LIBBY-NOVICK KUMARASWAMY DISTRIBUTION WITH ITS PROPERTIES AND APPLICATIONS

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INTRODUCTION
defined the Libby-Novick Kumaraswamy (LNK) distribution. The probability density function (pdf) and cumulative distribution function (cdf) are respectively defined as The Libby-Novick Kumaraswamy (LNK) distribution is a continuous probability distribution with double-bounded support. It is very similar, in many respects, to the Libby-Novick Beta (LNB, 1982) [2] distribution and Kumaraswamy (Kum, 1980) [3] distribution. The Kum distribution is a two parameter distribution like Beta distribution Where as M.C. Jones (2009) [4] find out numerous benefits of the Kum distribution over classical beta distribution.The Kum distribution is a special case of McDonald's (1984) [5] generalized Beta of the first kind distribution. One key difference between the Kum and Beta distributions is the availability for the former, but not for the latter, of an invertible closed form cumulative distribution function presented by Mitnick (2013) [6].
The two distributions LNK and LNB are very flexible and can take approximately the same shapes and the former distribution is also a generalized distribution of Kum distribution.
There are, however, important realistic differences between LNB and LNK distributions. On the one hand, the availability for the LNK , but not for the LNB distribution, of an invertible closedform cumulative distribution function makes the LNK distribution much better suited than the LNB for activities that require the generation of random varieties, in particular simulation modeling and simulation-based model estimation. In contrast, the lack of tractable-enough expressions for the mean and variance of the LNK distribution has stalled its utilization for modeling purposes; in spite of the advantages that the availability of an invertible closed-form cumulative distribution function entails, the LNK distribution may use rather sparingly in the modeling of stochastic phenomena and processes. Libby-Novick (1982) [2] derived two new multivariate probability density functions, which are the generalized forms of beta distribution and F distribution. They proved that these distributions that seem to be suitable in utility modeling. They reproduced the same distributional form in the cases of marginal and conditional distributions. Chen and Novick (1984) [7] used the Libby-Novick (1982) [2] univariate generalized beta distribution as a prior and estimated the parameters of Bernoulli and Binomial distributions.
These expressions are the generalized forms of the standard beta class. Ristic et al., (2013) [8] derived new family of skewed distributions such as Libby and Novick's generalized beta exponential distribution and found some useful properties of this family of distributions. Cordeiro et al., (2014) [9] defined a family of distributions, named the Libby-Novick beta family of distributions, which includes the classical beta generalized and exponentiated generators. This extended family gave reasonable parametric fits to real data in several areas because the additional shape parameters controlled the skewness and kurtosis simultaneously. Ali (2019) [10] worked on new form of Libby-Novick (NLN) distribution and explored some properties of NLN distribution. This model was compared with other distributions by fitting them to a real data set. Rashid et al., (2020) [11] derived different entropy measures and characterized Libby-Novick generalized Beta (LNGB) distribution through various methods. Iqbal et al., (2021) [12] derived mathematical properties of LNGB distribution and applied it to modeling on three real data sets.
In this paper we derive basic and advanced properties of LNK distribution and find applications of LNK distribution to three real data sets. In Section 2, a detailed remarks about the graphs of pdf, hazard rate function, reverse hazarz rate function and survival function are provided. Section 2 also contains the derivation and results of some important mathematical properties of LNK distribution. Maximum likelihood estimators of three parameters of LNK distribution are derived in section 3. In section 4, a simulation of the parameters is carried out for different sample sizes. A number of deduced models are shown in section 5. In section 6, LNK distribution is compared with some other models through three data sets. Some concluding remarks are presented in section 7.

Shape properties of the pdf
The LNK density function defined in (1) has real flexibility and it is shown through graphs w.r.t. some different combinations of values of the parameters.

Shapes of pdf
(i) For a > 0, c > 0 and 0 < b < 1 then LNK distribution is U-shaped.
(iv) For a = b = c = 1, the LNK distribution is uniform distribution.
(v) For a > 2, b > 2 and c ≥ 1 the LNK distribution is unimodel positively skewed with decreasing mode when c→ .
(vi) For a > 2, b = 1, 0 < c < 1, the form of LNK distribution is an increasing. (vii) For a > 2, b = 1, 1< c < 3.5, the LNK distribution increase but it increases slowly when c increases in the interval and for c ≥ 3.5 the LNK distribution again turns to unimodel.
(viii) For b = c = 1, the LNK distribution is Power distribution.
(ix) For a = c = 1, the LNK distribution is LNK is a special case of Kum-distribution or reflected exponentiated distribution.

Distribution function
The cumulative distribution function (cdf) of LNK distribution is

Quantile Function
The quantile function of LNK distribution is given by: And it After simplification we find the quantile function of the LNK distribution as under; ( ) iii. For c → 0, the quantile value of the LNK distribution increases sharply.

Hazard Rate Function
The hazard or instantaneous rate function is denoted by h(x). The hazard function of x can be interpreted as instantaneous rate or the conditional probability density of failure at time x, given that the unit has survived until x. The hazard rate function of LNK distribution as The hazard rate function of LNK distribution is of Bath-Tub shape, increasing shape and decreasing shape for specific sets of values of parameters.

Reverse Hazard rate function
The reverse hazard can be interpreted as an approximate probability of failure in given that the failure has occurred in   0, x . The reverse hazard function

Survival Function
The survival function or reliability function of LNK distribution is defined as

The r th Moment
The r th moments about origion of the LNK distribution is defined as

Moment generating function
The moment generating function of the LNK distribution about zero is

Factorial Moments
This section devotes to inceasing and decreasing factorial moments of LNK distribution as under:

Decreasing Factorial Moments of LNK distribution
The decreasing factorial moments of the LNK distribution is defined as: s n, r are the Stirling's numbers of first kind.

Increasing Factorial Moments of LNBD
The increasing factorial moments of the LNK distribution is defined as: can be deduced from the relation

Negative Moments
The negative moments of the LNK distribution is defined as: By applying the substitution (4), we have Incomplete Moments The Incomplete moments of the LNK distribution is defined as:

Scaled Total Time for Aging Properties
The scaled total time of the LNK distribution is defined as:

Conditional Moments
The conditional moments of the LNK distribution is defined as: By applying the substitution (4) and after some simplification, we have

Mean Residual Function
The mean residual function of the LNK distribution is defined as:

Vitality Function
The vitality function of the LNK distribution is defined as:

Geometric Vitality Function
The Geometric vitality function of the LNK distribution is defined as:

Characteristics Function
The characteristic function of the LNK distribution is defined as: Mode of LNK distribution is obtained by solving

Points of Inflection
The points of the inflection of the LNK distribution are found from fx  or equavalentally as

Bonferroni Curve
The Bonferroni curve of the LNK distribution is as:

Lorenz Curve
The Lorenz curve of the LNK distribution is as:

Gini Coefficient
The Gini coefficient of the LNK distribution is explained as:

ESTIMATION
Here, we consider

Maximum-likelihood estimation
The likelihood function of random sample 12 , ,..., n x x x of observation is given by

Asymptotes and Shapes
The asymptotes of (1.

APPLICATION
In order to prove that LNK distribution can be a better model than the Power distribution, Beta distribution with (a = 1), Beta distribution, Kumaraswamy distribution, let us use three real data sets.

Data Set 1:
The following right to skewed dataset presented by Cordeiro and Brito (2012) [13] is obtained from the measurements on petroleum rock samples. The data consists of 48 rock samples from a petroleum reservoir. The dataset corresponds to twelve core samples from petroleum reservoirs that were sampled by four cross-sections. Each core sample was measured for permeability and each cross-section has the following variables: the total area of pores, the total perimeter of pores and shape. We analyze the shape perimeter by squared (area) variable and the observations are:

Data Set 3:
The following second data set presented by Cordeiro and Brito (2012) [13], displays the skewed symmetric trend of data. This data discusses the total milk production in the first birth of 107 cows from SINDI race. These cows are property of the Carnaúba farm which belongs to the

CONCLUSION
The LNK distribution has some properties like the LNB distribution but, there are numerous benefits of the LNK distribution over LNB distribution: the LNK distribution has simple closed form of both its cdf and quantile function and that's why it is easy to use in simulation studies.The distribution and quantile function of LNK do not involve any special functions. To compare the proposed model with other models, we apply these models to three sets of real data from different fields of science and engineering and it is examined by using well-known statistics. We conclude that the LNK distribution is better than the power, Beta with (a=1), Beta distribution and Kumaraswamy distribution.

Appendix 2: Variance, Skewness and Kurtosis of LNK distribution
Following