7. Nonparametric inference

7. Nonparametric inference • Quantile function Q • Inference on F • Confidence bands for F • Goodness- of- fit tests 1

Quantile function X ≡ F and Θ=F Definition. (Quantile function) • (a) Inference can be done on: • The distribution function F. • The quantile function • The density function f, if it exists. Q. 2 NONPARAMETRIC INFERENCE

Quantile function (b) Another important nonparametric problem: estimate for a random variable (X, Y). Linear regression is the particular case with 3 NONPARAMETRIC INFERENCE

Inference on F Definition. (Empirical distribution function) (a) nFn(x) is B(n, F(x)) (b) Fn is the nonparametric maximum likelihood estimator of F. 4 NONPARAMETRIC INFERENCE

Inference on F Theorem. (Glivenko-Cantelli) ( Fundamental Theorem of Statistics) 5 NONPARAMETRIC INFERENCE

Confidence bands Theorem. (Doob-Donsker) If F is continuous, then has a distribution independent of F and 6 NONPARAMETRIC INFERENCE

Confidence bands with Remark. The confidence band is 7 NONPARAMETRIC INFERENCE

Goodness- of- fit tests 8 NONPARAMETRIC INFERENCE

Goodness- of- fit tests χ 2 - test (Pearson) under H0, 9 NONPARAMETRIC INFERENCE

Goodness- of- fit tests Goodness-of-fit to a family X ≡F; X1, . . . ,Xn i.i.d.; dim Θ=p; 10 NONPARAMETRIC INFERENCE

Goodness of fit tests Normality test (Lilliefors) Bootstrap 11 NONPARAMETRIC INFERENCE

7. Nonparametric inference