TL-Moments for Type-I Censored Data with an Application to the Weibull Distribution

This paper aims to provide an adaptation of the trimmed L (TL)-moments method to censored data. The present study concentrates on Type-I censored data. The idea of using TL-moments with censored data may seem conflicting. However, our perspective is that we can use data censored from one side and trimmed from the other side. This study is applied to estimate the two unknown parameters of the Weibull distribution. The suggested point is compared with direct L-moments and maximum likelihood (ML) methods. A Monte Carlo simulation study is carried out to compare these methods in terms of estimate average, root of mean square error (RMSE), and relative absolute biases (RABs).


Introduction
In the past fifty years, there has been great attention given to the use of unconventional estimation methods in the theory of estimation in addition to the classical methods.Classical estimation methods (e.g., method of moments, method of least squares, and maximum likelihood method) work well in cases where the distribution is exponential.However, in some applications, the data may contain some extreme observations, which can greatly influence the values of the estimator.Therefore, if there is a concern about outliers, one should use a robust method of estimation that has been developed to reduce the influence of outliers on the final estimates.Using a robust estimation techniques for estimating unknown parameters has great importance for investigators in many fields, such as in industrial, medical, and occasionally in business applications.In recent decades, much of the work on dealing with outliers has been focused on robust estimation methods (e.g., [1]).
The L-moments method has been noticed as an appealing alternative to the conventional moments method [2].To avoid the effect of outliers, Elamir and Seheult [3] introduced an alternative robust approach of L-moments which they called trimmed L-moments (TL-moments).TL-moments have some advantages over L-moments and the method of moments.TL-moments exist whether or not the mean exists (e.g., the Cauchy distribution), and they are more robust to the presence of outliers.
The idea of TL-moments is that the expected value E(X r−k:r ) is replaced with the expected value E(X r+t 1 −k:r+t 1 +t 2 ).Thus, for each r, we increase the sample size of a random sample from the original r to r + t 1 + t 2 , working only with the expected values of these r modified order statistics X t 1 +1:r+t 1 +t 2 , X t 1 +2:r+t 1 +t 2 , ..., X t 1 +r:r+t 1 +t 2 by trimming the smallest t 1 and largest t 2 from the conceptual random sample.This modification is called the rth trimmed L-moment (TL-moment) and marked as λ (t 1 ,t 2 ) r .
The TL-moment of the rth order of the random variable X is defined as: The expectation of the order statistics are given by: Its basic idea for the method of expectation is to take the expected values of some functions of the random variable of interest and extend them to a sample and equate the corresponding results and solve for the unknown parameters.
This paper is concerned with comparing the performance of three estimating methods-namely, TL-moments, direct L-moments, and maximum likelihood (ML)-with Type-I censored data.It is straightforward to adapt the methods for Type-II censored data.This study is applied to the estimation of the two unknown parameters of the Weibull distribution by a quantile function that takes the form: This article is organized as follows: TL-moments for censored data, in the general case, are introduced in Section 2. TL-moments for the Weibull distribution are presented in Section 3. A simulation study and real data analysis are presented in Sections 4 and 5, respectively.Concluding remarks are presented in Section 6.

TL-Moments for Censored Data
For the analysis of censored samples, Wang [4][5][6] introduced the concept of partial probability-weighted moments (PPWMs).Hosking [7] defined two variants of L-moments, which he used with right-censored data.Zafirakou-Koulouris et al. [8] extended the applicability of L-moments to left-censored data.Mahmoud et al. [9] introduced two variants of what they termed the method of direct L-moments, and used them of right-and left-censored data from the Kumaraswamy distribution.
The aim of this section is to introduce an adaptation of the TL-moments method to censored data.In fact, the idea of using TL-moments with censored data may seem to be conflicting, but the idea is that we may use data censored from one side and trimmed from the other side.

Right Censoring for Left Trim
Let x 1 , x 2 , ..., x n be a Type-I censored random sample of size n from a population with distribution function F(x) and quantile function q(u).From the formula of TL-moments (1), we know that TL-moments are defined as: We suppose a left trim t 1 (i.e., t 2 = 0).From Formula (3), we get In this case, let the censoring time T satisfy F(T) = c and c be the fraction of observed data.The random sample takes the form x t 1 +1 , x t 1 +2 , ..., x n .

TL-Moments for Right Censoring (Type-AT)
The quantile function of Type-AT TL-moments is ( Substitution into Equation ( 4) leads to the Type-AT TL-moments where: When we suppose that the value of the smallest trim is equal to one (i.e., t 1 = 1), from (6), we get: In this case, the first four Type-AT TL-moments are given by the following: When we suppose that the value of the smallest trim is equal to two (i.e., t 1 = 2), from (6), we get: Substituting r = 1, 2, 3, 4 in Equation ( 9), we get the first four Type-AT TL-moments: Using the method of expectations, Type-AT TL-moments estimators are given by: 2.1.2.TL-Moments for Right Censoring (Type-BT) The quantile function of Type-BT TL-moments is Substitution into the formula of left trimming in (4), the Type-BT TL-moments are given by µ Using the results in (A9), the second integration can be written as: where β c (a, b) is the upper incomplete beta function.
When we suppose the value of smallest trim is equal to one (i.e., t 1 = 1), from (12), we get: In this case, the first four TL-moments for Type-BT right censoring are calculated as follows: When we suppose that the value of the smallest trim is equal to two (i.e., t 1 = 2), from (12), we get In this case, the first four TL-moments for Type-BT right censoring are calculated as follows: Using the method of expectations, Type-BT TL-moments estimators are given by:

Left Censoring for Right Trim
Let x 1 , x 2 , ..., x n be a random sample of size n.We suppose right trim t 2 (i.e., t 1 = 0).From Formula (3) we get: In this case, the random sample becomes of the form x 1 , x 2 , ..., x n−t 2 .Type-I left censoring occurs when the observations below censoring time T are censored: Let censoring time T satisfy F(T) = h, where h is the fraction of censored data.

TL-Moments for Left Censoring (Type-A T)
The quantile function of Type-A T TL-moments is: Substitution into (18) leads to the Type-A T TL-moments where: When we suppose that the value of the largest trim is equal to one (i.e., t 2 = 1), from (19), we get: In this case, the first four TL-moments for Type-A T left censoring are calculated as follows: When we suppose that the value of the largest trim is equal to two (i.e., t 2 = 2), from (19), we get: In this case, the first four TL-moments for Type-A T left censoring are calculated as follows: Using the method of expectations, Type-A T TL-moments estimators are given by:

TL-Moments for Left Censoring (Type-B T)
The quantile function of Type-B T TL-moments is Substitution into Equation (18) leads to the Type-B T TL-moments where: Using the results in (A8), the first integration can be written as where β z (a, b) is the lower incomplete beta function.
When we suppose the value of largest trim is equal to one (i.e., t 2 = 1), from (25), we get: The first four TL-moments for Type-B T left censoring are calculated as follows: When we suppose the value of the largest trim is equal to two (i.e., t 2 = 2), from (25), we get The first four TL-moments for Type-B T left censoring are calculated as follows: Using the method of expectations, Type-B T TL-moments estimators are given by:

Application to the Weibull Distribution
In this section, the rth population TL-moments for the Weibull distribution are introduced.

Right Censoring with Left Trim
• Type-AT; t 1 = 1 From Equation ( 6), the rth population Type-AT TL-moments for Type-I right censoring for the Weibull distribution are: By taking that the value of smallest trim is equal to one (t 1 = 1), from (7) we get: Substituting r = 1, 2 in Equation (8a), the first two Type-AT TL-moments for Type-I right censoring with left trim for the Weibull distribution will be: Putting z = − log(1 − u), this equation becomes: Using the results in (A3), this equation can be written as: where γ(c, b) is the lower incomplete gamma function.
Similarly, from Equation (8b), we can also obtain the second Type-AT TL-moments, when t 1 = 1, for Type-I right censoring for the Weibull distribution as follows: When we suppose that the value of the smallest trim is equal to two (i.e., t 1 = 2), from (9), we get: Substituting r = 1, 2 in Equation ( 35), the first two Type-AT TL-moments for Type-I right censoring with left trim for Weibull distribution will be: and, • Type-BT; t 1 = 1 From Equation (12), the rth population Type-BT TL-moments for Type-I right censoring for the Weibull distribution are: By taking that the value of the smallest trim is equal to one (t 1 = 1), from (13) we get: Substituting r = 1, 2 in Equation ( 39), the first two Type-BT TL-moments for Type-I right censoring with left trim for Weibull distribution will be: and, When we suppose that the value of the smallest trim is equal to two (i.e., t 1 = 2), from (15), we get: Substituting r = 1, 2 in Equation ( 42), the first two Type-BT TL-moments for Type-I right censoring with left trim for Weibull distribution will be: and, (44)

Left Censoring with Right Trim
• Type-A'T; t 2 = 1 From Equation (19), the rth population Type-A T TL-moments for Type-I left censoring for the Weibull distribution are: When we suppose that the value of the largest trim is equal to one (i.e., t 2 = 1), from (20), we get: Substituting r = 1, 2 in Equation ( 46), the first two Type-A T TL-moments for Type-I left censoring with right trim for Weibull distribution will be: and, When we suppose that the value of the largest trim is equal to two (i.e., t 2 = 2), from ( 22), we get: Substituting r = 1, 2 in Equation ( 49), the first two Type-A T TL-moments for Type-I left censoring with right trim for Weibull distribution will be: • Type-B T; From Equation (25), the rth population Type-B T TL-moments for Type-I left censoring for the Weibull distribution are: When we suppose the value of largest trim is equal to one (i.e., t 2 = 1), from (26), we get: Substituting r = 1, 2 in Equation ( 53), the first two Type-B T TL-moments for Type-I left censoring with right trim for Weibull distribution will be: Table 1.The estimates, root of mean square error (RMSE) and relative absolute biases (RABs) for two parameters of the Weibull distribution using TL-moments, direct L-moments, and ML method based on left censoring (a = 0.5 and b = 5) in the presence of outliers.

Table 2 .
The estimates, root of mean square error (RMSE) and relative absolute biases (RABs) for two parameters of the Weibull distribution using TL-moments, direct L-moments, and ML method based on left censoring (a = 2 and b = 4) in the presence of outliers.

Table 4 .
The estimates, root of mean square error (RMSE) and relative absolute biases (RABs) for two parameters of the Weibull distribution using TL-moments, direct L-moments, and ML method based on right censoring (a = 0.5 and b = 5) in the presence of outliers.

Table 5 .
The estimates, root of mean square error (RMSE) and relative absolute biases (RABs) for two parameters of the Weibull distribution using TL-moments, direct L-moments, and ML method based on right censoring (a = 2 and b = 4) in the presence of outliers.