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Lecture 29: Convergence concepts
Asymptotic approach
In statistical analysis or inference, a key to the success of finding a
good procedure is being able to find some moments and/or
distributions of various statistics.
In many complicated problems we are not able to find exactly the
moments or distributions of given statistics.
When the sample size
n
is large, we may approximate the
moments and distributions of statistics, using asymptotic tools,
some of which are studied in this course.
In an asymptotic analysis, we consider a sample
X
= (
X
1
,...,
X
n
)
not for fixed
n
, but as a member of a sequence corresponding to
n
=
n
0
,
n
0
+
1
,...
, and obtain the limit of the distribution of an
appropriately normalized statistic or variable
T
n
(
X
)
as
n
→
∞
.
The limiting distribution and its moments are used as
approximations to the distribution and moments of
T
n
(
X
)
in the
situation with a large but actually finite
n
.
UWMadison (Statistics)
Stat 609 Lecture 29
2014
1 / 10
beamertulog
This leads to some asymptotic statistical procedures and
asymptotic criteria for assessing their performances.
The asymptotic approach is not only applied to the situation where
no exact method (the approach considering a fixed
n
) is available,
but also used to provide a procedure simpler (e.g., in terms of
computation) than that produced by the exact approach.
In addition to providing more theoretical results and/or simpler
procedures, the asymptotic approach requires less stringent
mathematical assumptions than does the exact approach.
Definition 5.5.1 (convergence in probability)
A sequence of random variables
Z
n
,
i
=
1
,
2
,...
, converges in
probability to a random variable
Z
iff for every
ε
>
0,
lim
n
→
∞
P
(

Z
n
−
Z
 ≥
ε
) =
0
.
A sequence of random vectors
Z
n
converges in probability to a random
vector
Z
iff each component of
Z
n
converges in probability to the
corresponding component of
Z
.
UWMadison (Statistics)
Stat 609 Lecture 29
2014
2 / 10
beamertulog
This leads to some asymptotic statistical procedures and
asymptotic criteria for assessing their performances.
The asymptotic approach is not only applied to the situation where
no exact method (the approach considering a fixed
n
) is available,
but also used to provide a procedure simpler (e.g., in terms of
computation) than that produced by the exact approach.
In addition to providing more theoretical results and/or simpler
procedures, the asymptotic approach requires less stringent
mathematical assumptions than does the exact approach.
Definition 5.5.1 (convergence in probability)
A sequence of random variables
Z
n
,
i
=
1
,
2
,...
, converges in
probability to a random variable
Z
iff for every
ε
>
0,
lim
n
→
∞
P
(

Z
n
−
Z
 ≥
ε
) =
0
.
A sequence of random vectors
Z
n
converges in probability to a random
vector
Z
iff each component of
Z
n
converges in probability to the
corresponding component of
Z
.