Estimation Parameter of R = P(Y < X) for Length-Biased Weighted Lomax Distributions in the Presence of Outliers

The concept of length-biased distribution is applied in expending proper models for lifetime data. The length-biased distribution is a special case of well-known weighted distribution. In this article, we introduce a length-biased weighted Lomax distribution (LBWLD) with k presence of outliers and estimate the parameter of R = P(Y < X) when the random variables X and Y are independent and have LBWLD in presence of outliers and without outliers, respectively. The bias and mean square error (MSE) of the estimator are examined with simulations of numerical and bootstrap resampling. Analysis of a real data set is considered for illustrative purposes.


Introduction
In recent years, the Lomax distribution (Pareto distribution II), has been applied in the field of life testing because it fits the business failure data.The Lomax distribution is presented by Lomax [1].In 2010, Abdullah and Abdullah [2] estimated the parameters of the Lomax distribution based on generalized probability weighted moments.The Lomax distribution was applied in the theoretical background in a number of ways.Weighted distribution theory gives a unified approach to deal with model specification and data interpretation problems.Weighted distributions are frequent in studies related to reliability, survival analysis, analysis of family data, biomedicine, ecology and several other areas; Stene [3] and Oluyede and George [4].Gupta and Tripathi [5], Patil and Rao [6] and Das and Roy [7] researched the length-biased Weighted Generalized Rayleigh distribution with its properties.Moreover, they devised a length-biased Weighted Weibull distribution [8].Many authors have researched important findings on weighted distributions.A unified concept of weighted distribution was proposed by Rao [9] where sampling situations example of being modeled are identified through weighted distributions.In the context of reliability, the life of a component which has a random strength X and is subject to a random stress Y is described by stress-strength models.The failed component at the instant means the stress is assigned to exceed the strength, and in case X > Y, the component functions properly.Consequently, R = P(Y < X) is a measure of component reliability.The parameter of R is the reliability parameter.This type of function can be of practical importance in many applications.For example, provided that X is the response for a control group, and Y is the treatment group, then R = P(Y < X), which is a measure of the effect of the treatment.This R = P(Y < X) can be applied when estimating heritability of a genetic trait.For more information on R, Halperin et al. [10], Simonoff et al. [11], Reiser and Farragi [12] and Bamber [13] are recommended.Bamber [13] provides a geometrical interpretation of A(X, Y) = P(X < .The random variable   is named the weight version of X and its distribution corresponds to that of X and is named the weighted distribution with weight function w(x).Consequently, at w(x) = x, the obtained distribution is named a length-biased distribution and the PDF of a length-biased random variable X yields

Lomax Distribution and Properties
A random variable X is distributed as Lomax distribution (LD) with two parameters of  (shape parameter) and  (scale parameter) if it is of the following PDF (; , ) It is easy to show that ;  > 2.
The corresponding cumulative distribution function (CDF) is exposed through It is assumed that ( 1 , … ,   ) is a random sample of size n from LD, where the maximum likelihood estimators (MLEs) of the parameters of  and , represented by  ̂ and  � , are obtained through direct maximization of the log-likelihood function.Different log-likelihood functions with respect to parameters are achieved through normal equations The MLEs are simultaneous solutions of the nonlinear normal equations (1).So, by simplification of (1), we have . The estimators cannot be written in a closed-form expression.

LBWLD and Properties
An LBWLD is derived by weights of w(x) = x in the weighted Lomax distribution [16] (; , ) = (−1)  2 (1 It is easy to show that The corresponding CDF is where  and  are the shape and scale parameters, respectively.Let ( 1 , … ,   ) be a random sample of size m from LBWLD with two parameters of  and .The log-likelihood function would be .
The partial derivatives of (, ) with respect to  and  are, respectively The MLE ̂= ( � ,  ̂) of  = (, ) is obtained with simultaneous solutions nonlinear system of normal equations, given by ), . (2) 2. Joint Distribution of Random Variables (  , . . .,   ) in Presence of k Outliers According to Dixit [17], let random variables ( 1 , … ,   ) be such that k of them (the number of outliers) are distributed as LBWLD with parameters of  (shape parameter) and  (scale parameter) and the remaining (n-k) random variables are distributed as LBWLD with parameters of  (shape parameter) and  (scale parameter).The joint distribution of ( 1 , … ,   ) with k outliers is expressed as It is easy to show that marginal distribution of X is expressed as [18] (; , , ) =  � ( − 1) The CDF is where

Method of Maximum Likelihood Estimation (MLE)
The MLEs of the parameters are achieved through direct maximization of the log-likelihood function.Partial derivations of log-likelihood functions with respect to parameters and setting the results equal to zero will produce normal equations.The MLE is the result of solving a nonlinear system of normal equations.It is more practical to apply nonlinear optimization algorithms like the quasi-Newton algorithm to maximize the log-likelihood function in its numerical sense.
Provided that ( 1 , . . .,   ) is a random sample of size n from LBWLD with presence of k outliers, then the log-likelihood function is presented as The partial derivatives of (, , ) with respect to ,  and  are as The MLE ̂= � �,  ̂,  ̂� of  = (, , ) is obtained as simultaneous solutions in a nonlinear system of normal equations, given by (6)

Maximum Likelihood Estimator of R = P(Y < X)
Let the random variables X and Y be independent and Y be LBWLD to parameters of (, ) and the random variables ( 1 , … ,   ) are such that k of them are distributed as LBWLD to parameters of (, ), and the remaining (n-k) random variables are distributed as LBWLD with parameters of (, ).The parameters of (, , ) are the shape parameters and parameter of  is the scale parameter.Using the marginal distribution of X given by (3), we obtain The value of R does not depend on parameter of .
If we insert k = 0, then X and Y will be LBWLD, without outliers, with different shape parameters and identical scale parameter.So Suppose ( 1 , … ,   ) is a random sample for random variable Y from LBWLD to parameters of (, ).The MLEs ( � ,  �,  ̂) of the parameters of (, , ) are yielded with simultaneous solutions in nonlinear system of normal equations, (2) and ( 4)- (6).If (, , ) replaces ( � ,  �,  ̂) into R, then they will obtain the MLE  � of the parameter of R.

Bootstrap Procedure
We consider the bootstrap resampling method for deriving the bias-corrected MLEs based on the idea of Efron [19].Let  = { 1 , … ,   } be a random sample of size n from the random variable X with distribution function F and ̂= ℎ() be an estimator of parameter  = () that is a function of F known.In this procedure, we generate a large number of pseudo-samples  * = { * 1 , … ,  *  } from the sample  and compute the corresponding bootstrap replicates , say ̂ * = ℎ( * ).Thus, the empirical distribution of ̂ * is used to estimate the distribution function of .Therefore, we generate N bootstrap samples independently from the sample  and compute the corresponding bootstrap estimates (̂ * (1) , … , ̂ * () ).The second-order bias-corrected MLEs for  can be obtained as =1 [20].

Results
Tables 1 and 2 present the values of the averages bias and the MSEs of the estimates for LBWLD with outliers in two cases of known and unknown λ and values of bootstrap method based on 500 resampling are reported within brackets.The values of MSEs of the estimations of α, β, θ, λ, and R often decrease while sample sizes n and m increase.Average biases and MSEs decrease by applying a bootstrap procedure.The sampling experiments and all computations were performed using R (V. 3.4.3)software.

Application
In order to illustrate the usefulness of the proposed model, we consider data reported by King et al. [21].A laboratory investigator interested in the relationship between diet and the development of tumors divided rats into two groups and fed them with saturated fat and unsaturated fat diets, respectively.The data represent the tumor-free time, the time from injection to the time that a tumor develops or to the end of the study.Group One contains m = 30 data (Unsaturated Fat) and Group Two n = 25 data (Saturated Fat).The data are as follows.The required numerical evaluations were carried out using the Package of R software.We fitted length-biased weighted Lomax distributions with presence of outliers for k = 0, 1, 2 to this data.These three distributions were fitted into the subject data using maximum likelihood estimation.The MLEs, estimated standard errors (SE) of the parameters and the corresponding log-likelihood values (−2Log−L), Akaike information criterions (AIC; [22]), and P-values for Kolmogorov-Smirnov test (K-S P-value) are displayed in Table 3.
The model with a minimum of AIC value was chosen as the best model to fit the data.From Table 3 we conclude that the LBWLD with presence of k = 1 outliers is better than others.

Conclusion
This paper proposed a length-biased weighted Lomax distribution with presence of outliers.The MLES of the parameters have been investigated.The problem of estimating R = P(Y < X) parameters for length-biased weighted Lomax distributions with Presence of Outliers have been addressed.The simulation experiment was carried out for  = 0, 1, 2;  = 2;  = 7;  = 5; and  = 4, and the results were given in Tables 1 and 2. The average bias and the MSE of the estimates based on 1000 replications and bootstrap method based on 500 resampling were reported.The results indicated that the MSEs of the estimations decrease when sample sizes increase and the average biases and MSEs of the estimations decrease when applying the bootstrap method.The usefulness of the distributions was illustrated in the analysis of real data.The results indicated that the LBWLD in presence of k = 1 outliers may be used for a wider range of statistical applications.

Table 2 .
Average Bias and MSEs for