Direct L-moments for Type-I censored data with application to the Kumaraswamy distribution

Abstract: This paper suggests a modification of L-moments method to make it suitable for censored data directly (namely: Direct L-moments). This study concentrates on Type-I censored data. The modification is applied to estimate the unknown parameters of Kumaraswamy (Kw) distribution. The suggested modification is compared with L-moments via partial probability-weighted moments (PPWM) method and maximum likelihood (ML) method. The results are achieved using a comparative numerical study in terms of estimate of the unknown parameters, relative bias and root of mean square error (RMSE) using Monte Carlo simulation.


Introduction
Finding robust and reliable estimates of unknown parameters has a great importance to the practitioner, whether he does this repeatedly as in industrial and medical applications, or occasionally as in business applications. For estimating the unknown parameters, conventional moments method considerable one of the most important methods of estimation. But it is often considerably less

PUBLIC INTEREST STATEMENT
The Kumaraswamy distribution is a very good alternative to the beta distribution for modeling random processes which values are bounded from below and above. It has been utilized to model certain measurements in genetics, hydrology and ecology. Fitting this model to a given dataset requires the estimation of the parameters of the distribution. Among the methods used for estimating the parameters we find, the method of moments is used frequently. The use of the L-moments has been suggested as an alternative to the method of moments. In this article, the L-moments are utilized to estimate the parameters of the Kumaraswamy distribution.
der of the random variable X is defined by Hosking (1990) as Now, the expectation of the order statistics are given by: The sample L-moments can be estimated unbiasedly from the sample order statistics by a formula suggested by Asquith (2007) as: For censored data, Hosking (1995) defined two variants of L-moment which are used with right censored observations. Similarly, Zafirakou-Koulouris et al. (1998) extended the applicability of L-moments ratio diagrams to left censored data via introducing a new partial probability weighted moment PPWM's based on L-moments definition. Wang (1990aWang ( , 1990b extended the concept of PWM, suggested by Greenwood, Landwehr, Matalas, and Wallis (1979), to PPWM for the analysis of censored samples. The aim of this study is introduced a modification of L-moments method to make it suitable for Type-I censored data directly with application from Kumaraswamy distribution. Kumaraswamy (1980) presented a distribution and called it the double bounded distribution to model hydrological random processes which are bounded at the lower and upper ends. The probability density function (pdf) of Kw distribution is and cumulative distribution function (cdf) is (1.4) f (x;a, b) = abx a−1 ( This article is organized as follows; Section 2 is concerned with L-moments for Type-I censored data via PPWM. A modification of L-moments, direct L-moments, is introduced in Section 3. direct L-moments for Kw distribution is presented in Section 4. L-moments via PPWM for Censored Data for Kw distribution are introduced in Section 5. ML method for censored data for Kw distribution is presented in Section 6. Simulation study and concluding remarks are presented in Section 7. Wang (1990a, b) introduced the concept of PPWM. Let X 1:n ≤ X 2:n ≤ ⋯ ≤ X n:n , the order statistics of a random sample of size n that comes from the distribution of the random variable X. Wang (1990a) defined a lower censoring PPWM as follows:

L-moments for censored data via PPWM
where c = F(T) is the random fraction of observations that are uncensored, T being the lower bound censoring threshold. He showed that the following statistic is an unbiased estimator of ′ r : where, Similar quantities for upper bound censoring are defined by Wang (1990b) as follows: where c = F(T), T now being the upper bound censoring threshold. Given a complete sample x (1) ≤ x (2) ≤ ⋯ ≤ x (n) , the following statistic is an unbiased estimator of ′′ r : where, Hosking (1995) introduced two different PPWM for using in L-moments with right censored data, Type-A PPWM and Type-B PPWM respectively.
• Type-A PPWM (1.5) F where s is censored data.
• Type-B ′ PPWM They presented unbiased estimators of Type-B ′ PPWM for left censoring as follows: For any distribution the rth L-moments is related to the rth PPWM, see Hosking (1990), via from which the first four L-moments in terms of PPWM are

Direct L-moments for censored data
All the research sighted above discussed L-moments method with censored data to estimate the parameters relied on using PPWM. The aim of this section is introducing a modification of L-moments (namely: Direct L-moments) method to make it suitable for censored data directly.

Direct L-moments for right censored data
Let x 1 , x 2 , … , x n be a Type-I censored random sample of size n from a distribution with distribution function F(x) and quantile function x(u). Let the threshold T satisfy F(T) = c and c is the fraction of observed data.

Direct L-moments for right censored data (Type-AD)
The quantile function of Type-AD L-moments is substitution into Equation (1.1) leads to the Type-AD L-moments where: The first four L-moments for Type-AD right censoring are calculated as follows: The standard method to compute L-moments estimator is equating the sample L-moments (m A r ) with the corresponding population L-moments ( A r ). Type-AD L-moments estimators of the uncensored data of m observations are given by (2.13) 1 = 0 , 2 = − 0 + 2 1 , 3 = 0 − 6 1 + 6 2 , 4 = − 0 + 12 1 − 30 2 + 20 3 .

Direct L-moments for right censored data (Type-BD)
The quantile function of Type-BD L-moments is substitution into Equation (1.1) leads to the Type-BD L-moments where: The first four L-moments for Type-BD right censoring are calculated as follows: The standard method to compute L-moments estimator is equating the rth sample L-moments (m

Direct L-moments for left censored data
Let x 1 , x 2 , … , x n be a random sample of size n. Type-I left censoring occurs when the observations below the fixed threshold T are censored: Let the threshold T satisfy F(T) = h and h is the fraction of censored data.

Direct L-moments for left censored data (Type-A'D)
The quantile function of Type-A ′ D L-moments is substitution into (1.1) leads to the Type-A ′ D L-moments where: The first four L-moments for Type-A ′ D left censoring are calculated as follows: The standard method to compute L-moments estimator is equating the sample L-moments (m A ′ r ) with the corresponding population L-moments ( A ′ r ). Type-A ′ D L-moments estimators for left censored data given by:

Direct L-moments for left censored data (Type-B'D)
The quantile function of Type-B

Direct L-moments for censored data for Kw distribution
In this section, the population L-moments of order r for Kw distribution is introduced.
• Type-AD From 0 (3.2), the first four L-moments for Type-I right censoring with Type-AD for Kw distribution are calculated as follows: , the first four L-moments for Type-I right censoring with Type-BD for Kw distribution are calculated as follows: • Type-A (4.2)

L-moments via PPWM for censored data for Kw distribution
In this section, different types PPWM of Kw distribution are introduced which we use in determining the L-moments of a Type-I censored data.
• Type-A From Equation (2.5), the first four PPWM for Type-I right censoring with Type-A for Kw distribution are calculated as follows: • Type-B From Equation (2.6), the first four PPWM for Type-I right censoring with Type-B for Kw distribution are calculated as follows: • Type-A ′ From Equation (2.8), the first four PPWM for Type-I right censoring with Type-A ′ for Kw distribution are calculated as follows: • Type-B ′ From Equation (2.10), the first four PPWM for Type-I right censoring with Type-B ′ for Kw distribution are calculated as follows:

ML method for right censored data for Kw distribution
Let x 1 , x 2 , … , x n is a sample size of Kw distribution. The likelihood function of the parameters (a and b) of Kw distribution, based on Type-I right censoring, is given by: Therefore, the log-likelihood function is To estimate the unknown parameters, a and b, the first partial derivations of the log likelihood function, , with respect to a and b respectively is needed. Setting a = 0 and b = 0, we get the likelihood equations: These equations constitute a system of two nonlinear equations must be solved in a and b to get the maximum likelihood estimator (MLEs) for right censored data of these parameters. It is obvious that the system of nonlinear equation has no closed form solution. So, a numerical technique is required to get the estimates of the unknown parameters.

ML method for left censored data for Kw distribution
Let x 1 , x 2 , … , x n is a sample size of Kw distribution. The likelihood function of the parameters (a and b) of Kw distribution, based on Type-I left censoring, is given by: Therefore, the log-likelihood function is To estimate the unknown parameters, a and b, the first partial derivations of the log likelihood function, , with respect to a and b respectively is needed. Setting a = 0 and b = 0, we get the likelihood equations: and, These equations constitute a system of two nonlinear equations must be solved in a and b to get the MLEs for left censored data of these parameters. It is obvious that the system of nonlinear equation has no closed form solution. So, a numerical technique is required to get the estimates of the unknown parameters.

Simulation study and concluding remarks
This section is devoted to present the modification of L-moments method (namely: Direct L-moments) in estimation process using a comparative numerical study. The two unknown parameters of Kumaraswamy distribution are estimated using direct L-moments method, L-moments via PPWM method and ML method for Type-I censored data(right and left censoring). A comparative study based on relative bias (RB) and root of mean square errors. All computations are performed using Mathematica-10 programs. The simulation study is conducted according to the following steps: