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Definition
[ edit ]
For X 1 , X 2 , ... X n independent and identically-distributed random variables in R with common cumulative distribution function F (x ), the empirical distribution function is defined by
F
n
(
x
)
=
1
n
∑
i
=
1
n
I
(
−
∞
,
x
]
(
X
i
)
,
{\displaystyle F_{n}(x )={\frac {1}{n}}\sum _{i=1}^{n}I_{(-\infty ,x]}(X_{i}),}
where IC is the indicator function of the set C .
For every (fixed) x , F n (x ) is a sequence of random variables which converge to F (x ) almost surely by the strong law of large numbers . That is, F n converges to F pointwise . Glivenko and Cantelli strengthened this result by proving uniform convergence of F n to F by the Glivenko–Cantelli theorem .[2]
A centered and scaled version of the empirical measure is the signed measure
G
n
(
A
)
=
n
(
P
n
(
A
)
−
P
(
A
)
)
{\displaystyle G_{n}(A )={\sqrt {n}}(P_{n}(A )-P(A ))}
It induces a map on measurable functions f given by
f
↦
G
n
f
=
n
(
P
n
−
P
)
f
=
n
(
1
n
∑
i
=
1
n
f
(
X
i
)
−
E
f
)
{\displaystyle f\mapsto G_{n}f={\sqrt {n}}(P_{n}-P)f={\sqrt {n}}\left({\frac {1}{n}}\sum _{i=1}^{n}f(X_{i})-\mathbb {E} f\right)}
By the central limit theorem ,
G
n
(
A
)
{\displaystyle G_{n}(A )}
converges in distribution to a normal random variable N (0, P (A )(1 − P (A ))) for fixed measurable set A . Similarly, for a fixed function f ,
G
n
f
{\displaystyle G_{n}f}
converges in distribution to a normal random variable
N
(
0
,
E
(
f
−
E
f
)
2
)
{\displaystyle N(0,\mathbb {E} (f-\mathbb {E} f)^{2})}
, provided that
E
f
{\displaystyle \mathbb {E} f}
and
E
f
2
{\displaystyle \mathbb {E} f^{2}}
exist.
Definition
(
G
n
(
c
)
)
c
∈
C
{\displaystyle {\bigl (}G_{n}(c ){\bigr )}_{c\in {\mathcal {C}}}}
is called an empirical process indexed by
C
{\displaystyle {\mathcal {C}}}
, a collection of measurable subsets of S .
(
G
n
f
)
f
∈
F
{\displaystyle {\bigl (}G_{n}f{\bigr )}_{f\in {\mathcal {F}}}}
is called an empirical process indexed by
F
{\displaystyle {\mathcal {F}}}
, a collection of measurable functions from S to
R
{\displaystyle \mathbb {R} }
.
A significant result in the area of empirical processes is Donsker's theorem . It has led to a study of Donsker classes : sets of functions with the useful property that empirical processes indexed by these classes converge weakly to a certain Gaussian process . While it can be shown that Donsker classes are Glivenko–Cantelli classes , the converse is not true in general.
Example
[ edit ]
As an example, consider empirical distribution functions . For real-valued iid random variables X 1 , X 2 , ..., X n they are given by
F
n
(
x
)
=
P
n
(
(
−
∞
,
x
]
)
=
P
n
I
(
−
∞
,
x
]
.
{\displaystyle F_{n}(x )=P_{n}((-\infty ,x])=P_{n}I_{(-\infty ,x]}.}
In this case, empirical processes are indexed by a class
C
=
{
(
−
∞
,
x
]
:
x
∈
R
}
.
{\displaystyle {\mathcal {C}}=\{(-\infty ,x]:x\in \mathbb {R} \}.}
It has been shown that
C
{\displaystyle {\mathcal {C}}}
is a Donsker class, in particular,
n
(
F
n
(
x
)
−
F
(
x
)
)
{\displaystyle {\sqrt {n}}(F_{n}(x )-F(x ))}
converges weakly in
ℓ
∞
(
R
)
{\displaystyle \ell ^{\infty }(\mathbb {R} )}
to a Brownian bridge B (F (x )) .
See also
[ edit ]
References
[ edit ]
Further reading
[ edit ]
Billingsley, P. (1995). Probability and Measure (Third ed.). New York: John Wiley and Sons. ISBN 0471007102 .
Donsker, M. D. (1952). "Justification and Extension of Doob's Heuristic Approach to the Kolmogorov- Smirnov Theorems" . The Annals of Mathematical Statistics . 23 (2 ): 277–281. doi :10.1214/aoms/1177729445 .
Dudley, R. M. (1978). "Central Limit Theorems for Empirical Measures" . The Annals of Probability . 6 (6 ): 899–929. doi :10.1214/aop/1176995384 .
Dudley, R. M. (1999). Uniform Central Limit Theorems . Cambridge Studies in Advanced Mathematics. Vol. 63. Cambridge, UK: Cambridge University Press.
Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference . Springer Series in Statistics. doi :10.1007/978-0-387-74978-5 . ISBN 978-0-387-74977-8 .
Shorack, G. R. ; Wellner, J. A. (2009). Empirical Processes with Applications to Statistics . doi :10.1137/1.9780898719017 . ISBN 978-0-89871-684-9 .
van der Vaart, Aad W. ; Wellner, Jon A. (2000). Weak Convergence and Empirical Processes: With Applications to Statistics (2nd ed.). Springer. ISBN 978-0-387-94640-5 .
Dzhaparidze, K. O.; Nikulin, M. S. (1982). "Probability distributions of the Kolmogorov and omega-square statistics for continuous distributions with shift and scale parameters". Journal of Soviet Mathematics . 20 (3 ): 2147. doi :10.1007/BF01239992 . S2CID 123206522 .
External links
[ edit ]
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Empirical_process&oldid=1208525194 "
C a t e g o r i e s :
● E m p i r i c a l p r o c e s s
● N o n p a r a m e t r i c s t a t i s t i c s
● T h i s p a g e w a s l a s t e d i t e d o n 1 7 F e b r u a r y 2 0 2 4 , a t 2 1 : 4 5 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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