- 1. Study of the basic concepts of nonparametric statistics. 2. Study mathematical background of the nonparametric statistical methods.
- Understanding the methodology of the probability density estimation
- Understanding the methods of nonparametric regression
- Knowing the nonparametric tests for solving various statistical problems
- Understanding the wavelet approach
- Part I: Probability density estimation1. Statement of the problem. Estimation of the distribution function. 2. Histogram as a density estimate. Bias-variance tradeoff. General concept and particular results for the histogram. Bias-variance decomposition for histograms. Minimization of AMISE for histogram: Scott and Friedman- Diaconis rules for bandwidth selection. Other ideas for the choice of the amount of bins: Sturges rule. “Pretty” procedure in the R language. 3. Kernel density (Parzen-Rosenblatt) estimates. Bias-variance decomposition for kernel estimates. Minimization of AMISE for kernel estimates with respect to the kernel: Epanechnikov kernel. The notion of the kernel efficiency. Minimization of AMISE for kernel estimates with respect to the bandwidth: nrd and nrd0 options. (Unbiased) cross-validation for the probability density estimates. 4. Rates of convergence for histogram and kernel density estimates. Lower bounds for density estimates: van der Vaart’s theorem.
- Part II. Nonparametric regression1. Statement of the problem. 2. The notion of linear smoother. Regressogram. 3. Nearest neighboors algorithm, local averaging. Method “Super smoother”. Cross-validation approach and the bass parameter. 4. Local regression. Method “Loess” (“Lowess”). 5. The Nadaraya-Watson kernel estimator. MISE for this estimator (without proof). 2 6. Nadaraya-Watson kernel estimator as a solution of the optimization problem and local polynomial estimate. Gasser-Muller estimate. 7. Generalized cross-validation. Motivation of the algorithm: theorem about the closed form of the cross-validation error for linear regression. 8. Akaike criterion. Kullback- Leibler discrepancy
- Part III. Nonparametric tests1. Tests for independence I: Kendall’s tau. Unbiased estimate for Kendall’s tau. Exact distribution of this estimate for n=3. Large-sample approximation for the constructed estimate. Calculation of the mean and the variance. Construction of asymptotic confidence intervals. The notion of bootstrap. Relation between Kendall’s tau and the Pearson correlation coefficient. 2. Tests for independence II: Spearman’s rho. Equivalent form of the Spearman’s rho. Exact distribution of Spearman’s rho for n=3. Large-sample approximation for the constructed estimator. Calculation of the mean and the variance. 3. Paired replicates data. Wilcoxon test. Exact distribution for n=3. Large-sample approximation. Calculation of the mean and the variance. 4. 2 independent samples. Wilcoxon statistics and Mann-Whitney statistics. Mann-Whitney test. Exact distribution for n=3 and m=2. Large-sample approximation. Calculation of the mean and the variance. 5. Many independent samples. Kruskal-Wallis test. Relation to the ANOVA test. Large-sample approximation (without proof). 6. Two-way layout. Friedman’s test (only general idea)
- Part IV. Bonus lectureWavelets. Haar basis. The notion of resolution. Application of this idea to the regression problem
- Wasserman, L. All of nonparametric statistics. – Springer Science & Business Media, 2006. – 270 pp.
- Сдвижков О.А. - Непараметрическая статистика в MS Excel и VBA - Издательство "ДМК Пресс" - 2014 - 172с. - ISBN: 978-5-94074-917-2 - Текст электронный // ЭБС ЛАНЬ - URL: https://e.lanbook.com/book/58695