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THE EMPIRICAL CHARACTERISTIC FUNCTION AND LARGE
SAMPLE HYPOTHESIS TESTING
Sayfullayeva Gulnoz Sayfullayevna
Navoiy State University (Uzbekistan)
Zayniddinova Dinara Xusnitdin qizi
Fayzullayeva Sevara Akbar qizi
Navoiy State University (Uzbekistan)
https://doi.org/10.5281/zenodo.14506988
Annotation:
Many properties of distribution functions can be characterized
in terms of their characteristic functions. This suggests that statistical inference
about such properties might utilize the characteristic function. This requires
that some means be found to gather information about the characteristic
function from a random sample. In this investigation, the empirical
characteristic function is proposed to do this.
Key words:
empirical characteristic function, characteristic function,
probability space, independent and identically distributed random variables.
INTRODUCTION.
Let
, ,
P
be a fixed but otherwise arbitrary probability space. Let
1
2
,
,....
n
X
X
X
be independent and identically distributed (i.i.d) random variables
defined on
with common distribution function F and characteristic function c.
Then the empirical distribution function is
/ ,
,
F
x
N x
n
x
R
n
where
N(x) is the number of
j
X
x
for
1
j
n
. The empirical characteristic function
(e.c.f) is defined by
1
,
1
itX
n
j
itx
c
t
e
dF
x
e
t
R
n
n
n j
To justify the use of the e.c.f. for statistical inference we should know
something about the convergence of the e.c.f. to the characteristic function. In
this chapter we shall discuss this convergence briefly and give a class of
hypothesis testing problems to which the e.c.f. may be applied.
CONVERGENCE OF THE E.C.F.
We first note that for fixed t,
c
t
n
is average of bounded i.i.d. random
variables. Therefore, by the strong law of large numbers
c
t
n
converges almost
surely to
c t
for every
t
R
. Furthermore, we have the following result of
Feuerverger and Mureika (1977).
THEOREM. For fixed
T
,
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lim sup
( )
0
1
n
n
t T
P
c t
c t
Let
1
2
,
,....
n
X
X
X
be independent and identically distributed random variables
with common law X. For given
2
and
consider the problem of testing
2
0
:
,
.
H
X
This hypothesis is
equivalent to
'
'
0
:
0,1
/ .
H
X
where
X
X
Hence without loss of generality we will
take
2
0
1
and
.
Let
c
t
n
be the e.c.f. of
1
2
,
,....
n
X
X
X
. Noting that
2
1
exp
2
t
is the
characteristic function of a
0,1
distribution, we take as our test statistic the
following
2
1
sup
exp
2
n
n
t S
s
c t
t
where S is a bounded subset of the real line. Here we will take S to be the
points of maximization of
2
1
exp
2
t
t
c
t
Where
c
t
is the characteristic function of some close alternative
distribution. Taking
c
t
to be the characteristic function of a
,1
distribution with
chosen small, we then have
2
2
2
1
2
2
1
1
1
exp
exp
exp
exp
2
2
2
1
exp
2 2 cos
2
t
t
i t
t
t
i t
t
t
It can be seen that the e.c.f. based test performs significantly better than the
Kolmogorov-Smirnov test for these alternatives. This is to be expected since the
point
0
t
was chosen to make the test sensitive to location shift alternatives.
However it is significant that the e.c.f. test performs better than the Kolmogorov-
Smirnov test for values of the mean less than 0.1 since that is the specific
alternative used in determining
0
t
.
Bibliography:
1.
Billingsley, P., Convergence of Probability Measures, Wiley, New York,
1968.
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2.
Cramér, H., Mathematical Methods of Statistics, Princeton Univ. Press,
Princeton, New Jersey, 1946.
3.
Csurgo, S., The empirical characteristic process when parameters are
estimated, 1979 (to appear).
4.
Feuerverger, A., and Mureika, R. A., The empirical characteristic function
and its applications, Ann. Statis., 5(1977).
