Length biased Weighted New Quasi Lindley Distribution: Statistical Properties and Applications

In this Paper, we have introduced a new version of new quasi Lindley distribution known as the length-biased weighted new quasi Lindley distribution (LBWNQLD). The length biased distribution is a special case of weighted distribution. The different mathematical and statistical properties of the newly proposed distribution are derived and discussed in detail. The model parameters of the newly executed distribution are estimated by using the method of maximum likelihood estimation and also the Fisher’s information matrix have been discussed. Finally, an application of the proposed distribution is demonstrated with two real life data sets for illustrating its supremacy and applicability.


Introduction
The study of weighted distributions are useful in distribution theory because it provides a new understanding of the existing standard probability distributions and it provides methods for extending existing standard probability distributions for modeling lifetime data due to the introduction of additional parameter in the model which creates flexibility in their nature. Weighted distributions occur in modeling clustered sampling, heterogeneity, and extraneous variation in the dataset. The concept of weighted distributions was firstly introduced by Fisher (1934) to model ascertainment biases which were later formalized by Rao (1965) in a unifying theory for problems where the observations fall in non-experimental, non-replicated and non-random manner. When observations are recorded by an investigator in the nature according to certain stochastic model, the distribution of the recorded observations will not have the original distribution unless every observation is given an equal chance of being recorded. Weighted models were formulated in such situations to record the observations according to some weighted function. The weighted distribution reduces to length biased distribution when the weight function considers only the length of the units. The concept of length biased sampling was first introduced by Cox (1969) and Zelen (1974). Warren (1975) was the first to apply the size biased distributions in connection with sampling wood cells. Patil and Rao (1978) studied weighted distributions and size biased sampling with applications to wildlife populations and human families. Van Deusen (1986) arrived at size biased distribution theory independently and applied it in fitting assumed distributions to data arising from horizontal point sampling .More generally, when the sampling mechanism selects units with probability proportional to some measure of the unit size, resulting distribution is called sizebiased. There are various good sources which provide the detailed description of weighted distributions. Different authors have reviewed and studied the various weighted probability models and illustrated their applications in different fields. Weighted distributions are applied in various research areas related to reliability, biomedicine, ecology and branching processes. Afaq et al (2016) have obtained the length biased weighted version of lomax distribution with properties and applications. Reyad et al. (2017) obtained the length biased weighted frechet distribution with properties and estimation. Mudasir and Ahmad (2018) discussed the characterization and estimation of length biased Nakagami distribution. Para and Jan (2018) introduced the Weighted Pareto type II Distribution as a new model for handling medical science data and studied its statistical properties and applications. Rather and Subramanian (2019) obtained the length biased Pakistan Journal of Statistics and Operation Research erlang truncated exponential distribution with lifetime data. Rather and Ozel (2020) discussed on the weighted power Lindley distribution with application of real life time data. Hassan, Dar, Peer and Para (2019) obtained the weighted version of Pranav distribution with real life data. Hassan, Wani and Para (2018) discussed on the weighted three parameter quasi Lindley distribution with properties and applications. Ganaie, Rajagopalan and Rather (2019) discussed on the length biased Aradhana distribution with applications. Recently, Ganaie, Rajagopalan and Rather (2020) discussed on the weighted two parameter quasi Shanker distribution with properties and its applications, which shows more reliable and efficient than the classical distribution.
Shanker and Ghebretsadik (2013) introduced a new quasi Lindley distribution is a newly proposed two parametric probability distribution and derive its various mathematical and statistical properties as moments, skewness, kurtosis, failure rate function, mean residual life function and stochastic ordering. It is observed that the expression for failure rate function, mean residual life function and stochastic ordering of the new quasi Lindley distribution shows its flexibility over Lindley distribution, exponential distribution and quasi Lindley distribution of Shanker and Mishra (2013). Also, the new quasi Lindley distribution is a particular case of one parameter Lindley distribution.The parameter estimation is also discussed by using the method of moments and method of maximum likelihood estimation. The goodness of fit of new quasi Lindley distribution has been fitted to number of data sets related to survival times, grouped mortality data and waiting times to test its goodness of fit and it is observed that the new quasi Lindley distribution provides closer fit than those by the Lindley and quasi Lindley distribution.

Length Biased Weighted New Quasi Lindley Distribution (LBWNQLD)
The probability density function of new quasi Lindley distribution is given by (1) and the cumulative distribution function of the new quasi Lindley distribution is given by Suppose X is a non-negative random variable with probability density function ( ).Let w(x) be the nonnegative weight function, then, the probability density function of the weighted random variable is given by , > 0.
We should note that the different choices of the weight function w(x) gives different weighted distributions. When w(x) = x c , the result is known as weighted distributions and when w(x) = x, the result is known as length biased distribution. In this paper, we have to obtain the length biased version of new quasi Lindley distribution. The length biased weighted new quasi Lindley distribution is obtained by taking c =1 in weights x c to the weighted new quasi Lindley distribution in order to obtain the length biased weighted new quasi Lindley distribution. Therefore, the probability density function of length biased weighted new quasi Lindley distribution is given by Substitute the equations (1) and (4) in equation (3), we will obtain the probability density function of length biased weighted new quasi Lindley distribution ( ; , ) = 3 ( 2 + 2 ) ( + ) − (5) and the cumulative distribution function of length biased weighted new quasi Lindley distribution is given by After the simplification of equation (6), we obtain the cumulative distribution function of length biased weighted new quasi Lindley distribution

Reliability Analysis
In this sub section, we obtain the Reliability function, hazard function and Reverse hazard rate function for the proposed length biased weighted new quasi Lindley distribution.
The reliability function or the survival function of length biased weighted new quasi Lindley distribution is given by The hazard function is also known as hazard rate or instantaneous failure rate or force of mortality and is given by The reverse hazard function of length biased weighted new quasi Lindley distribution is given by

Statistical properties
In this section we shall discuss the structural properties of length biased weighted new quasi Lindley distribution, especially its moments, harmonic mean, moment generating function and characteristic function.

Moments
Let X denotes the random variable of length biased weighted new quasi Lindley distribution with parameters θ and α, then the r th order moment E(X r ) of the length biased weighted new quasi Lindley distribution is obtained as After simplification, we obtain Substitute r = 1, 2, 3, 4 in equation (8), we obtain the first four moments of length biased weighted new quasi Lindley distribution.

Harmonic mean
The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. The harmonic mean for the proposed length biased weighted new quasi Lindley distribution is given by After simplification, we obtain

Moment generating function
The

Characteristic function
The characteristic function is defined as the function of any real valued random variable completely defines the probability distribution of a random variable and the characteristics function exists always even if moment generating function does not exists. The characteristic function of length biased weighted new quasi Lindley distribution is given by

Order Statistics
Let X (1) , X (2) , ,…. , X (n) denotes the order statistics of a random sample X 1 , X 2 ,… . ,X n drawn from the continuous population with cumulative distribution function F x (x) and probability density function fx(x), then the probability density function of rth order statistics X (r) is given by For r = 1, 2… n.

Likelihood Ratio Test
Let X 1 , X 2 , …….X n be a random sample of size n from the new quasi Lindley distribution or length biased weighted new quasi Lindley distribution. We test the hypothesis. Thus for testing the hypothesis, whether the random sample of size n comes from new quasi Lindley distribution or length biased weighted new quasi Lindley distribution, the following test statistic is used.
We should reject the null hypothesis if = ( ( 2 + ) ( 2 + 2 ) ) ∏ > =1 Equivalently, we reject the null hypothesis if * = ∏ > ( ( 2 + 2 ) ( 2 + ) ) =1 * = ∏ > * ,Where Thus if the sample is large of size n, 2log Δ is distributed as chi-square distribution with one degree of freedom and also p-value is obtained from the chi-square distribution. Also, we reject the null hypothesis, when the probability value is given by ( * > * ), Where * = ∏ is less than a specified level of significance and ∏ =1 =1 is the observed value of the statistic Δ *

Bonferroni and Lorenz Curves
The Bonferroni and Lorenz curves are used not only in economics to study the distribution of income or wealth or income or poverty, but it is also being used in other fields like reliability, medicine, insurance and demography. The Bonferroni and Lorenz curves are given by

Entropies
The concept of entropies is important in different areas such as probability and statistics, physics, communication theory and economics. Entropy is also called the degree of randomness or disorder in a system.
Entropies quantify the diversity, uncertainty, or randomness of a system. Entropy of a random variable X is a measure of variation of the uncertainty.

Renyi Entropy
The Renyi entropy is important in ecology and statistics as index of diversity. The entropy is named after Alfred Renyi. The Renyi entropy is important in quantum information, where it can be used as a measure of entanglement. For a given probability distribution, Renyi entropy is given by Where, > 0 and ≠ 1 Using Binomial expansion in (10), we obtain

Tsallis Entropy
A generalization of Boltzmann-Gibbs (B.G) statistical properties initiated by Tsallis has focused a great deal to attention. This generalization of B-G statistics was proposed firstly by introducing the mathematical expression of Tsallis entropy (Tsallis, 1988) for a continuous random variable is defined as follows Using Binomial expansion in equation (11), we get

Maximum likelihood Estimation and Fisher's Information Matrix
In this section, we will discuss the maximum likelihood estimation for estimating the parameters of length biased weighted new quasi Lindley distribution and also discuss its Fisher's information matrix. Let X1, X2,…., Xn be a random sample of size n from the length biased weighted new quasi Lindley distribution, then the likelihood function of length biased weighted new quasi Lindley distribution is given by The log likelihood function is given by Differentiating the log likelihood equation (12) with respect to θ and α and equating to zero, we obtain the normal equations. Because of the complicated form of the likelihood equations, algebraically it is very difficult to solve the system of non-linear equations. Therefore we use R and wolfram mathematics for estimating the required parameters of the proposed distribution.
To obtain confidence interval we use the asymptotic normality results. we have that if ̂= (̂,̂) denotes the MLE of = ( , ) we can state the results as follows: Where I(λ) is the Fisher's Information matrix .i.e., Where Since λ being unknown, we estimate −1 ( ) by −1 (̂) and this can be used to obtain asymptotic confidence intervals for θ and α

Application
In this section, here we analyse and evaluate the two real life data sets for fitting length biased weighted new quasi Lindley distribution and the model has been compared with new quasi Lindley, quasi Lindley, Lindley and exponential distributions. In order to show that the length biased weighted new quasi Lindley distribution is better than the new quasi Lindley, quasi Lindley, Lindley and exponential distributions, the results obtained from the two real life data sets are used. The two real life data sets are given below as: The first data set denotes the time to failure of turbocharger (103h) of one type of engine studied by Xu et al. (2003). The first data set is given as follows Where k is the number of parameters in the statistical model, n is the sample size and -2logL is the maximized value of the log-likelihood function under the considered model.

Conclusion
In the present study, we have introduced the length biased weighted new quasi Lindley distribution as a new generalization of new quasi Lindley distribution. The subject distribution is generated by using the length biased technique and taking the two parameter new quasi Lindley distribution as the base distribution. The different mathematical and statistical properties of the newly executed distribution along with the reliability measures are discussed. The parameters of the proposed distribution are obtained by using the method of maximum likelihood estimator and also the Fisher's information matrix has been discussed. Finally, the application of the new distribution has also been illustrated by demonstrating with two real life data sets. The results of the two data sets are used by comparing the length biased weighted new quasi Lindley distribution over new quasi Lindley, quasi Lindley, Lindley and exponential distributions and the results indicate that the length biased weighted new quasi Lindley distribution provides a better fit than the new quasi Lindley, quasi Lindley, Lindley and exponential distributions.