1. Introduction

The statistical analysis of time-to-event data shows a main role in many of various application areas such as, industrial (reliability testing), medical (survival analysis), and demographic (migration analysis, life events research). The hazard rate is used greatly as a methodological tool in these types of usages to define the instantaneous risk of observing the event of interest over time.

The L-moments and TL-moments have an appealing property that the system of moments does not have; see Hosking (1990). L-moments of any order k, for example, exist. The entire sequence of L-moments is available for heavy-tailed distributions under only a finite first moment assumption. The weaker assumption of L-moments is used to describe TL-moments, which does not require a finite first moment assumption.

The estimation of the hazard function is crucial in many statistical applications, especially in the data-analytic and functional statistical methods advocated by Nair and Sanakaran 2009.

2. Quantile Function in Term of TL-Moments

In statistical analysis, the quantile function method is a valuable tool.

The method of moments is widely used to measure descriptive features of a univariate distribution, but its application is limited to sufficiently light-tailed distributions; see Bera and Bilias2002. The sequence of L-moments and TL-moments, which take the form of expectations of selected linear functions of order statistics, offer an appealing alternative.

is defined as:

(1)

is the population cdf. If F is an absolutely continuous with density f = and is one-to -one Q is differentiable on the open unite intrveral,

And

is called the quantile density function (qdf). In this case

. Differentiation on both sides give

the function

is called the density quantile function; see Parzen (1979), Cheng and Parzen (1997) and Cheng (2002). Thus, the density quantile function is another related quantity which obtained from the pdf, by substituting for x with the quantile function, as see Parzen (1979).

(2)

As

and for any paire of values it follows from the deffinition of differention that

Hence and, therefore, expressing all in terms of

The two functions

are thus reciprocals of each other. Thus, the density-quantile function can be defined in terms of quantile density function as

Otherwise, the quantile density function can be defined in terms of density quantile function as:

(3)

3. Representation of Density Quantile Function in Terms of TL-Moments

Hosking (2007), derived the TL-moments as the coefficients in the expansion of the quantile function in the form of a weighted sum of shifted Jacobi polynomials as

(4)

where is shifted Jacobi polynomials which is given by

(5)

and,

(6)

This is convergent in the weighted mean square with weight function , i.e.

(7)

the remainder after stopping the sum, after s terms, satisfies

It also represents of density function in terms of TL moments by the following theorem

Let X be a continuous real-valued variable such that its trimmed mean is finite, quantile function.

density function

And Quantile density function

and trimmed than the

Representation

(8)

This is convergent in the weighted mean square with weight function This imply that

(9)

4. Hazard Quantile in Term of TL- Moments

Nair and Sankaran introduced hazard quantile function as

Where q(u) is the quantile density function defined by

From equation (8) and (9)

(10)

This is convergent in the weighted mean square with weight function i.e.

(11)

proof:

let

5. Estimation of the Hazard Quantile Function

From a random sample x_1,x_{2,……………,}x_n of size n from the population where its corresponding order statistics is x_{1:n <} x_{2:n <……………<} x_n:n by replacing the population TL-moments by it sample version sample TL-moments

From (4) we can estimate the quantile function as

(12)

While, from (8) we can estimate the quantile density function

(13)

Thus, from (10) the hazard quantile function can be estimated as

(14)

6. Optimal Trimming Depends on the Hazard Quantile Function

There is method for obtaining optimal trimming depends on the hazard quantile function.

The error between the hazard quantile function and its TL- moments representation, in (14) can be written as

(15)

The optimal values of t₁ and t₂ can be chosen as the values which have less sum of the absolute error

Where

(16)

7. Application

We study the approximation of hazard quantile, h(u), based on TL-moments in this section, using some known distributions and real data. Using the formulas (16) and (15) that meet the two requirements in (10) and respectively (11) for different values of trimming (t₁, t₂) and terms (r), we compute the sum absolute error which is given in (16).

As the basis of our comparisons, we simulate 50,100, and150 observations from log normal and normal distributions. We take the hazard quantile which have and, in the same time, best fitting to distribution. In addition, we will compare this new approximation of the hazard quantile with the corresponding histogram and kernel density.

7.1. Lognormal Distribution

X ∼ log normal is used to indicate that the random variable has the log normal distribution with parameters and . A log normal random variable with parameters and has probability density function

Table 1. Simulated data from lognormal distribution with µ = 0 and σ = 1 using n = 50, 100, 150

Figure 1. Approximation of the hazard and density functions of lognormal distribution (µ = 0 and σ = 1), using terms (r=3), trimming (t1=1, t2=1) and sample size n=50

Figure 2. Approximation of the hazard and density functions of lognormal distribution (µ = 0 and σ = 1), using terms (r=6), trimming (t1=1, t2=1) and sample size n=100

Figure 3. Approximation of the hazard and density functions of lognormal distribution (µ = 0 and σ = 1), using terms (r=4), trimming (t1=1, t2=1) and sample size n=150

7.2. Normal Distribution

Normal distributions are useful in statistics and are often used in the natural and social science for real valued random variables whose distributions are not known.

If X is distributed according to the normal distribution, then

Table 2. Simulated data from normal distribution with µ = 0 and σ = 1 using n = 50, 100 and 150

Figure 4. Approximation of the hazard and density functions of normal distribution (µ = 0, σ = 1), using terms (r=6), trimming (t1=1, t2=0) and sample size n=50

Figure 5. Approximation of the hazard and density functions of normal distribution (µ = 0, σ = 1), using terms (r=6), trimming (t1=0, t2=0) and sample size n=100

Figure 6. Approximation of the hazard and density functions of normal distribution (µ = 0, σ = 1), using terms (r=6), trimming (t1=1, t2=1) and sample size n=150

7.3. Real Data

We add an approximation to the hazard quantile function for real data using the same technique of approximation to know distributions.

Table 3 shows the ages of 155 patients with Stomach Tumors who were admitted to (Zagazig University Hospital, Pain Clinic, Egypt) between June and December 2020.

Table 3. Data of the ages for 155 patients of Stomach Tumors taken from (June-December 2020) whose entered in (Zagazig University Hospital, Pain Clinic, Egypt)

Figure 7. Approximation of hazard quantile function of Stomach Tumors size n=155, trimming (t1=1,t2=2) and using two terms (r=2)

Figure 8. Approximation of hazard quantile function of Stomach Tumors size n=155, trimming (t1=1,t2=2) and using four terms (r=4)

Figure 9. Approximation of hazard quantile function of Stomach Tumors size n=155, trimming (t1=1,t2=2) and using six terms (r=6)

8. Conclusions

The population hazard function (which indicated the approximation of hazard quantile function) is approximated using a nonparametric methodology based on the TL-moments and the orthogonal Jacobi polynomial as an approximation. This methodology has the capacity to acquire more information regarding the data's arbitrary distribution. This method is focused on reducing the sum absolute error between the population hazard quantile function and its portrayal in TL-moments.

Unlike its parametric counterpart, there is no need to make any assumptions about the underlying distribution. We also used symmetric and asymmetric distributions to demonstrate the benefits of the suggested technique.

The performance of the proposed estimator is tested by applications to real-life and simulated data. Also, a comparison of its performance to that of the normal estimator indicates that the lognormal estimator performs better than that of the normal estimator near the boundary.