Sahar Khairyfarag, Zohdy Mohamed Nofal, Mohamed Goda Hindawy
Department of Statistics, Mathematics and Insurance, Banha University, Egypt
Correspondence to: Sahar Khairyfarag, Department of Statistics, Mathematics and Insurance, Banha University, Egypt.
Email:  
Copyright © 2022 The Author(s). Published by Scientific & Academic Publishing.
This work is licensed under the Creative Commons Attribution International License (CC BY).
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Abstract
When a parametric form for the underlying distribution is not available, the estimation of hazard quantile is of major interest. Furthermore, the hazard quantile function can be employed instead of the hazard function. This work describes a new nonparametric hazard quantile function estimator based on TLmoments approaches in detail.
Keywords:
Quantile, Hazard, Hazard quantile, TLmoments
Cite this paper: Sahar Khairyfarag, Zohdy Mohamed Nofal, Mohamed Goda Hindawy, On Nonparametric Methods for Estimating the Hazard Function with Application, International Journal of Probability and Statistics , Vol. 11 No. 1, 2022, pp. 918. doi: 10.5923/j.ijps.20221101.02.
1. Introduction
The statistical analysis of timetoevent 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 Lmoments and TLmoments have an appealing property that the system of moments does not have; see Hosking (1990). Lmoments of any order k, for example, exist. The entire sequence of Lmoments is available for heavytailed distributions under only a finite first moment assumption. The weaker assumption of Lmoments is used to describe TLmoments, which does not require a finite first moment assumption.The estimation of the hazard function is crucial in many statistical applications, especially in the dataanalytic and functional statistical methods advocated by Nair and Sanakaran 2009.
2. Quantile Function in Term of TLMoments
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 lighttailed distributions; see Bera and Bilias2002. The sequence of Lmoments and TLmoments, 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 oneto 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 thatHence and, therefore, expressing all in terms of The two functions are thus reciprocals of each other. Thus, the densityquantile function can be defined in terms of quantile density function asOtherwise, the quantile density function can be defined in terms of density quantile function as:  (3) 
3. Representation of Density Quantile Function in Terms of TLMoments
Hosking (2007), derived the TLmoments 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, satisfiesIt also represents of density function in terms of TL moments by the following theoremLet X be a continuous realvalued variable such that its trimmed mean is finite, quantile function. density function And Quantile density function and trimmed than theRepresentation  (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 asWhere q(u) is the quantile density function deﬁned byFrom 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 TLmoments by it sample version sample TLmoments 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 t_{1 }and t_{2 }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 TLmoments 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 (t_{1}, t_{2}) 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 functionTable 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 (t_{1}=1, t_{2}=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 (t_{1}=1, t_{2}=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 (t_{1}=1, t_{2}=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 (t_{1}=1, t_{2}=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 (t_{1}=0, t_{2}=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 (t_{1}=1, t_{2}=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 (JuneDecember 2020) whose entered in (Zagazig University Hospital, Pain Clinic, Egypt) 
 

 Figure 7. Approximation of hazard quantile function of Stomach Tumors size n=155, trimming (t_{1}=1_{,}t_{2}=2) and using two terms (r=2) 
 Figure 8. Approximation of hazard quantile function of Stomach Tumors size n=155, trimming (t_{1}=1_{,}t_{2}=2) and using four terms (r=4) 
 Figure 9. Approximation of hazard quantile function of Stomach Tumors size n=155, trimming (t_{1}=1_{,}t_{2}=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 TLmoments 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 TLmoments.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 reallife 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.
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