M Robustified Additive Nonparametric Regression

Recently, the rapid growth of users has considered ending their Facebook use or actually done so due to privacy issues, concerns about overexposing personal information online, or interpersonal interventions or distress. This study conducted research on users of Facebook, the most representative social networking site in the world, to determine the effect of Big Five personality traits, self-monitoring, and Facebook usage behaviors on continuance usage intention. A total of 498 valid samples were collected from a questionnaire survey and a multivariate nonparametric regression technique -multivariate adaptive regression splines-was employed to screen critical variables and predict users' continuance usage intention in the future. The results revealed that when the main motivation for Facebook use is not documenting life or expressing emotions, continuance usage intention notably decreases. In addition, the degree of agreeableness has a moderating effect on the relationship between Facebook usage behaviors and continuance usage intention.

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.

For tting penalized Fourier regression, Wu (2013) provide some R-functions in the settings of univariate nonparametric regression, partial linear models, additive models, and nonparametric logistic and Poisson regression. We improve these functions to follow the R-function format so that users could apply generic R-functions summary, plot, lines, fitted, predict, and residuals. In addition, we write some R-function to select penalty parameters by AICc or BIC. Some data examples are used to illustrate our R-functions. A small simulation study is conducted to compare the penalized Fourier to spline approaches.

How well can a regression function be estimated if the distribution of the (random) design is concentrated on a finite set?

Major specialization at High School is aimed to gives opened opportunity for students to choose subject that are interest and develop their potential in accordance with the abilities, interests, talents, and personality. Major specialization at High school is influenced by some factors. To detect those factors, used biner logistic regression method and Multivariate Adaptive Regression Spline (MARS). Biner Logistic Regression is method that describes relationship between dependent variable and some independent variable, with independent variable has been coded 1 as representing the presence of the characteristic, and 0 as representing the absence of the characteristic. MARS is multivariate nonparametric regression method that development of Recursive Partitioning Regression (RPR) method and Spline method for high dimensional data that produces accurate prediction and continuous models on knots. Both of the methods are compared to know the best method used in research. From the result of analysis using biner logistic regression method and MARS, concluded that major specialization has been influenced by mathematic score, science score and relationship between students and friends. From proportion test, concluded classification that formed by regression logistic is as good as by MARS. Keywords : Major specialization at High School, Biner Logistic Regression, Mutlivariate Adaptive Regression Spline (MARS), Clasification

Variance function estimation in multivariate nonparametric regression with fixed design

In the application of extreme value analysis it is usually assumed that the size of the samples from which the extreme values are obtained is sufficiently large for the asymptotic extreme value distribution to be used. The necessary sample size depends upon the population distribution and this is generally not known; but assuming a Weibull distribution, which is often fitted to wind speed and wave height data, it is shown that the rate of convergence is rapid and that the asymptotic distribution may be used for a sample size as small as ten. An ‘exponential approximation’ for the distribution of maxima is sometimes confused with the extreme value distribution. This approximate form is derived for the Weibull distribution and the essential difference between it and the asymptotic extreme value distribution is explained.

In investigations on the behaviour of robust estimators, typically their consistency and their asymptotic normality are studied as a necessity. Their rates of convergence, however, are often given less weight. We show here that the rate of convergence of a multivariate robust estimator to its true value plays an important role when using the estimator in procedures for identifying outliers in multivariate data.

This paper is about optimal estimation of the additive components of a nonparametric, additive isotone regression model. It is shown that asymp- totically up to first order, each additive component can be estimated as well as it could be by a least squares estimator if the other components were known. The algorithm for the calculation of the estimator uses backfitting. Conver- gence of the algorithm is shown. Finite sample properties are also compared through simulation experiments. In this paper we discuss nonparametric additive monotone regression models. We discuss a backfitting estimator that is based on iterative application of the pool adjacent violator algorithm to the additive components of the model. Our main result states the following oracle property. Asymptotically up to first order, each additive component is estimated as well as it would be (by a least squares estimator) if the other components were known. This goes beyond the classical finding that the estimator achieves the same rate of convergence, independently of the number of additive components. The result states that the asymptotic distribution of the estimator does not depend on the number of components. We have two motivations for considering this model. First of all we think that this is a useful model for some applications. For a discussion of isotonic additive regres- sion from a more applied point, see also Bacchetti (1), Morton-Jones et al. (32) and De Boer, Besten and Ter Braak (7). But our main motivation comes from statistical theory. We think that the study of nonparametric models with several nonparamet- ric components is not fully understood. The oracle property that is stated in this paper for additive isotone models has been shown for smoothing estimators in some other nonparametric models. This property is expected to hold if the estimation of the different nonparametric components is based on local smoothing where the lo- calization takes place in different scales. An example are additive models of smooth functions where each localization takes place with respect to another covariate. In Mammen, Linton and Nielsen (28) the oracle property has been verified for the lo- cal linear smooth backfitting estimator. As local linear estimators, also the isotonic least squares is a local smoother. The estimator is a local average of the response variable but in contrast to local linear estimators the local neighborhood is chosen � Research of this paper was supported by the Deutsche Forschungsgemeinschaft project MA

Consistent hypothesis testing in semiparametric and nonparametric models for econometric time series

Olfactometer experiments are used to determine the effect of odours on the behaviour of organisms such as insects or nematodes, and typically result in data comprising many groups of small counts, overdispersed relative to the multinomial distribution. Overdispersion reflects a lack of independence or heterogeneity among individuals and can lead to statistics having larger variances than expected and possible losses of efficiency. In this thesis, some distributions which consist of generalisations of the multinomial distribution have been developed. These models are based on non-homogeneous Markov chain theory, take the overdispersion into account, and potentially provide a physical interpretation of the overdispersion seen in olfactometer data. Some inference aspects are considered, including comparison of the asymptotic relative efficiencies of three different sampling schemes. The fact that the empirical distributions well approximate the corresponding asymptotic distributions is checked. Observable differences in parameter estimates between data generated under different hypotheses are also studied. Finally, different models intended to shed light on various aspects of the data and/or the experiment procedure, are applied to three real olfactometer datasets.

This thesis contains three essays on inference in econometric models. Chapter 1 considers the question of bootstrap inference for Propensity Score Matching. Propensity Score Matching, where the propensity scores are estimated in a first step, is widely used for estimating treatment effects. In this context, the naive bootstrap is invalid (Abadie and Imbens, 2008). This chapter proposes a novel bootstrap procedure for this context, and demonstrates its consistency. Simulations and real data examples demonstrate the superior performance of the proposed method relative to using the asymptotic distribution for inference, especially when the degree of overlap in propensity scores is poor. General versions of the procedure can also be applied to other causal effect estimators such as inverse probability weighting and propensity score subclassification, potentially leading to higher order refinements for inference in such contexts. Chapter 2 tackles the question of inference in incomplete econometric models. In many economic and statistical applications, the observed data take the form of sets rather than points. Examples include bracket data in survey analysis, tumor growth and rock grain images in morphology analysis, and noisy measurements on the support function of a convex set in medical imaging and robotic vision. Additionally, nonparametric bounds on treatment effects under imperfect compliance can be expressed by means of random sets. This chapter develops a concept of nonparametric likelihood for random sets and its mean, known as the Aumann expectation, and proposes general inference methods by adapting the theory of empirical likelihood. Chapter 3 considers inference on the cumulative distribution function (CDF) in the classical measurement error model. It proposes both asymptotic and bootstrap based uniform confidence bands for the estimator of the CDF under measurement error. The proposed techniques can also be used to obtain confidence bands for quantiles, and perform various CDF-based tests such as goodness-offit tests for parametric models of densities, two sample homogeneity tests, and tests for stochastic dominance; all for the first time under measurement error.

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.

We propose a dynamic differential Bayesian networks (DDBNs) and nonparametric regression model. This model is an extended model of traditional dynamic Bayesian networks (DBNs), which can incorporate temporal information in a natural way and directly handle real-valued data obtained from microarrays without any transformation. In addition, it can cope with differential information between gene expression levels, without any loss to the traditional advantage, i.e., the capability of estimating non-linear relationships between genes. We apply DDBNs to analyze simulated data and real data, i.e., Saccharomyces cerevisiae cell cycle gene expression data. We have confirmed the effectiveness of our approach in the sense that some edges have been successfully detected only by DDBNs, not by DBNs.

In theory, recent results in nonparametric regression show that neural network estimates are able to achieve good rates of convergence provided suitable assumptions on the structure of the regression function are imposed. However, these theoretical analyses cannot explain the practical success of neural networks since the theoretically studied estimates are defined by minimizing the empirical L_2 risk over a class of neural networks and in practice, solving this kind of minimization problem is not feasible. Consequently, the neural networks examined in theory cannot be implemented as they are defined. This means that neural network in applications differ from the ones that are analyzed theoretically. In this thesis we narrow the gap between theory and practice. We deal with neural network regression estimates for (p,C)-smooth regression functions m that satisfy a projection pursuit model. We construct three implementable neural network estimates and show that each of them achieve up to a logarithmic factor the optimal univariate rate of convergence. Firstly, for univariate regression functions with p contained in [-1/2,1] we construct a neural network estimate with one hidden layer where the weights are learned via gradient descent. The starting weights are randomly chosen from an interval independently of the data. The interval is large enough to guarantee that the estimate is close to a piecewise constant approximation. Secondly, for multivariate regression functions with p contained in (0,1] we construct a neural network estimate with one hidden layer where the weights are learned via gradient descent. The initial weights are chosen from specific intervals dependently on the data and the projection directions. This choice guarantees that the estimate is close to a piecewise constant approximation. The projection directions are repeatedly chosen randomly. Lastly, for multivariate regression functions with p>0 we construct a multilayer neural network estimate. The value of the inner weights are prescribed dependently on the projection directions by a new approximation result for a projection pursuit model by piecewise polynomials. The outer weights are chosen by solving a linear equation system. The projection directions are repeatedly chosen randomly. Since we are able to show a rate of convergence that is independent of the dimension of the data our second and third estimates are able to circumvent the curse of dimensionality.