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Wavelet has rapid development in the current mathematics new areas. It also has a double meaning of theory and application. In signal and image compression, signal analysis, engineering technology has a wide range of applications. In this paper, we use wavelet method, for estimating the density function for censoring data. We evaluate the mean integrated squared error, convergence ratio of given estimator. Also, we obtain empirical distribution of given estimator and verify the conclusion by two simulation examples.

One of data types, which researchers are extremely interested in, is caring to the time interval till the occurrence of certain events such as death etc. Any process waiting for a specific event produces survival data. Survival function, which is shown by

Wavelets can be used for transient phenomena analysis or functions analysis which sometimes changes rapidly, and they are symmetrical and have limited period unlike rugged Sine waves, thus the signals with radical changes are analyzed better. The close relationship between wavelet coefficients and some spaces, wavelet bases being orthogonal and also useful properties of them in wavelet issues simplify the computational algorithms.

Wavelets theory was proposed by Alfred Harr [

Such that

Also for mother wavelet and father wavelets the following:

Definition 1-1: Assume that

Spaces

1-

4-

6-

If we consider the scale function in the interval_{j} is defined as

thus

Let the nested sequence of closed subspaces; …

The term wavelets are used to refer to a set of basis functions with very special structure. The special of wave- lets basis for function

Given above Wavelet basis, a function

where

As for general orthogonal series estimator, Daubechies [

where the obvious coefficient estimator can be written:

We divide time axis into two parts, the intervals and the number of events in each interval. We determine number of events and hazard function according to the observations. Then we flatten them separately via linear wavelet density estimation on the whole time and then we calculate the function estimator and evaluate the asymptotic distribution.

In this paper we obtain estimator density for censoring data by using wavelet method and evaluate mean integral square error with convergence ratio and empirical distribution of given estimator.

Wavelets can be used for transient phenomena analysis or functions analysis which sometimes changes rapidly, and they are symmetrical and have limited period unlike rugged Sine waves, thus the signals with radical changes are analyzed better. The close relationship between wavelet coefficients and some spaces, wavelet bases being orthogonal and also useful properties of them in wavelet issues simplify the computational algorithms. As a result, numerous articles have been published about density function estimation. The mathematical theorem of wavelets and their application in statistics have been studied as a technique for nonparametric curve estimators by Antoniadys [

Afshari [

Suppose

Assuming independency of failure times and censored time of the observed random variable,

Such that

Also we definite as follows:

To estimate

Estimation procedures of

Select

We figure estimators on the finite interval

tistic

Suppose that

Suppose that

The

Now we define the following indicator function that indicates the number of uncensored failures in the time interval

interval

Theorem 2-1: Suppose that the sub density

Proof: see [

We smooth the data

We can write,

where,

The complex structural polymorphism analysis causes an efficient tree construction algorithm for analysis of functions in

As a result a reasonable estimate for image of

If we assume that the collected values

That it is the orthogonal image of

Theorem 2-2: Suppose that the sub density

Proof: by using theorem (2-1) we can write:

Since,

So Equations (9) can be written as follows:

By using Equation (1) we have:

By using Equations (10) and (11) we have:

By using theorem (2-1) we can writhe as follows:

Using this fact that

Since

According Equation (13), we can write:

In this section we evaluate mean integral square error and convergence ratio is investigated.

Definition 3-1: The mean integrated square error (MISE) of kernel estimator of a density function

Theorem 3-1: Suppose that the sub density

Proof:

By using Equation (15) and theorem (2-2) for

Because

So by using Equations (16) and (17), we can write:

For evaluate

Also we can write:

By using theorem (2-1) and expectation of Equation (19), we can write as the following:

By using theorem (2-1) we have:

By using Equation (22) and this fact that

The second part of Equation (20) can be written as the following:

By using

In this section we investigate empirical distribution of estimator under some condition.

Theorem 4-1 Suppose that the sub density

Proof:

By using theorems (2-1) and (2-2), we can write as the following:

So by using equation of (23) and (24) we can write as the following:

We prove that II has asymptotically normal distribution and also I, III tend to zero when

First, we show that I, III tend to zero when

By using Equation (23) we have:

So by using Equation (24) and (25), the phrase I, III tend to zero when

So we have:

Such that for each fixed

By using cushy Schwartz inequality:

So we can write as the following:

Using this fact that

Thus, the Equation (26) state is convergent in

Also by using Theorem (2-2), we have:

Thus we have:

We control the Lindberg condition in order to prove that II is asymptotically normal. For this purpose, we

set:

By using cushy Schwartz inequality:

following:

In this section we simulate,

Example 1: We generate

The results in

The panel in

Example 2: Suppose that

The results in

The panel in

. The average mean square errors of subdensity function estimator by wavelet method

17.9 10.1 7.2 | 26.1 19.2 18.6 | 8 16 32 |

. The average mean square errors of subdensity function estimator by wavelet method

610 275 278 | 680 420 379 | 8 16 32 |

The wavelet subdensity and true density estimator

The wavelet subdensity and true density estimator

In this paper we obtain density estimation for censoring data by using wavelet method and evaluate mean integral square error. We show that convergence ratio is acceptable and empirical distribution of given estimator under some condition is normal.

The support of Research Committee of Persian Gulf University is greatly acknowledged.