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

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

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

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

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

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

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

8. NONPARAMETRIC INFERENCE Goodness- of- fit tests 8

9. NONPARAMETRIC INFERENCE Goodness- of- fit tests 9 χ 2 - test (Pearson) under H 0,

10. NONPARAMETRIC INFERENCE Goodness- of- fit tests 10 Goodness-of-fit to a family X ≡ F; X 1,...,X n i.i.d.; dim Θ =p;

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