Partial Generalized Probability Weighted Moments for Exponentiated Exponential Distribution

The generalized probability weighted moments are widely used in hydrology for estimating parameters of flood distributions from complete sample. The method of partial generalized probability weighted moments was used to estimate the parameters of distributions from censored sample. This article offers new method called partial generalized probability weighted moments (PGPWMs) for the analysis of censored data. The method of PGPWMs is an extended class from partial generalized probability weighted moments. To illustrate the new method, estimation of the unknown parameters from exponentiated exponential distribution based on doubly censored sample is considered. PGPWMs estimators for right and left censored samples are obtained as special cases. Simulation study is conducted to investigate performance of estimates for exponentiated exponential distribution. Comparison between estimators is made through simulation via their biases and mean square errors. An illustration with real data is provided.


Introduction
, introduced the generalized exponential distribution which, also known as exponentiated exponential (EE) distribution. It is observed that the EE distribution can be considered for situation where a skewed distribution for a nonnegative random variable is needed. Also, it is observed that it can use quite effectively to analyze lifetime data in place of gamma, Weibull and log-normal distributions.
The probability density function of the two parameter EE distributions given by; The corresponding cumulative distribution function is given by: The probability weighted moments are useful for estimating the unknown parameters and quantiles of a distribution from complete sample. Wang (1990a) introduced extended class of probability weighted moments which called partial probability weighted moments (PPWMs) to estimate the parameters of a distribution from censored sample. Wang (1990aWang ( , b, 1996 originally introduced that concept for the purpose of estimating the upper quantiles of flood flows when one's interest is in the right tail of the distribution and there is some benefit to censoring some of the smaller observations in the left tail.
where X is a random variable and ) (x F is cumulative distribution function.
We introduce the method of PGPWMs as an extension class of GPWMs and PPWMs methods. The general form of PGPWMs with double bound censoring for the random variable X is defined as follows: It is noted when 1  d and the lower bound 0  c then the PGPWMs with double bound censoring is reduced to GPWMs in complete samples as defined in (2).
The PGPWMs with lower bound (left) censoring, denoted by , can be obtained as a special case from (4)by putting Therefore, the general form of PGPWMs with left censoring for the random variable X takes the following form The PGPWMs with upper bound (right) censoring, denoted by , can be defined as a special case from (4) by putting . 0  c Therefore, the general form of PGPWMs with right censoring for the random variable X takes the following form ,..., , 2 1 be a random sample of size n from the distribution function   X F , ,... 2 1 be the corresponding ordered sample. According to Hosking (1986), the estimates of , .
Note that, the PGPWM estimators with doubly, left or right censoring is obtained by equating the theoretical PGPWM with the corresponding sample PGPWM.

PGPWMs Estimation of the EE
In this section, the PGPWMs estimation method will be used to estimate the unknown parameters from EE distribution with doubly censored samples. Furthermore, the PGPWMs estimators with left and right censored samples are obtained as special cases.
From Equations (10) and (11), the two parameters,  and  , can be expressed as follows: The PGPWMs estimators of  and  from doubly censored samples, can be obtained by solving equations (12) From Equations (16) and (17), the two parameters,  and  , can be expressed as follows: To compute the PGPWMs estimators of  and  , denoted by L ˆ and L ˆ, from left censored samples, Solving numerically by iteration Equation (20)

(3.3) PGPWMs Estimation of the EE from Right Censored Sample
The theoretical PGPWMs with right bound censored, for the EE distribution can be obtained from (6) after setting 1 u u  and 2 u u  as the following From Equations (22) and (23), the two parameters,  and  , can be expressed as follows: The A numerical procedure is required to estimate based on iterative technique. Thus the value of

Numerical Experiments and Discussions
An extensive numerical study is carried out to investigate the properties of the PGPWMs method of estimation for the EE distribution from censored samples. The investigated properties are biases and mean square error (MSE) of the PGPWMs estimators of the two parameters and  . The simulation procedure can be summarized as the following steps: Step (1): , of sizes n = 15, 20, 25, 30, 35and 50 are generated from the EE distribution.

Step (3):
For each combination of values of sample size n,  and  , the parameters of distribution are estimated using three different estimation methods; PGPWMs with doubly, left and right censored samples.

Step (4):
PGPWMs of the unknown parameter  for EE distribution with doubly censored samples are obtained by solving numerically the non-linear Equation (15). The estimate of the scale parameter  is obtained by substituting the value of ˆin (14).

Step (5):
Based on left censored samples, PGPWMs of the shape parameter  for EE distribution is obtained by solving iteratively the non-linear Equation Step (6) Step ( Tables (1-6) and described in Figures (1-18). From simulation study many observations can be made on the performance of PGPWMs estimators with doubly, left and right censored samples. These observations are summarized as follows: 1.

Data Analysis
This section presents a real data set for illustrative purposes. Lawless (1982) Tables 7 and 8 and some observations are summarized as follows: 1.
Considering the MSE of the estimator for  , it is clear from