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205
THE STUDY OF DEVIATIONS FROM THE NORMAL DISTRIBUTION OF
SUMS OF INDEPENDENT OR DEPENDENT RANDOM VARIABLES
USING GENERAL ESTIMATES OF DIFFERENCE PSEUDO MOMENTS
Akhmedov Sokhibjon Akbarovich
Associate professor ofAndijan State University
Tursunov Bekzod Burkhan ugli
The base doctoral student ofAndijan State University
Abstract
. In this article, using the method of semi invariants under the
Statulevicius condition, general uneven estimates of approximation by a normal
distribution are obtained and estimates of absolute pseudo moments are obtained on
the basis of this. In order to obtain similar estimates in limit theorems for sums of
independent or dependent random variables, it is sufficient to obtain an estimate of
semi invariants of the Statulevicius type. These results can be used in tasks related to
the analysis of rare events and in statistical testing tasks, where we want to test, for
example, the hypothesis that the data is distributed normally.
Keywords
: Cumulants, the Statulevicius condition,uneven estimate, difference
pseudo-moment, probabilities of large deviations.
Introduction.
The publication of non-classical estimates in limit theorems for sums of
independent random variables began in the second half of the last century with the
fundamental works of V. M. Zolotorev (see, for example, [9]).Out of the many
problems,one of the most relevant issues of theoretical and practical interest was
obtaining estimates using difference pseudo-metrics.Here , pseudo moments are used
to compare two distributions when constructing estimates of the accuracy of the
approximation of distributions. For example, such estimates in the case when one of
these distributions is normal are obtained in [5].
In this paper, general non-uniform estimates of the approximation to the normal
distribution and estimates of difference pseudo moments in the zones of large
deviations are obtained by the method of semi invariants under the condition of
Statulyavichus on the semi invariants of the random variables. In order to obtain
similar estimates in the limit theorems for sums of independent or dependent random
variables, it is sufficient to obtain an estimate of seven Statulevicius-type invariants.
Research methods and main results.
Consider a random variable (r.v)
𝜂 = 𝜂
∆
, depending on the parameter
∆,
with a
distribution function
𝐹
𝜂
(𝑥) = 𝑃 (𝜂 < 𝑥)
with mean
𝐸𝜂 = 0
and variance
𝐷𝜂 = 1
.
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We denote
Г
𝑘
{𝜂}𝑎𝑠 𝑎 𝑠𝑒𝑣𝑒𝑛 −
invariant
𝑘 −
of the kth order sl. in
𝜂,
Ф(𝑥) =
𝑃(𝑁 < 𝑥) − (0,1)
the
normal
distribution,
𝜒
𝑠
(𝜂, 𝑁) = 𝑠 ∫
|𝑥|
𝑠−1
|𝐹
𝜂
(𝑥) −
∞
− ∞
Ф(𝑥)| 𝑑𝑥
are difference pseudo moments of order
𝑠 ≥ 1.
Suppose that there are constants
𝛾 ≥ 0, 𝐻 > 0, ∆> 0
such thattheStatulevicius
condition is satisfied:
|Г
𝑘
{𝜂}| ≤ 𝐻(𝑘!)
1+𝛾
/∆
𝑘−2
, 𝑘 = 3,4, … , 𝑙
(𝑆
𝛾
)
.
In this paper, we obtain some general estimates for the values
𝜒
𝑠
(𝜂, 𝑁)
under the
condition
(𝑆
𝛾
)
.The estimate for
𝜒
𝑠
,
(𝜂, 𝑁)
is based on general non uniform estimates
for the value
|𝐹
𝜂
(𝑥) − Ф(𝑥)|
, obtained using the general lemma on the probabilities
of large deviations
[15]
, as well as on exponential estimates of the distribution r.v
[14].
Lemma 1
. Let sl. in
𝜂
with
𝐸𝜂 = 0
and
𝐷𝜂 = 1
satisfy the condition
(𝑆
𝛾
).
Then
1)
At
0 ≤ 𝑥 ≤ 1
|𝐹
𝜂
(−𝑥) − Ф(−𝑥)| = |𝐹
𝜂
(𝑥) − Ф(𝑥)| ≤ 𝑐(𝛿, 𝛾)/∆
1/(1+2𝛾)
(1)
𝑐(𝛿, 𝛾) =
2160
𝛿
(
√6
2
)
1/(1+2𝛾)
(1 +
2560
𝛿
4
𝑒
4
) , 0 < 𝛿 < 1;
2)
For
1 ≤ 𝑥 ≤ √∆
𝛾
3
|𝐹
𝜂
(−𝑥) − Ф(−𝑥)| = |𝐹
𝜂
(𝑥) − Ф(𝑥)| ≤
𝑐(𝛿,𝛾)
√2𝜋
𝑥
3
𝑒𝑥𝑝{−𝑥
2
/2}
∆
1/(1+2𝛾)
(2)
Proof
. Using the general lemma on the probabilities of large deviations
[15]
, in
the interval
1 ≤ 𝑥 ≤ √∆
𝛾
3
, we can obtain the following estimate:
|𝐹
𝜂
(−𝑥) − Ф(−𝑥)| = |𝐹
𝜂
(𝑥) − Ф(𝑥)| ≤ (1 − Ф(𝑥)) {|1 − 𝑒𝑥𝑝{𝐿
𝛾
(𝑥)}| +
|1 − 𝑒𝑥𝑝{𝐿
𝛾
(𝑥)}| ∙ |𝑓
𝑗
(𝑥)| ∙
𝑥+1
∆
𝛾
+ |𝑓
𝑗
(𝑥)| ∙
𝑥+1
∆
𝛾
}
(3)
Here
𝐿
𝛾
(𝑥) = ∑
𝜆
𝑖
𝑥
𝑖
+ 𝜃(𝑥/∆
𝛾
)
3
3≤𝑖<𝑞
, 𝑞 = {
1
𝛾
+ 2, 𝛾 > 0 ∞, 𝛾 = 0, |𝜃| <
1
𝜆
𝑖
are expressed in terms of the semi invariants r.v in
𝜂,
where
𝐿
𝛾
(±𝑥) ≤
𝑥
3
2(𝑥+8∆
𝛾
)
,
(4)
𝑓
𝑗
(𝑥) =
60(1+10∆
𝛾
2
exp{1−𝑥/∆
𝛾
}√∆
𝛾
)
1−𝑥/∆
𝛾
, 𝑗 = 1,2 ∆
𝛾
is given in
(1)
Let then
0 ≤ 𝑥 ≤ √∆
𝛾
3
. Then using (4) it is easy to check that
𝐿
𝛾
(±𝑥) ≤
1
2
and
hence
|1 − exp{𝐿
𝛾
(±𝑥)}| ≤ 2|𝐿
𝛾
(±𝑥)| ≤
𝑥
3
𝑥+8∆
𝛾
(5)
Let's denote
𝛿 = (1 − 1/√∆
𝛾
2
3
)
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moreover, for sufficiently large
∆> 0
, it can be shown that
0 < 𝛿 < 1.
Further, for
0 ≤ 𝑥 ≤ √∆
𝛾
3
,
we obtain in an elementary way
|𝑓
𝑗
(𝑥)| ≤
60
𝛿
(1 +
2560
𝛿
4
𝑒
4
) , 𝑗 = 1,2
(6)
Now let
0 ≤ 𝑥 ≤ 1,
then using (5) and (6) from (3) we obtain (1). If
0 ≤ 𝑥 ≤
√∆
𝛾
3
,
meaning that
Ф(−𝑥) = 1 − Ф(𝑥) ≤
1
𝑥√2𝜋
exp{−𝑥
2
/2}
(7)
(see
[8]
.page 192) using also (5) and (6) from (3), we obtain (2)
Lemma 1 is proved.
Lemma 2.
Let r.v in
𝜂
with
𝐸𝜂 = 0
and
𝐷𝜂 = 1
satisfy the condition
(𝑆
𝛾
)
.
Then there is a constant
𝑐(𝛿, 𝛾)
, which is estimated by:
𝜒
𝑠
(𝜂, 𝑁) ≤ 𝑐(𝛿, 𝛾)/∆
1/(1+2𝛾)
(8)
Proof
. We have
𝜒
𝑠
(𝜂, 𝑁) = 𝑠|𝐹
𝜂
(𝑥) − Ф(𝑥)| 𝑑𝑥 + 𝑠|𝐹
𝜂
(𝑥) − Ф(𝑥)| 𝑑𝑥 ≤
≤ 𝑠|𝐹
𝜂
(𝑥) − Ф(𝑥)| 𝑑𝑥 + 𝑠𝑃(𝑁 ≥ 𝑥) 𝑑𝑥 + 𝑠𝑃(𝜂 ≥ 𝑥) 𝑑𝑥 = 𝐽
1
+ 𝐽
2
+ 𝐽
3
(9)
Next
𝐽
1
, we estimate by (1) and (2).
𝐽
2
by (7),
𝐽
3
and by the estimate obtained
in
[14]:
if the condition
(𝑆
𝛾
)
is satisfied, then for all
𝑥 ≥ 0
𝑃(±𝜂 ≥ 𝑥) ≤ exp{−𝑥
2
/4𝐻}
if
0 ≤ 𝑥 ≤ (𝐻
1+𝛾
∆)
1/(1+2𝛾)
:
(10)
𝑃(±𝜂 ≥ 𝑥) ≤ exp {−
1
4
(𝑥 ∆)
1/(1+2𝛾)
}
if
𝑥 ≥ (𝐻
1+𝛾
∆)
1/(1+2𝛾)
(11)
Let's evaluate
𝐽
1
. Using (1) and (2) we have
𝐽
1
= 𝑠 ∫ |𝑥|
𝑠−1
|𝐹
𝜂
(𝑥) − Ф(𝑥)|𝑑𝑥 + 𝑠|𝐹
𝜂
(𝑥) − Ф(𝑥)|
1
−1
≤
𝑠𝑐(𝛿,𝛾)
∆
1/(1+2𝛾)
(
2
𝑠
+
1
√2𝜋
∫ |𝑥|
𝑠+2
exp{−𝑥
2
/2}𝑑𝑥
∞
−∞
) ≤
4𝑠𝑐(𝛿,𝛾)
∆
1/(1+2𝛾)
1
√2𝜋
∫ |𝑥|
𝑠+2
exp{−𝑥
2
/2}𝑑𝑥
∞
−∞
.
Since
𝐸|𝑁|
𝑚
= {
𝑚!
2
𝑚/2
(
𝑚
2
)!
, 𝑚 − 𝑖𝑠 𝑒𝑣𝑒𝑛; √
2
𝜋
2
(𝑚−1)/2
(
𝑚−1
2
) !, 𝑚 − 𝑖𝑠 𝑜𝑑𝑑.
(12)
Then taking (12)
𝑚 = 𝑠 + 2
we get
𝐽
1
≤
𝑐
𝑖
(𝑠,𝛾)
∆
1/(1+2𝛾)
, 𝑖 = 1,2
(13)
Where
𝑐
𝑖
(𝑠, 𝛾) = {
4𝑠𝑐(𝛿,𝛾)(𝑠+2)!
2
(𝑠+2)
2
(
𝑠+2
2
)!
, 𝑠 − 𝑖𝑠 𝑒𝑣𝑒𝑛, 𝑖 = 1,4 𝑐(𝛿, 𝛾)√
2
𝜋
2
(𝑠+1)
2
(
𝑠+1
2
) !,
𝑠 − 𝑖𝑠 𝑜𝑑𝑑, 𝑖 = 1;
Let's evaluate
𝐽
2
. Using (7), we obtain
𝐽
2
≤
2𝑠
√2𝜋
∫
𝑥
𝑠−2
𝑒𝑥𝑝{−𝑥
2
/2}𝑑𝑥
∞
√∆
𝛾
3
It is known (see page 176 in
[7]
) that
∫ 𝑥
𝑝
exp{−𝑥
2
/2}𝑑𝑥~𝑎
𝑝−1
exp{−𝑎
2
/2}, 𝑎 → ∞
∞
𝑎
(14)
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Therefore
∫
𝑥
𝑠−2
exp{−𝑥
2
/2}𝑑𝑥~ (√∆
𝛾
3
)
𝑠−3
exp {−√∆
𝛾
2
3
/2}
∞
√∆
𝛾
3
for sufficiently large
∆
𝛾
, the expression
𝐽
2𝑠
(∆
𝛾
) = (√∆
𝛾
3
)
3−𝑠
exp {−√∆
𝛾
2
3
/2} ∫
𝑥
𝑠−2
exp{−𝑥
2
/2}𝑑𝑥
∞
√∆
𝛾
3
restricted.
We denote
𝐶
2𝑠
= 𝐽
2𝑠
(∆
𝛾
)
Then
𝐽
2
≤
2𝑠𝐶
2𝑠
√2𝜋∆
𝛾
(√∆
𝛾
3
)
𝑠
exp {−√∆
𝛾
2
3
/2}
.
(15)
The function
𝑀
2𝑠
(∆
𝛾
) = (√∆
𝛾
3
)
𝑠
exp {−√∆
𝛾
2
3
/2}
reaches its maximum at
∆
𝛾
≥
1
. Let's denote
𝑀
2𝑠
= 𝑀
2𝑠
(∆
𝛾
)
(16)
Given (16) from (15), we have
𝐽
2
≤
2𝑠𝐶
2𝑠
𝑀
2𝑠
√2𝜋𝑐
𝛾
1
∆
1/(1+2𝛾)
(17)
Let us now evaluate
𝐽
3
. We use (10) and (11). Let
√∆
𝛾
3
≤ (𝐻
1+𝛾
∆)
1
(1+2𝛾)
Then
𝐽
3
≤ 2𝑠 ∫
𝑥
𝑠−1
𝑃(𝜂 ≥ 𝑥)𝑑𝑥
(𝐻
1+𝛾
∆)
1
(1+2𝛾)
√∆
𝛾
3
+ 2𝑠 ∫
𝑥
𝑠−1
𝑃(𝜂 ≥ 𝑥)𝑑𝑥
∞
(𝐻
1+𝛾
∆)
1
(1+2𝛾)
≤
≤ 2𝑠 ∫
𝑥
𝑠−1
𝑒𝑥𝑝 {
−𝑥
2
4𝐻
} 𝑑𝑥 + 2𝑠
∞
√∆
𝛾
3
∫
𝑥
𝑠−1
𝑒𝑥𝑝 {−
1
4
(𝑥∆)
1/(1+2𝛾)
} 𝑑𝑥 =
∞
(𝐻
1+𝛾
∆)
1
(1+2𝛾)
= 𝐽
3
′
+ 𝐽
3
′′
To evaluate
𝐽
3
′
, we use (14) again. We have
∫
𝑥
𝑠−1
exp{−𝑥
2
/4𝐻}𝑑𝑥
∞
√∆
𝛾
3
= (√2𝐻)
𝑠
∫
(
𝑥
√2𝐻
)
𝑠−1
exp {− (
𝑥
√2𝐻
)
2
/2} 𝑑 (
𝑥
√2𝐻
)
∞
√∆
𝛾
3
~
~(√2𝐻)
𝑠
(√∆
𝛾
3
)
𝑠−2
exp {−√∆
𝛾
2
3
/2}
Let's denote
𝐶
3𝑠
′
= 𝐽
3𝑠
′
(∆
𝛾
) , 𝑀
3𝑠
′
= 𝑀
3𝑠
′
(∆
𝛾
)
where
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209
𝐽
3𝑠
′
(∆
𝛾
) = (√∆
𝛾
3
)
2−𝑠
exp {−√∆
𝛾
2
3
/2}
×
×
∫
(
𝑥
√2𝐻
)
𝑠−1
exp {− (
𝑥
√2𝐻
)
2
/2}
∞
√∆
𝛾
3
(
𝑥
√2𝐻
)
𝑠−1
exp {− (
𝑥
√2𝐻
)
2
/2} 𝑑 (
𝑥
√2𝐻
)
𝑀
3𝑠
′
(∆
𝛾
) = (√∆
𝛾
3
)
𝑠+1
𝑒𝑥𝑝 {−√∆
𝛾
2
3
/2}
Then
𝐽
3
′
≤ 2𝑠(√2𝐻)
𝑠
𝐶
3𝑠
′
(∆
𝛾
)𝑀
3𝑠
′
/𝑐
𝛾
∆
1/(1+2𝛾)
(18)
To estimate
𝐽
3
"
, we use the following asymptotic relation (see[13], page 35)
∫ 𝑥
𝑝−1
exp {−𝛽𝑥
𝜇
} 𝑑𝑥~
1
𝑝
𝑎
𝑝
𝑒𝑥𝑝{−𝛽𝑎
𝜇
}, 𝑎 → ∞
∞
𝑎
(19)
Taking
𝑝 = 𝑠, 𝛽 =
1
4
, 𝑎 = (𝐻
1+𝛾
∆)
1/(1+2𝛾)
, 𝜇 =
1
1+2𝛾
from (19), we have
𝐽
3
′′
~
2𝑠
∆
𝑠
1
𝑠
(𝐻
1+𝛾
∆)
1/(1+2𝛾)
𝑒𝑥𝑝 {−
1
4
(𝐻
1+𝛾
∆)
1/(1+2𝛾)
2
}
Let's denote
𝐶
3𝑠
"
= 𝐽
3𝑠
"
(∆), 𝑀
3𝑠
"
= 𝑀
3𝑠
′′
(∆),
where
𝐽
3
′′
(∆)
=
𝑠 exp {
1
4
(𝐻
1+𝛾
∆)
1/(1+2𝛾)
2
}
(𝐻
1+𝛾
∆)
1/(1+2𝛾)
∫
(𝑥∆)
𝑠−1
exp {−
1
4
(𝑥∆)
1/(1+2𝛾)
} 𝑑(𝑥∆)
∞
(𝐻
1+𝛾
∆)
1/(1+2𝛾)
,
𝑀
3
′′
(∆) = ∆
(1−2𝑠𝛾)/(1+2𝛾)
𝑒𝑥𝑝 {−
1
4
(𝐻
1+𝛾
∆)
1/(1+2𝛾)
2
}
Then
𝐽
3
"
≤ 2𝑠𝐻
𝑠(1+𝛾)
1+2𝛾
𝐶
3𝑠
"
𝑀
3𝑠
"
/∆
1/(1+2𝛾)
.
(20)
As a result, from (18) and (20) we obtain
𝐽
3
≤ 2𝑠 (
(√2𝐻)
𝑠
𝑐
𝛾
𝐶
3𝑠
′
′𝑀
3𝑠
′
+ 𝐻
𝑠(1+𝛾)
1+2𝛾
𝐶
3𝑠
"
𝑀
3𝑠
"
)
1
∆
1/(1+2𝛾)
(21)
Finally, using (13), (17) and (21) from (9), we obtain (8):
𝜒
𝑠
, (𝜂, 𝑁) ≤ ((𝑐
𝑖
(𝑠, 𝛾) +
2𝑠𝐶
2𝑠
𝑀
2𝑠
√2𝜋𝑐
𝛾
+ 2𝑠 (
(√2𝐻)
𝑠
𝑐
𝛾
С
3𝑠
′
𝑀
3𝑠
′
+ 𝐻
𝑠(1+𝛾)
1+2𝛾
С
3𝑠
′′
𝑀
3𝑠
′′
))
×
×
1
∆
1/(1+2𝛾)
= 𝑐(𝑠, 𝛾)/∆
1/(1+2𝛾)
Lemma 2 is proved
Consequences of general lemmas.
Let
𝑋
𝑡
, 𝑡 = 1,2, …
with
𝐸𝑋
𝑡
= 0
and
𝐷𝑋
𝑡
= 1
is a random process defined on
(𝛺, 𝐹, 𝑃)
,
{𝐹
𝑠
𝑡
; 1 ≤ 𝑠 ≤ 𝑡 ≤ ∞}
-family
𝜎 −
of algebras:
𝐹
𝑠
𝑡
⊂ 𝐹, ∀𝑠 ≤ 𝑡; 𝐹
𝑠
1
𝑡
1
≤
𝐹
𝑠
2
𝑡
2
; ∀[𝑠
1
; 𝑡
1
] ⊂ [𝑠
2
; 𝑡
2
]; 𝜎{𝑋
𝑛
: 𝑠 ≤ 𝑛 ≤ 𝑡} ⊂ 𝐹
𝑠
𝑡
,
𝑋
𝑡
satisfies the Rosenblatt
condition with the
𝛼 −
displacement coefficient:
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210
𝛼(𝑠, 𝑡) = |𝑃(𝐴𝐵) − 𝑃(𝐴)𝑃(𝐵)|
,
𝑍
𝑛
=
𝑆
𝑛
𝐵
𝑛
, 𝑆
𝑛
= ∑
𝑋
𝑡
𝑛
𝑡=1
, 𝐵
𝑛
2
= 𝑀𝑆
𝑛
2
>
𝑛𝜎
0
, 𝜎
0
> 0
Conclusion.
From
Lemmas 1 and 2 we can obtain
similar statements of theorems 1 and 2
using the statements of theorem 4.23; 4.25-4.31 in the monograph [ 11 ]. If we take
into account that these estimates take place in zones of large deviations, the rough
constants do not greatly affect the deviation error by
n
.
We can use the theorems obtained in this way , for example , in statistical testing
problems, where we want to test, for example, the hypothesis that the data are
normally distributed. From this we can construct control charts to test the stability of
the distribution parameters when studying random processes.
Literature:
1.
A.Dembo, O.Zeitouni. “Large Deviations Techniques and Applications”,
1998
2.
A.J.McNeil, R.Frey, P.Embrechts "Quantitative Risk Management:
Concepts, Techniques, and Tools",2015.
3.
F.Hollander. “Large Deviations”, 2008.
4.
J.G.Kalbfleisch “Statistical Analysis of Stochastic Processes in
Time”,2005.
5.
S.Asmussen, H.Albrecher "Risk Theory and Large Deviations" (2010)
6.
А.А.Боровков, А.А Магульский. “Большие уклонения и проверка
статистических гипотез.Новосибирск”.»Наука»,1992.
7.
В.В Петров. “Предельные теоремы для сумм независимых
случайных величин”. – М.: Наука, 1987. – 320 с.
8.
В.Феллер. “Введение и теорию вероятностей и ее приложения”. Т.I.
– М., Мир. 1984. 528 с.
9.
В.М.Золоторев “Современная теория суммировния незавысимых
случайных величин”. М.Наука,1986.
10.
В.Паулаускас
.
“Одна
оценка
скорости
сходимости
с
использованием псевдомоментов”. Литов.мат.сб.Т-XI, № 2 ,1971,с.317-327.
11.
Л.Саулис , В.Статулявичус . “Предельные теремы о больших
уклонениях”.Вильнюс,Мокслас,1989
12.
Л.С
.
Ярославцевая
. “Автореферат диссертаци на тему "О точности
аппроксимации
нормальным
распределением
и
асимптотическими
разложениями в терминах псевдомоментов”.
.МГУ имени М.В.Ломоносова,
М.:2008.
International scientific journal
“Interpretation and researches”
Volume 2 issue 22 (44) | ISSN: 2181-4163 | Impact Factor: 8.2
211
13.
М.В.Федорюк. “Асимптотика: Интегралы и ряды”. – М.: Наука,
1987. – 544 с
14.
Р.Бенткус,
Р.Рудзкис
“Об
экспоненциальных
оценках
распределения случайных величин”. Литовск. матем. сб. – 1980. – Т. XX, №1. –
С. 15 – 30.
15.
Р.Рудзкис, Л.Саулис., В.Статулявичус Обшая лемма о вероятностях
больших уклонений. Литовск. матем. сб. – 1978. – Т. XVIII, №2. – С. 99 – 116.
