The Weighted New Weibull Pareto Distribution: Some Characteristics and Applications

Received:06/Oct /2018, Accepted:15/Oct/2018, Online: 31/Oct/2018 Abstract-In this paper, a weighted version of New Weibull Pareto (NWP) distribution known as weighted new Weibull Pareto (WNWP) distribution is obtained. Some structural properties of the new model are studied. Applications are provided using two real life data sets. It is shown that our new model performs better as compared to other models.


I. INTRODUCTION
In real life, there exist some situations when for an investigator it is not possible to select a sample with equal probability. In such situations, sampling frames are not properly defined and recorded observations are biased and do not follow the original distribution. Modeling of these observations gives birth to the theory of weighted distributions which was given by Fisher [1] and then studied by Rao [2].There are many authors who have presented important results on weighted distributions among them are Jain et al. [3] introduced the weighted version of gamma distribution, Abd El-Moonsef and Ghoneim [4] studied the weighted version of Kumaraswamy distribution, Fatima and Ahmad [5] introduced the weighted form of inverse Rayleigh distribution and study its various properties, Sofi Mudasir and Ahmad [6] proposed the weighted version of Nakagami distribution and finds its application to real life, Sofi Mudasir and Ahmad [7] estimate the scale parameter of weighted Erlang distribution through classical and Bayesian methods of estimation, Jan et al. [8] Where Z is the normalizing constant.
then the distribution is called the weighted distribution of order .
The probability density function of NWP distribution given by Nasiru and Luguterah [9] is given as (1.2) By using eq. (1.2) and in eq. (1.1), we get the required pdf of WNWP distribution and is given by

III. FUNCTIONS RELATED TO WNWP DISTRIBUTION
Proposition 1. Let V be a r.v. with pdf given in (1.3). The associated cumulative distribution function (cdf) is given by: Proof. Using the definition of cdf, we find that This completes the proof. © 2018, IJSRMSS All Rights Reserved 155 The survival and hazard rate functions follow immediately: , then the pdf also tends to zero. Hence the WNWP distribution has mode.

V. STATISTICAL PROPERTIES
This section deals with the statistical properties of WNWP distribution.

Mode of WNWP distribution
The mode of the WNWP distribution can be found by solving the equation Proposition 2. Let V be a r.v. with pdf given by (3). Then the th r non-central moment is given by Proof. According to (3) © 2018, IJSRMSS All Rights Reserved 157 After the simplification of the above integral, we get This ends the proof. By using eq. (5.2.1), the mean and variance of WNWP distribution are given by After the simplification, we get © 2018, IJSRMSS All Rights Reserved 158 This completes the proof. 5.5. Standard deviation and coefficient of variation Standard deviation of WNWP distribution is given by And the coefficient of variation (C.V.) is given as

Skewness and Kurtosis
The coefficient of skewness (C.S.) and kurtosis (C.K.) of WNWPD are given by Where  is the mean.
Proposition 5. If V follows WNWP distribution with pdf given in (1.3), then its Lorenz curve is given by: On solving the above integral and substituting the value of , Hence proved.

VII. SHANNON'S ENTROPY
Shannon's entropy is the most popular measure of entropy and is defined for a random variable V having pdf (7.1) © 2018, IJSRMSS All Rights Reserved 160 Proposition 6. For a r.v. V with pdf given in (1.3), the Shannon's entropy is given by Proof. Using equation (7.1), we have By substituting the value of eq. (1.3) in eq. (7.3), we get On substituting the value of eq. (7.4) and eq. (7.5) in eq. (7.2), we get This proves the theorem.

VIII. ENTROPY ESTIMATION OF WNWP DISTRIBUTION
Suppose that we have a statistical model having likelihood function L and N be the number of parameters. Then Akaike information criteria (AIC) and Bayesian information criteria (BIC) of the model is given by On comparing eq. (7.2) and (8.3), we get Thus from eq. (8.1) and eq. (8.2), we get )) (    Table 4. Maximum likelihood estimates, standard error in parentheses and statistics for model selection using data set 1.

XI. CONCLUSION
This paper deals with the weighted new Weibull Pareto (WNWP) distribution and studies its different statistical properties include reliability analysis, mode, moments, moment generating function, incomplete moments,standared deviation, coeffecient of variation, skewness, kurtosis, Lorenz curve, Shannon's entropy. Graphs were plotted using R-software. The superiority of the new model over some other models viz LNWPD, NWPD and WD were checked. An application to real life data sets shows that the fit of WNWP distribution is superior to the fits using LNWPD, NWPD and WD.