Variance function estimation in multivariate nonparametric regression with fixed design

Pricing of American options in discrete time by means of regression-based Monte Carlo methods is considered in this thesis. Thereby, the options are allowed to be based on several underlyings (so-called basket options). It is assumed that the underlyings price processes satisfy the Markov property and consequently are Markov processes. In this dissertation, Monte Carlo methods are used to generate artificial sample paths of these price processes, and subsequently adaptive nonparametric least squares regression estimates are used to estimate the so-called continuation values from these data. The continuation values describe the mean values of the American options for given values of the underlyings at time t, subject to the constraint that the options are not exercised at time t, but optimally exercised in the future. Adaptive least squares neural networks as well as adaptive least squares spline estimates are used as nonparametric regression estimates. In conjunction with the pricing of American options, these estimates are theoretically analysed, and results with respect to consistency and rate of convergence are derived. Finally, the pricing of American options on simulated data by means of the estimates is illustrated.

In the present paper, we extend the work of Slaoui (Stat Sin 30:417–437, 2020) in the case of strong mixing data. Since, we are interested in nonparametric regression estimation, we focus on well adapted dependence structures based on mixing type conditions. We study the properties of these regression estimators and compare them with the nonparametric non-recursive regression estimator. The bias, variance and mean squared error are computed explicitly. We showed that using a selected wild bootstrap bandwidth procedure and a special stepsize, our proposed recursive regression estimators allowed us to obtain quite similar results compared to the non-recursive regression estimator under $$\alpha $$ -mixing condition in terms of estimation error and much better in terms of computational costs.

Nonparametric curves occur in a wide variety of contexts, including repeated measurements mean profiles, survivor functions and in different types of smoothing techniques such as nonparametric regression and density estimation. In some cases confidence bands can be attached to these curves as an indication of the variability of estimation. This is more difficult in the case of nonparametric regression and density estimation where bias is present. In the important case of the comparison of two curves, attention can be focused instead on whether there are differences between the curves. In this paper, the idea of a reference band for the comparison of two curves is introduced The band is derived from the standard error of the difference between the two curves at each point. The bands have a simple hypothesis testing interpretation which applies equally well to smoothing methods where bias occurs. It does not remove the need for a global test of effects of interest, but it can be very useful as a graphical means of exploring where any identified differences might lie, or of explaining why apparent differences do not actually contribute strong evidence of statistical significance to the global comparison of curves. Reference bands for equality are derived and explored in a variety of settings. Reference bands for parallelism are also derived for nonparametric regression models. Reference bands for parametric models, against which nonparametric regression and density estimates can be compared, are derived for linearity and for normality.

Estimation of heterogeneous causal effects - i.e., how effects of policies and treatments vary across subjects - is a fundamental task in causal inference. Many methods for estimating conditional average treatment effects (CATEs) have been proposed in recent years, but questions surrounding optimality have remained largely unanswered. In particular, a minimax theory of optimality has yet to be developed, with the minimax rate of convergence and construction of rate-optimal estimators remaining open problems. In this paper we derive the minimax rate for CATE estimation, in a Hölder-smooth nonparametric model, and present a new local polynomial estimator, giving high-level conditions under which it is minimax optimal. Our minimax lower bound is derived via a localized version of the method of fuzzy hypotheses, combining lower bound constructions for nonparametric regression and functional estimation. Our proposed estimator can be viewed as a local polynomial R-Learner, based on a localized modification of higher-order influence function methods. The minimax rate we find exhibits several interesting features, including a non-standard elbow phenomenon and an unusual interpolation between nonparametric regression and functional estimation rates. The latter quantifies how the CATE, as an estimand, can be viewed as a regression/functional hybrid.

Nonparametric regression models offer a way to understand and quantify relationships between variables without having to identify an appropriate family of possible regression functions. Although many estimation methods for these models have been proposed in the literature, most of them can be highly sensitive to the presence of a small proportion of atypical observations in the training set. A review of outlier robust estimation methods for nonparametric regression models is provided, paying particular attention to practical considerations. Since outliers can also influence negatively the regression estimator by affecting the selection of bandwidths or smoothing parameters, a discussion of robust alternatives for this task is also included. Using many of the “classical” nonparametric regression estimators (and their robust counterparts) can be very challenging in settings with a moderate or large number of explanatory variables, so recently proposed robust nonparametric regression methods that scale well with a growing number of covariates are also discussed.

We consider the problem of testing significance of predictors in multivariate nonparametric quantile regression. A stochastic process is proposed, which is based on a comparison of the responses with a nonparametric quantile regression estimate under the null hypothesis. It is demonstrated that under the null hypothesis this process converges weakly to a centered Gaussian process and the asymptotic properties of the test under fixed and local alternatives are also discussed. In particular we show, that - in contrast to the nonparametric approach based on estimation of $L^{2}$-distances - the new test is able to detect local alternatives which converge to the null hypothesis with any rate $a_{n}\\to 0$ such that $a_{n}\\sqrt{n}\\to\\infty$ (here $n$ denotes the sample size). We also present a small simulation study illustrating the finite sample properties of a bootstrap version of the corresponding Kolmogorov-Smirnov test.

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.

No. Most goodness-of-fit (GOF) tests attempt to discern a preferred weighting using either absolute or relative errors in the back-calculated calibration x values. However, the former are predisposed to select constant weighting and the latter 1/x2 or 1/y2 weighting, no matter what the true weighting should be. Here, I use Monte Carlo simulations to quantify the flaws in GOF tests and show why they falsely prefer their predisposition weighting. The weighting problem is solved properly through variance function (VF) estimation from replicate data, conveniently separating this from the problem of selecting a response function (RF). Any weighting other than inverse-variance must give loss of precision in the RF parameters and in the estimates of unknowns x0. In particular, the widely used 1/x2 weighting, if wrong, not only sacrifices precision but even worse, appears to give better precision at small x, leading to falsely optimistic estimates of detection and quantification limits. Realistic VFs typically become constant in the low-x, low-y limit. Thus, even when 1/x2 weighting is correct at large signal, the neglect of the constant variance component at small signal again gives too-small detection and quantification limits. VF estimation has been disparaged as too demanding of data. Why this is not true is demonstrated with Monte Carlo simulations that show only a few percent increase in calibration parameter uncertainties when the VF is estimated from just three replicates at each of six calibration x values. This point is further demonstrated using examples from the recent literature.

Tests used for such purposes as determining educational quality, defining educational needs, hiring an employee, student selection and placement and performing guidance and clinic services have an important place in education and psychology. Of course, they should have certain psychometric features related to test scores' validity and reliability. Various test theories have helped to create more valid and reliable measurements and, as a result, to make better decisions regarding individuals. In education and psychology, Classical Test Theory (CTT) and Item Response Theory (IRT) are both widely used. CTT assumes that an individual's observed score is the total of the true score and the error score, while IRT estimates an individual's ability or latent trait from responses to test items (Embretson & Reise, 2000).When IRT assumptions and model-data fit are ensured, item and ability parameters' invariance occurs; this is known as the most important advantage IRT has over CTT. Item and ability parameters' invariance means estimating ability parameters independently of item sample and estimating item parameters independently of ability sample. IRT's invariance feature makes it very practicable in many applications, for instance, test development, computerized adaptive testing, bias studies, test equating and item mapping (Hambleton & Swaminathan, 1985). IRT is classified under two main categories as parametric IRT (PIRT) and nonparametric IRT (NIRT) (Olivares, 2005; Sijtsma & Molenaar, 2002).To analyse ordered items, such as Likert-type attitude items, partial credit cognitive items or not ordered graded items such as multiple-choice test items, item response models are developed towards polytomous items in IRT (Ostini & Nering, 2006). In these models developed for polytomous items, a non-linear relationship between an individual's latent trait and the possibility of choosing a certain category of item answer is explained (Embretson & Reise, 2000). Graded Response Model (GRM), part of IRT models developed for polytomous items, is often preferred by researchers for applications since it is more useful in presentations, portfolios, essays and Likert-type items with ordered item categories (DeMars, 2010; Ostini & Nering, 2006). To scale tests that consist of polytomous items by making true estimates according to GRM, evaluating PIRT's assumptions and model-data fit is necessary. And to provide these assumptions and model-data fit, large samples are needed. At this point, NIRT models draw attention because they provide a practical advantage in determining psychometric properties of tests with fewer items and respondents (Stout, 2001).NIRT models are defined as statistical scaling methods that require fewer assumptions than PIRT models for measuring persons and items (Stochl, 2007). With their wide application area, NIRT models are used in ordinal scales, applied research areas, sociology, marketing research and health research on quality of life (Sijtsma, 2005). The literature reveals that two models, namely, the Mokken model and nonparametric regression estimation models, are employed. These two models are themselves divided into sub-models. The Mokken model consists of the sub-models Monotone Homogeneity Model (MHM) and the Double Monotonicity Model (DMM). Nonparametric regression estimation models consist of such sub-models as the Kernel Smoothing Approach Model (KSAM), the Isotonic Regression Estimation and the Smoothed Isotonic Regression Estimation models (Lee, 2007; Sijtsma & Molenaar, 2002). Along with theoretical studies being conducted, new sub-models are being added to nonparametric regression estimation models.As a NIRT model, MHM requires unidimensionality, local independence and monotonicity assumptions, and it defines the relationship that latent variables and items with homogeneous (unidimensional) and monotone item characteristic curve (ICC) have (Meijer & Baneke, 2004; Sijtsma & Molenaar, 2002). …

Bagging cross-validated bandwidth selection in nonparametric regression estimation with applications to large-sized samples

Despite its great importance in practice, nonparametric regression estimation in continuous time has not been much studied up to now. The current chapter is perhaps the first general work on that topic.KeywordsContinuous TimeSample PathRegression EstimationKernel EstimatorDensity CaseThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Consider three random variables, Y , X1 and X2, having some unknown trivariate distribution and let n2j (j = 1, 2) be some measure of the strength of association between Y and Xj. When n2j is taken to be Pearson’s correlation numerous methods for testing Ho : n21 = n22 have been proposed. However, Pearson’s correlation is not robust and the methods for testing H0 are not level robust in general. This article examines methods for testing H0 based on a robust fit. The first approach assumes a linear model and the second approach uses a nonparametric regression estimator that provides a flexible way of dealing with curvature. The focus is on the Theil-Sen estimator and Cleveland’s LOESS smoother. It is found that a basic percentile bootstrap method avoids Type I errors that exceed the nominal level. However, situations are identified where this approach results in Type I error probabilities well below the nominal level. Adjustments are suggested for dealing with this problem.

Performance criteria and discrimination of extreme undersmoothing in nonparametric regression

The problem of estimating variance due to regression and due to error in the context of nonparametric regression is considered. An estimator is proposed on the basis of the difference of the mean total sum of squares of the data and a nonparametric estimate for the residual variance. Asymptotic expressions are derived for the expectation and variance of the estimator, and a number of simulations have been performed to assess its finite-sample behaviour. An adaptation of the estimator for residual variance to reduce bias in situations with a high signal-to-noise ratio is also proposed and evaluated. An application of the method to a data analysis problem concerning ‘tracking’ in the growth of children, the motivation behind this work, is then demonstrated. Possible extensions of the estimator to more complicated situations are also considered: nonparametric regression estimation in higher dimensions and estimating that part of variance in a regression function which is orthogonal to a linear parametric model.

The single-index model is an important tool in multivariate nonparametric regression. This paper deals with M-estimators for the single-index model. Unlike the existing M-estimator for the single-index model, the unknown link function is approximated by B-spline and M-estimators for the parameter and the nonparametric component are obtained in one step. The proposed M-estimator of unknown function is shown to attain the convergence rate as that of the optimal global rate of convergence of estimators for nonparametric regression according to Stone (Ann Stat 8:1348–1360, 1980; Ann Stat 10:1040–1053, 1982), and the M-estimator of parameter is $$\sqrt{n}$$ -consistent and asymptotically normal. A small sample simulation study showed that the M-estimators proposed in this paper are robust. An application to real data illustrates the estimator’s usefulness.