Estimation of Hazard Function for Censoring Random Variable by Using Wavelet Decomposition and Evaluation of MISE , AMSE with Simulation

Wavelet analysis is one of the mostly new methods of pure and applied mathematics science. In this paper, we use the wavelet method to estimate the hazard function for censoring random variable. We consider the convergence ratio of given estimator. Also we present the simulation in order to test purpose estimator by calculating the mean integrated squared error (MISE) and average mean squared error (AMSE).


Introduction
One of data types, which researchers are extremely interested in, is carrying to the time interval till the occurrence of certain events such as death etc.Every process waiting for a specific event produces survival data.Failure in survival analysis means the occurrence of the event that we were waiting for.The time, which survival is measured after that point, is called the start time.
The failure time which is denoted by i T , is the time that failure occurs for each individual.It's not always possible to observe the failure time for each individual in such cases that censorship occurs.
Survival function, which is shown by ( ) S t , indicates the ratio of people who survived since the base time which is the point they enter the experiment to the time unit t analysis.Hazard function for the failure continuous time is as follows: In this paper, we obtain estimator hazard function for censoring data by using wavelet method.We evaluate convergence ratio of given estimator by simulation.

Estimation of Hazard Function by Using Wavelet Method
Wavelets can be used for transient phenomena analysis or function analysis which sometimes changes rapidly.
They are symmetrical and have limited period.A close relationship between wavelet coefficients and some spaces is wavelet bases orthogonally.Also useful properties of them in wavelet issues simplify the computational algorithms.As a result, numerous articles have been published about in statistical science.The mathematical theorem of wavelets and their application in statistics have been studied as a technique for density function estimator, by Harr [1], Doukhan [2], Antoniadys [3], nonparametric curve estimators by Malat [4], Meyer [5], Daubechies [6], Donoho [7], Kyacharyan and Picard [8], Hall and Patil [9] have found a formula for the Mean Integrated Squared Error of Nonlinear Wavelet based on density estimators.Antoniadys et al. [10] achieved the density function estimator and the hazard function for right-censored data with the wavelets.Daubechies [11] studied and discussed the compactly supported wavelets which produce orthogonal bases.Afshari et al. [12][13][14] studied about density, derivative density function estimator, regression function for the mixing random variables.
Let the nested sequence of closed subspaces; , , Given above Wavelet basis, a function ( ) where As for general orthogonal series estimator, Daubechies [4], density estimator can be writhen as: where the obvious coefficient estimator can be written: In this article, 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.
Suppose 1 2 , , , n X X X ⋅⋅⋅ are failure time of n tests that are studied.They are non-negative, independent, identically distributed, with the density function f and distribution function F. Also suppose that 1 2 , , , n C C C ⋅⋅⋅ are corresponding to censored times, non-negative, indepen-dent, identically distributed, with the density function g and distribution function G .
Assuming independency of failure times and censored time of the observed random variable, i Z and the function i δ and hazard function are shown as below: We assume that, then we can write as follows: To estimate ( ) h t we need the estimator of ( ) , we divide the time axis into two parts of small intervals and the amounts of events (0 or 1) in each interval, and then we divide these values to the length of intervals.
Estimation procedures of ( ) can be summarized as the following: Select 0 ∆ > and collect the observed failures in 1 k + intervals with the length ∆ and using wavelet estimation on the collected data.We find an estimate of sub density.This means that we calculate the collected wavelet coefficients data on the scale of ( ) j n by choosing the decomposition level ( ) j n and then we estimate ( ) . It is necessary to state the following symbols to show the details: We figure estimators on the finite interval [ ] In fact we suppose Suppose that N is an integer that could be dependent to n and the estimated points are as follows: and we divide the interval [ ] 0,τ of time axis to The k-th interval is marked by k J so: . Now we define the following indicator function that indicates the number of uncensored failures in the time interval k J : We assume that k U is the observed failures ratio in the interval k J , in other words: We smooth the data k U ∆ by an appropriate wavelet smoother to find the estimation of f * .We can write as the following: .
where, ( ) ( ) The complex structural polymorphism analysis causes an efficient tree construction algorithm for analysis of functions in V N with theoretic scale wavelet coefficients not well available and we need an initial value for a fast wavelet transform.Antoniadis [4] suggested the following initial amount: τ and ϕ is regular of degree m .We esti- mate the unknown function f * as follows to level the data with a better rate for the sample size n and the sequence ( ) That it is the orthogonal image of N f *  on the leveler approximation space ( ) V .Now we consider an appropriate consistent estimator of ( ) L t , and finally we estimate the Hazard function.We assume that 1 , we can write: Suppose that ( ) as n → ∞ , then we define: so we can write as the following: ( ,0 0 1 By substituting Equation (9) in Equation ( 8), we obtain 1 Theorem: Suppose that the sub density ( ) By using Chung-Smirnov property and Taylor's theorem we can write as the following: By using Equations ( 10) and ( 11), we can write: then the proof is completed.

Numerical Computation and Simulation
In this section, we simulate ( ) ˆn k h t on the data of size n by using Semlayt's wavelet.We consider convergence ratio of given estimator by computing of average mean square error of given estimators.We use R software and wavelet package for simulations.
Example 1: We generate ( ) , , , , ~6 The results in Table 2 display the average mean square

W
. Other wavelets in the basis are then generated by translation of the scaling function and dilations of the mother wavelet by using the relationships:

,
The solid line in the Figure1displays the wavelet estimate of hazard function with the denoted line representing the true hazard rateThe results in Table1display the average mean square errors of hazard function estimator for sample sizes 400 n from sample size of n = 400 and n = 600 with K = 16, K = 32, K = 64 and 0.05 ∆ = .The solid line in the Figure2displays the wavelet estimate of hazard function with the denoted line representing the true hazard rate.

Figure 1 .
Figure 1.The panel in Figure 1 displays the wavelet estimator of hazard function ( ) ˆn k h t with the denoted line representing the true hazard rate.

Figure 2 .
Figure 2. The solid line in the panel displays the wavelet estimate of hazard function with the denoted line representing the true hazard rate.
function estimator for sample sizes n = 400 and n = 600.