Introduction
Introduction
• The talk concerns various goodness-of-fit procedures in
parametric regression models, non-parametric regression
models and time series.
• Tests based on discrepancy between data and the expected
statistical model.
• Basic formulation:
X1 , . . . , Xn are i.i.d. random variables with distribution function F
H0 : F = F 0
versus
H1 : F 6= F0
F0 is a given distribution (e.g. N(0, 1))
empirical distribution function
Fbn (x) =
n
1X
n
Goodness-offit tests based
on empirical
characteristic
functions
Marie
Hušková
Outline
Introduction
Some classical
test statistics
Tests based on
empirical
characteristic
function
Sequential
monitoring
based on
empirical
characteristic
functions
General
scheme
I {Xj ≤ x}
Procedures
j=1
Asymptotic
results
Marie Hušková (Charles University, Prague) Goodness-of-fit tests based on empirical characteristic
Salt Lake
functions
City, February 2012
Simulations
4 / 50
Some classical test statistics
Outline
1
Introduction
2
Some classical test statistics
Goodness-offit tests based
on empirical
characteristic
functions
Marie
Hušková
Outline
Introduction
3
Tests based on empirical characteristic function
4
Sequential monitoring based on empirical characteristic functions
5
General scheme
6
Procedures
7
Asymptotic results
8
Simulations
Some classical
test statistics
Tests based on
empirical
characteristic
function
Sequential
monitoring
based on
empirical
characteristic
functions
General
scheme
Procedures
Asymptotic
results
Marie Hušková (Charles University, Prague) Goodness-of-fit tests based on empirical characteristic
Salt Lake
functions
City, February 2012
Simulations
5 / 50
Some classical test statistics
Classical test statistics:
Kolmogorov- Smirnov test:
√
n sup |Fbn (x) − F0 (x)|
x
n
Marie
Hušková
Outline
Cramér-von-Mises test:
Z
Goodness-offit tests based
on empirical
characteristic
functions
Introduction
R
|Fbn (x) − F0 (x)|2 dF0 (x)
Anderson-Darling test
Z R
n
|Fbn (x) − F0 (x)|2 w (x)dF0 (x)
where w (.) is a positive weighted function, e.g.
w (x) =
1
F0 (x)(1 − F0 (x))
χ2 tests
Marie Hušková (Charles University, Prague) Goodness-of-fit tests based on empirical characteristic
Salt Lake
functions
City, February 2012
Some classical
test statistics
Tests based on
empirical
characteristic
function
Sequential
monitoring
based on
empirical
characteristic
functions
General
scheme
Procedures
Asymptotic
results
Simulations
6 / 50
Some classical test statistics
Large values of the above test statistics indicate that the null
hypothesis H0 is violated.
Critical values corresponding to the chosen level α are needed,
e.g. cα for Kolmogorov-Smirnov test under H0
√
P0 ( n sup |Fbn (x) − F0 (x)| ≥ cα ) ≈ α.
x
F0 usually continuous, in this case cα does not depend on F0 .
Quite often the testing problem is:
H0θ : F ∈ F with F = {F (x; θ), θ ∈ Θ}
against alternatives that the null hypothesis is not true
θ –nuisance parameter
Goodness-offit tests based
on empirical
characteristic
functions
Marie
Hušková
Outline
Introduction
Some classical
test statistics
Tests based on
empirical
characteristic
function
Sequential
monitoring
based on
empirical
characteristic
functions
General
scheme
Procedures
Asymptotic
results
Marie Hušková (Charles University, Prague) Goodness-of-fit tests based on empirical characteristic
Salt Lake
functions
City, February 2012
Simulations
7 / 50
Some classical test statistics
Goodness-offit tests based
on empirical
characteristic
functions
bn
In this case θ is estimated from the data – θ
New test statistics:
Marie
Hušková
Modified Kolmogorov - Smirnov test:
√
bn )|
n sup |Fbn (x) − F0 (x; θ
Outline
x
Important: The limit distribution of this statistics under
bn !!!!
depends on θ
Introduction
H0θ
Question: Choice of the estimators-not discussed here.
Some classical
test statistics
Tests based on
empirical
characteristic
function
Sequential
monitoring
based on
empirical
characteristic
functions
General
scheme
Procedures
Asymptotic
results
Marie Hušková (Charles University, Prague) Goodness-of-fit tests based on empirical characteristic
Salt Lake
functions
City, February 2012
Simulations
8 / 50
Some classical test statistics
Other type of testing problems
Two-sample problem and tests:
X1 , . . . , Xn and Y1 , . . . , Ym – two independent random samples
from F1 and F2 , respectively.
H0 : F 1 = F 2
versus H1 : F1 6= F2
Kolmogorov -Smirnov two-sample test:
r
mn
b
e
sup |Fn (x) − Fm (x)|
m
+n
x
Fbn and Fen empirical distribution functions
n
1X
I {Xj ≤ x}
Fbn (x) =
n
j=1
Goodness-offit tests based
on empirical
characteristic
functions
Marie
Hušková
Outline
Introduction
Some classical
test statistics
Tests based on
empirical
characteristic
function
Sequential
monitoring
based on
empirical
characteristic
functions
General
scheme
Procedures
Asymptotic
results
Marie Hušková (Charles University, Prague) Goodness-of-fit tests based on empirical characteristic
Salt Lake
functions
City, February 2012
Simulations
9 / 50
Some classical test statistics
Hypothesis symmetry and tests
H0S : F (x) = 1 − F (−x),
∀x
Kolmogorov -Smirnov test for symmetry:
√
Goodness-offit tests based
on empirical
characteristic
functions
Marie
Hušková
Outline
Introduction
n sup |Fbn (x) − (1 − Fbn− (x)|
x
Tests based on
empirical
characteristic
function
Independence test
(X1 , Y1 ), . . . , (Xn , Yn ) – i.i.d. pairs with the d.f. F (x, y ), Xi d.f. F1 ,
Yi d.f. F2
H0 : F (x, y ) = F1 (x)F2 (y ),
based on empirical d.f.
Some classical
test statistics
∀x
Sequential
monitoring
based on
empirical
characteristic
functions
General
scheme
Procedures
Asymptotic
results
Salt Lake City, February 2012
10 /
Simulations
Marie Hušková (Charles University, Prague) Goodness-of-fit tests based on empirical characteristic functions
50
Tests based on empirical characteristic function
Outline
1
Introduction
2
Some classical test statistics
Goodness-offit tests based
on empirical
characteristic
functions
Marie
Hušková
Outline
Introduction
3
Tests based on empirical characteristic function
4
Sequential monitoring based on empirical characteristic functions
5
General scheme
6
Procedures
7
Asymptotic results
8
Simulations
Some classical
test statistics
Tests based on
empirical
characteristic
function
Sequential
monitoring
based on
empirical
characteristic
functions
General
scheme
Procedures
Asymptotic
results
Salt Lake City, February 2012
11 /
Simulations
Marie Hušková (Charles University, Prague) Goodness-of-fit tests based on empirical characteristic functions
50