ABSTRACT

We consider n observations from the GARCH-type model: Z = UY, where U and Y are independent random variables. We aim to estimate density function Y where Y have a weighted distribution. We determine a sharp upper bound of the associated mean integrated square error. We also make use of the measure of expected true evidence, so as to determine when model leads to a crisis and causes data to be lost.

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.

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)

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.

REFERENCES