

Nonparametric geometric outlier detection

Outlier detection is a major topic in robust statistics due to the high practical significance of anomalous observations. Many existing methods, however, either are parametric or cease to perform well when the data are far from linearly structured. In this paper, we propose a quantity, Delaunay outlyingness, that is a nonparametric outlyingness score applicable to data with complicated structure. The approach is based on a well‐known triangulation of the sample, which seems to reflect the sparsity of the pointset to different directions in a useful way. We derive results on the asymptotic behavior of Delaunay outlyingness in case of a sufficiently simple set of observations. Simulations and an application to empirical data are also discussed.     

Nonparametric control chart based on change-point model

A change-point control chart for detecting shifts in the mean of a process is developed for the case where the nominal value of the mean is unknown but some historical samples are available. This control chart is a nonparametric chart based on the Mann–Whitney statistic for a change in mean and adapted for repeated sequential use. We do not require any knowledge of the underlying distribution such as the normal assumption. Particularly, this distribution robustness could be a significant advantage in start-up or short-run situations where we usually do not have knowledge of the underlying distribution. The simulated results show that our approach has a good performance across the range of possible shifts and it can be used during the start-up stages of the process.      

Nonparametric Methods in Continuous Time Model Specification

Some popular parametric diffusion processes have been assumed as such underlying diffusion processes. This paper considers an important case where both the drift and volatility functions of the underlying diffusion process are unknown functions of the underlying process, and then proposes using two novel testing procedures for the parametric specification of both the drift and diffusion functions. The finite-sample properties of the proposed tests are assessed through using data generated from four popular parametric models. In our implementation, we suggest using a simulated critical value for each case in addition to the use of an asymptotic critical value. Our detailed studies show that there is little size distortion when using a simulated critical value while the proposed tests have some size distortions when using an asymptotic critical value in each case.     

Nonparametric weighted symmetry tests

Weighted symmetry is an extension of the classical notion of symmetry in which the tails of a distribution are similar, up to a scaling factor. The authors develop test statistics of weighted symmetry based on empirical processes. The finite‐dimensional distributions of the proposed statistics are either non‐parametric or conditionally nonparametric, according as the parameters of weighted symmetry are known or estimated. Asymptotically, the distributions of the processes behave like Brownian bridges or motions, leading to familiar distributions for the proposed test statistics. The authors also establish the asymptotic normality of Hodges‐Lehmann type estimators based on a generalization of the Wilcoxon signed rank test. Furthermore, they propose density estimators in mat setting.     

Nonparametric Methods in Continuous Time Model Specification

Some popular parametric diffusion processes have been assumed as such underlying diffusion processes. This paper considers an important case where both the drift and volatility functions of the underlying diffusion process are unknown functions of the underlying process, and then proposes using two novel testing procedures for the parametric specification of both the drift and diffusion functions. The finite-sample properties of the proposed tests are assessed through using data generated from four popular parametric models. In our implementation, we suggest using a simulated critical value for each case in addition to the use of an asymptotic critical value. Our detailed studies show that there is little size distortion when using a simulated critical value while the proposed tests have some size distortions when using an asymptotic critical value in each case.     

Nonparametric estimation of copula functions for dependence modelling

Copulas characterize the dependence among components of random vectors. Unlike marginal and joint distributions, which are directly observable, the copula of a random vector is a hidden dependence structure that links the joint distribution with its margins. Choosing a parametric copula model is thus a nontrivial task but it can be facilitated by relying on a nonparametric estimator. Here the authors propose a kernel estimator of the copula that is mean square consistent everywhere on the support. They determine the bias and variance of this estimator. They also study the effects of kernel smoothing on copula estimation. They then propose a smoothing bandwidth selection rule based on the derived bias and variance. After confirming their theoretical findings through simulations, they use their kernel estimator to formulate a goodness-of-fit test for parametric copula models.     

Nonparametric analysis of aggregate loss models

This paper describes a nonparametric approach to make inferences for aggregate loss models in the insurance framework. We assume that an insurance company provides a historical sample of claims given by claim occurrence times and claim sizes. Furthermore, information may be incomplete as claims may be censored and/or truncated. In this context, the main goal of this work consists of fitting a probability model for the total amount that will be paid on all claims during a fixed future time period. In order to solve this prediction problem, we propose a new methodology based on nonparametric estimators for the density functions with censored and truncated data, the use of Monte Carlo simulation methods and bootstrap resampling. The developed methodology is useful to compare alternative pricing strategies in different insurance decision problems. The proposed procedure is illustrated with a real dataset provided by the insurance department of an international commercial company.     

Nonparametric estimation of discrete hazard functions

A smoothing procedure for discrete time failure data is proposed which allows for the inclusion of covariates. This purely nonparametric method is based on discrete or continuous kernel smoothing techniques that gives a compromise between the data and smoothness. The method may be used as an exploratory tool to uncover the underlying structure or as an alternative to parametric methods when prediction is the primary objective. Confidence intervals are considered and alternative techniques of cross validation based choices of smoothing parameters are investigated.     

Nonparametric estimation of mean residual quantile function under right censoring

In this paper, we develop non-parametric estimation of the mean residual quantile function based on right-censored data. Two non-parametric estimators, one based on the empirical quantile function and the other using the kernel smoothing method, are proposed. Asymptotic properties of the estimators are discussed. Monte Carlo simulation studies are conducted to compare the two estimators. The method is illustrated with the aid of two real data sets.     

Two non parametric methods for change-point detection in distribution

The change-point detection problem is determining whether a change has taken place. Two non parametric methods based on empirical likelihood and the likelihood ratio are proposed for detecting a change-point problem in distributions for independent observations. Numerical studies are carried out to evaluate the performance of the proposed methods. The simulation results demonstrate that the proposed methods are robust, that is, they perform well regardless of whether the observations are from the same distribution family.     

Estimation of a change point in the variance function based on the χ2-distribution

ABSTRACT

Let us consider that the variance function or its νth derivative in a regression model has a change/discontinuity point at an unknown location. To use the local polynomial fits, the log-variance function which break the positivity is targeted. The location and the jump size of the change point are estimated based on a one-sided kernel-weighted local-likelihood function which is provided by the χ²-distribution. The whole structure of the log-variance function is then estimated using the data sets split by the estimated location. Asymptotic results of the proposed estimators are described. Numerical works demonstrate the performances of the methods with simulated and real examples.     

Nonparametric estimation of the hazard function under dependence conditions

The estimation of the hazard rate has a great number of practical appli¬cations in dependence situations (seismicity analysis, reliability, economics), Based on kernel estimates of the density and the distribution function, we study the properties of the nonparametric estimator of the hazard function as-sociated with a strongly mixing time series. We prove consistency and asymp¬totic normality properties, and a cross-validation method for the smoothing parameter selection is studied. Some simulations and a practical application to real data are also shown.