Statistical Modeling Using Local Gaussian Approximation

Description

Statistical Modeling using Local Gaussian Approximation extends powerful characteristics of the Gaussian distribution, perhaps, the most well-known and most used distribution in statistics, to a large class of non-Gaussian and nonlinear situations through local approximation. This extension enables the reader to follow new methods in assessing dependence and conditional dependence, in estimating probability and spectral density functions, and in discrimination. Chapters in this release cover Parametric, nonparametric, locally parametric, Dependence, Local Gaussian correlation and dependence, Local Gaussian correlation and the copula, Applications in finance, and more. Additional chapters explores Measuring dependence and testing for independence, Time series dependence and spectral analysis, Multivariate density estimation,  Conditional density estimation, The local Gaussian partial correlation, Regression and conditional regression quantiles, and a A local Gaussian Fisher discriminant. Front Cover Statistical Modeling using Local Gaussian Approximation Copyright Contents Biography Preface 1 Introduction 1.1 Computer code References 2 Parametric, nonparametric, locally parametric 2.1 Introduction 2.2 Parametric density models 2.2.1 The Gaussian distribution 2.2.2 The elliptical distribution 2.2.3 The exponential family 2.3 Parametric regression models 2.3.1 Linear regression 2.3.2 Nonlinear regression and some further modeling aspects 2.4 Time series 2.5 Nonparametric density estimation 2.5.1 Nonparametric kernel density estimation 2.5.2 Bandwidth selection 2.5.3 Multivariate and conditional density estimation 2.6 Nonparametric regression estimation 2.6.1 Kernel regression estimation 2.6.2 Local polynomial estimation 2.6.3 Choice of bandwidth in regression 2.7 Fighting the curse of dimensionality 2.7.1 Additive models 2.7.2 Regression trees, splines, and MARS 2.8 Quantile regression 2.9 Semiparametric models 2.9.1 Partially linear models 2.9.2 Index models and projection pursuit 2.10 Locally parametric References 3 Dependence 3.1 Introduction 3.2 Weaknesses of Pearson's ρ 3.2.1 The non-Gaussianity issue 3.2.2 The robustness issue 3.2.3 The nonlinearity issue 3.3 The copula 3.4 Global dependence functionals and tests of independence 3.4.1 Maximal correlation 3.4.2 Measures and tests based on the distribution function 3.4.3 Distance covariance 3.4.4 The HSIC measure of dependence 3.4.5 Density-based tests of independence 3.5 Test functionals generated by local dependence relationships References 4 Local Gaussian correlation and dependence 4.1 Introduction 4.2 Local dependence 4.2.1 Quadrant dependence 4.2.2 Local measures of dependence 4.3 Local Gaussian correlation 4.4 Limit theorems 4.4.1 Asymptotic theory for b fixed 4.4.2 Asymptotic theory as b –>0 4.5 Properties 4.5.1 Some general properties of the local Gaussian correlation 4.5.2 Independence and functional dependence 4.5.3 The Rényi criteria 4.5.4 Linear transformation and symmetries 4.6 Examples 4.6.1 Simulated examples 4.6.2 A real-data example 4.7 Transforming the marginals: Normalized local correlation 4.7.1 Examples of the normalized LGC 4.8 Some practical considerations 4.8.1 Estimating the standard error of estimates 4.8.2 Choosing the bandwidth 4.9 The p-dimensional case 4.10 Proof of asymptotic results 4.10.1 Non-normalized 4.10.2 Normalized 4.10.3 Proof of the linear transformation result References 5 Local Gaussian correlation and the copula 5.1 Introduction 5.2 Local Gaussian correlation for copula models 5.2.1 Tail behavior 5.2.2 Normalized local Gaussian correlation 5.3 Examples 5.3.1 Archimedean copulas 5.3.2 Elliptical copulas 5.4 Recognizing copulas by goodness-of-fit 5.4.1 Uniform pseudo-observations 5.4.2 Gaussian pseudo-observations 5.4.3 A goodness-of-fit test based on local Gaussian correlation 5.4.4 Choice of bandwidth 5.4.5 Simulation study 5.4.6 Visualizing departures from H0 5.5 A real-data study References 6 Applications in finance 6.1 Introduction 6.2 Conditional correlation and the bias problem 6.2.1 Why local Gaussian correlation is better 6.3 Empirical analysis of dependence of financial returns 6.3.1 Daily stock index returns 6.3.2 Monthly stock index returns 6.3.3 Dependence between commodities, bonds, and stocks 6.4 The portfolio allocation problem 6.4.1 Portfolio allocation using the LGC 6.4.2 Empirical example 6.5 Financial contagion 6.5.1 A review of measures of interdependence and contagion 6.5.2 A bootstrap test for contagion based on the LGC 6.5.3 Example: Testing for contagion in the 1987 US stock market crash References 7 Measuring dependence and testing for independence 7.1 Introduction 7.2 Testing of independence in iid pairs of variables using local correlation functionals 7.2.1 Bootstrap test of independence 7.2.2 Local independence testing 7.2.3 Example: Bivariate t-distribution 7.2.4 Example: Aircraft data 7.3 Testing for serial independence in time series 7.3.1 The bootstrap 7.3.2 Example: GARCH(1,1) 7.3.3 Example: Exchange rates 7.3.4 Validity of the bootstrap 7.4 Describing nonlinear dependence and tests of independence for two time series 7.4.1 Local Gaussian cross-correlation 7.4.2 Asymptotic theory of the parameter estimates 7.4.3 Test of independence 7.4.4 The bootstrap and its validity 7.4.5 Example: Bivariate GARCH(1,1) 7.4.6 Example: Financial returns 7.5 Proofs 7.5.1 Proof of Theorems 7.9 and 7.10 7.5.2 Proof of Theorems 7.11 and 7.12 and Corollary 7.1 References 8 Time series dependence and spectral analysis 8.1 Introduction 8.2 Local Gaussian spectral densities 8.2.1 The local Gaussian correlations 8.2.1.1 Local Gaussian correlation, general version 8.2.1.2 Local Gaussian correlation, normalized version 8.2.2 The local Gaussian spectral densities 8.2.3 Estimation 8.2.4 Asymptotic theory for f̂xm(ω) and f̂zm(ω) 8.2.4.1 An assumption for Xt and the function u(v,θ(x)) 8.2.4.2 Assumptions for n, m, and b 8.2.5 Convergence theorems for f̂xm(ω) 8.3 Visualizations and interpretations 8.3.1 The input parameters and some other technical details 8.3.2 Estimation aspects for the given parameter configuration 8.3.3 Sanity testing the implemented estimation algorithm 8.3.3.1 Gaussian white noise 8.3.3.2 Some trigonometric examples 8.3.4 Real data and a fitted GARCH-type model 8.3.4.1 The real data example 8.3.4.2 A heatmap/distance plot for the dmbp-data 8.3.4.3 A GARCH-type model 8.3.4.4 Local testing of fitted models References 9 Multivariate density estimation 9.1 Introduction 9.2 Description of the estimator 9.2.1 Estimation of the marginals 9.2.2 Estimation of the joint dependence function 9.3 Asymptotic theory 9.4 Bandwidth selection 9.5 An example 9.6 Investigating performance in the multivariate case 9.7 A more flexible version of the LGDE 9.8 Proofs 9.8.1 Proof of Theorem 9.1 9.8.2 Proof of Theorems 9.2 and 9.3 9.8.3 Proof of Theorem 9.4 References 10 Conditional density estimation 10.1 Introduction 10.2 Estimating the conditional density 10.3 Asymptotic theory for dependent data 10.4 Examples 10.4.1 Stock data 10.4.2 Simulations 10.4.2.1 Simulated data with relevant variables 10.4.3 Simulated data from a heavy-tailed distribution 10.4.4 Simulated data with irrelevant variables 10.5 Proof of theorems 10.5.1 Proof of Theorem 10.1 10.5.2 Proof of Theorem 10.2 References 11 The local Gaussian partial correlation 11.1 Introduction 11.2 The local Gaussian partial correlation 11.2.1 Definition 11.3 Properties 11.4 Estimation of the LGPC by local likelihood 11.4.1 Estimation of R(z) when p = 3 and X(2) is a scalar 11.4.2 Estimation of R(z) when X(2) is a vector 11.5 Asymptotic theory 11.6 Examples 11.7 Testing for conditional independence 11.7.1 The recent fauna of nonparametric tests 11.7.2 A test for conditional independence 11.7.3 Comparing with other tests 11.8 The multivariate LGPC 11.8.1 Definition 11.8.2 Some particular cases References 12 Regression and conditional regression quantiles 12.1 Introduction 12.2 Comparison with additive regression modeling 12.3 Local Gaussian regression estimation 12.4 Asymptotic normality 12.5 Example 12.6 Conditional quantiles 12.6.1 Distribution of the conditional empirical distribution function 12.6.2 Convergence rate 12.6.3 Distribution of conditional quantiles 12.7 Proof References 13 A local Gaussian Fisher discriminant 13.1 Introduction 13.1.1 Background 13.1.2 Estimating densities and discriminants 13.2 A local Gaussian Fisher discriminant 13.3 Some asymptotics of Bayes risk 13.4 Choice of bandwidth 13.5 Illustrations 13.5.1 Simulations 13.5.2 Illustration: Fraud detection 13.6 Summary remark References Author index Subject index Back Cover