Wavelet Density Estimation and Statistical Evidences Role for a GARCH Model in the Weighted Distribution ()
1. Introduction
We suppose that 
is a sample of a strictly stationary and exponentially strongly mixing process 
where, for any
,
(1)
is a sequence of identically distributed random variables with common known density 
and 
is a sequence of identically distributed random variables with common unknown density
. For any
, 
and 
are independent. We suppose that 
is a weighted density of the form
(2)
where 
is a known positive function, 
an unknown density of a random variable 
and 
is the unknown normalization parameter:
Our goal is to estimate 
when only 
are observed. The Equation (1) is a GARCH-type time series model classically encountered in financial models see [1] and practical examples of Equation (2) can be found in e.g. [2-4].
In this article, we construct a linear wavelet estimator and measure its performance by determining upper bounds of the mean integrated squared error (MISE) over Besov space.
In what follows, we have also surveyed the role of data and evidential inference in the model. The data play a very important essential role in statistical analysis, to the extent that many statistical researchers believe in the famous saying: “Ask the data.” We consider the Test
(3)
for the model and we evaluate the sensitivity of the value in the test hypotheses. In this test, the evaluation criterion is the area between the curves of the cumulative distribution functions under 
and 
hypotheses. Details on evidential inference can be found in [5,6]. Also [7] have studied about Comparing of record data and random observation based on statistical evidence.
Through the rest of the paper, at first assumptions and then an introduction about wavelets are presented in Section 2. The estimators and results are given in Section 3. In Section 4, general explanations regarding evidential inference and its application in a test. The proofs are gathered in Section 5.
2. Assumptions and Wavelets
2.1. Assumptions
We formulate the following assumptions:
• Without loss of generality, we assume that 
and
have the support 
and 
where
• 
• We suppose that for any
, the 
-th strongly mixing coefficient of 
by
where, for any
, let 
be the 
-algebra generated by 
and 
is the 
-algebra generated by
.
We suppose that there exist three (known) constants, 
and 
such that
This assumption is satisfied by a large class of GARCH processes. See e.g. [8-10].
• For any
, it follows from the independence of 
and 
that the density of 
is
• 
• We suppose that there exists two constants, 
and
, such that
(4)
and
(5)
2.2. Wavelets and Besov Balls
Let 
be a positive integer, and 
and 
be the Daubechies wavelets 
which satisfy
.
Set
Then, there exists an integer 
such that, for any integer
, the collection
is an orthonormal basis of 
(the space of squareintegrable functions on 0,1). We refer to [11].
For any integer
, any 
can be expanded on 
as
where 
and 
are the wavelet coefficients of 
defined by
(6)
Let 
and
. A function 
belongs to 
if and only if there exists a constant 
(depending on
) such that the associated wavelet coefficients Equation (6) satisfy
We set
. Details on Besov balls can be found in [12].
3. Estimators and Results
Firstly, we consider the following estimator for 
(7)
Then, for any integer 
and any
, we estimate
(8)
where, for any
, 
is the operator
(9)
and 
are similar with multiplicative censoring model (see [13]).
We are now in the position to define the considered estimators for
. Suppose that
. We define the linear estimator 
by
(10)
where 
is defined by Equation (8) and 
is the integer satisfying
(11)
• Lemma 3.1
• Let 
be Equation (7) and
. Then we have
• 
• Let 
be Equation (1), 
be Equation (9) and for any integer 
and any
,
. Then we have
• 
• For any integer 
and any
, let
be Equation (8) and
.
Then, under the assumptions of Subsection 2.1, there exists a constant 
such that
Proposition 3.1 For any integer 
and any 
Then, under the assumptions of Subsection 2.1• 
Proposition 3.2 Let
, for any integer 
and any
, let 
be Equation (8) and
. Then• there exists a constant 
such that
• 
• there exists a constant 
such that
• 
• there exists a constant 
such that
Theorem 3.1 (Upper bound for
) Consider Equation (1) under the assumptions of Subsection 2.1. Suppose that 
with
. For any
and 
be Equation (10), then there exists a constant 
such that
Remark that 
is the slower than the optimal one in the standard density estimation problem i.e. 
(see e.g. [14, Chapter 10]). This deterioration is due to the presence of GARCH model and weighted distribution.
4. Statistical Evidence
4.1. Statistical Inference
The evidential approach to statistical inference concerns a novel approach in statistical analysis. Evidential inference is solely based on data as evidence and calculation of the evidence strength. It is not influenced by mental and personal components and factors such as former beliefs and loss functions. Using evidential inference in the model Equation (1), we will survey when censoring of data will lead to considerable data loss, and we will determine the time when data is lost by determining an appropriate criterion. In the model
for
, the data observed from the variable 
are denoted by the subscript (cen), and the data observed from the variable 
are denoted by the subscript (ncen). Considering the Test Equation (3) in the above model, due to the symmetry of the test hypotheses in evidential methods and without losing the generality of the problem, the value of is assumed to be
. In order to support 
and 
hypotheses, we now use the following criterion:
(12)
where 
and 
are the measure of expected true evidence in the censored and uncensored data respectively, and 
is the criterion of the support of data from 
hypothesis against 
hypothesis. This support criterion is optimal when the area between the two curves of 
cumulative functions under 
and 
is maximum, please see [15]. This area which is denoted by 
in the form of
where 
is the cumulative distribution function of 
and 
is the mean value of 
under 
hypotheses. In view of [6], the support criterion 
is defined as follows:
(13)
where 
is the likelihood ratio and for the two censored and uncensored cases we have
(14)
where 
and 
are likelihood functions for 
and 
variables respectively, in the Equation (1) under 
hypotheses.
4.2. Measuring Statistical Evidence
We consider i.i.d case for variables in the Equations (1) and (2), also set 
and then, we investigate the behavior of 
by means of simulation. In addition, we analyze 
and 
hypotheses in Test Equation (3) by determining support criterion 
of the measure of expected true evidence
. The programming codes of this part are written in the MAPLE (15) environment.
Example 1. In this example, we generate data from gamma and uniform distribution as follows, considering multiplicative censoring model:
Then, according to Equations (13) and (14), we calculate the support criterion and the likelihood ratio via below relations:
such that
and
such that
For different values of 
and
, we calculate the value of 
according to Equation (12). The results can be observed in Table 1. By carefully considering this table, it is observed that as the value of 
increases (which implies the distance growth between 
and
), the value of 
gets closer to one. In other words, 
approaches 
more and more. This fact can be interpreted in this way that if the distance between 
and 
is large, the data lost in censored data 
is negligible. That is to say, evidential inference draws our attention to the time when 
is close to
. The above analysis can also be observed in Figure 1.
In what follows, the variations of sample volume ratio increase against 
are investigated, and the value of
in the 
equation is determined for different values of 
and
.
If 
then
. The result can be viewed in Figure 2, If 
then
. The result can be viewed in Figure 3, If 
then
. The result can be viewed in Figure 4.
The above results can be interpreted in this way that as the sample volume increases from a certain stage on, the value of 
remains constant. In other words, it can be intuitively said that the 
ratio tends to the constant 
value, which this also leads to an increase in evidential strength.
5. Proofs
In this section, 
denotes any constant that does not depend on 
and
. Its value may change from one term to another and may depends on 
or
.
Table 1. Computed values for γ.
Figure 1. γ computed from gamma distribution for different values of n.
Figure 2. The most relative changes when sample size n1 = 20 increases to n2 = 25 happens in α = 1.6.
Figure 3. The most relative changes when sample size n1 = 10 increases to n2 = 15 happens in α = 2.1.
Proof of Lemma 3.1.
Proof can be found in [13].
Proof of proposition 3.1.
1. By Equation (9) and Equation (5), we have
(15)
Figure 4. The most relative changes when sample size n1 = 10 increases to n2 = 20 happens in α = 2.2.
Therefore
(16)
2. By Equation (9) and Equation (5), we have
(17)
By Equation (4) and under the assumptions of Subsection 2.1 and also doing the change of variables
, we have
(18)
Using again Equation (4) and the assumptions of Subsection 2.1, the equality 
and doing the change of variables
, we obtain
(19)
Combining Equations (17)-(19), we obtain
(20)
The proof of Proposition 3.1 is complete.
Proof of Proposition 3.2.
1. We have
(21)
Using Equation (20) we obtain
(22)
also
(23)
By the Davydov inequality (see [16]) and for any 
we obtain
(24)
Putting Equation (24), Equation (16) and Equation (20) together, we have
(25)
Since
, we obtain
(26)
Putting Equations (21) and (22) and Equation (26) together, we have
(27)
2. We have
(28)
By Equation (5), we have
(29)
By the Davydov inequality (see [16]) we obtain
(30)
Combining Equations (28)-(30), we obtain
(31)
3. Using Lemma (3.1), Equation (27) and Equation (31), then
(32)
The proof of Proposition 3.2 is complete.
Proof of Theorem 3.1. For any integer
, any 
can be expanded on 
as
where
We obtain
where
Using Proposition 3.2 and inequalitity Equation (11)
and since
, we have
. Hence
Hence we have
This ends the proof of Theorem 3.1.