7. Nonparametric inference

Quantile function Q Inference on F Confidence bands for F Goodness- of- fit tests

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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.

NONPARAMETRIC INFERENCE

Quantile function

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(b) Another important nonparametric problem: estimate

for a random variable (X, Y).

Linear regression is the particular case with

NONPARAMETRIC INFERENCE

Inference on F

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Definition. (Empirical distribution function)

(a) nFn(x) is B(n, F(x))(b) Fn is the nonparametric maximum likelihood estimator of F.

NONPARAMETRIC INFERENCE

Inference on F

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Theorem. (Glivenko-Cantelli)

( Fundamental Theorem of Statistics)

NONPARAMETRIC INFERENCE

Confidence bands

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Theorem. (Doob-Donsker) If F is continuous, then

has a distribution independent of F and

NONPARAMETRIC INFERENCE

Confidence bands

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with

Remark. The confidence band is

NONPARAMETRIC INFERENCE

Goodness- of- fit tests

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NONPARAMETRIC INFERENCE

Goodness- of- fit tests

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χ 2 - test (Pearson)

under H0,

NONPARAMETRIC INFERENCE

Goodness- of- fit tests

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Goodness-of-fit to a family

X ≡ F; X1, . . . ,Xn i.i.d.; dim Θ =p;

NONPARAMETRIC INFERENCE

Goodness of fit tests

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Normality test (Lilliefors)

Bootstrap