Cubic Transformation Kappa Distribution: Properties and Applications

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1. Introduction

The Statistical Researcher faces many statistical difficulties during the,development of analyzing the data as well as Estimating the parameters related to the distribution, and among those problems that the researcher,directs is the process of determining the, appropriate and appropriate,distribution of the data singularity. The Researchers Specialized in the ground specialized in the Statistical field have functioned to progress, the Probability Distributions and move them to,the phase of the Transformed probability Distributions, in order to obtain, the best representation of the data with the least errors, Particularly,when the Researcher appearances the Problem of Choosing The,Sample With an Equal Probability, which Made The Original Distribution Limited to the Modelling of Phenomena It is unserviceable and when it develops necessary, to suggest a Specific Modification to be Done finished the Transformed System in Order to Obtain, Vocabulary That Has The Same Appearance of Accidents, " now a Days Transmuted Distributions And Their Mathematical Properties are Widely Studied For Applied Sciences Experimental Data Sets. transmuted rayleigh distribution (merovci, 2013) [13], Transmuted Inverse, Rayleigh Distribution (Ahmad et al., (2014) [2], Transmuted Generalized Inverse weibull distribution (khan and king, 2013) [9], transmuted modified inverse weibull distribution (elbatal, 2013) [8], transmuted log-logistic distribution (aryal, (2013) [4], transmuted modified weibull dsistribution & transmuted lomax distribution (ashour Aand eltehiwy, (2013) [5], transmuted frechet "distribution (mahmoud & mandouh, (2013) [12], transmuted pareto distribution (merovci & puka,( 2014), transmuted generalized gamma distribution (lucena et al., (2015) [11], transmuted weibull lomax distribution (afify et al.,( 2015) [1] are reported with their various structural properties including explicit expressions for the Moments, Quantiles, Entropies, Mean Deviations And Order Statistics. all the Above Transmuted Distributions are Derived By Using quadratic Rank transmutation map(qRTM) Studied By Shaw & Buckley (2007) [16]. Report reveals that some properties of these distributions along with their parameters Are Estimated By Using Maximum Likelihood And bayesian Methods. usefulness Of some of These New Distributions Are Also illustrated With Experimental Data Sets. transmuted gumbel distribution (tGD) Along With Several Mathematical Properties Has Studied By aryal and tsokos (2009) [3] Using quadratic rank transmutation map(qRTM)and Reported That( TGD) Can be Used to Model Climate Data. Therefore an Attempt Has Been Made to Developed transmuted exponentiated gumbel distribution (tEGD) Using exponentiated Gumbel distribution (EGD),and The quadratic rank transmutation map(qRTM). the parameters of The (TEGD) are Estimated by The Method Of Maximum Likelihood and Applied to The Water Quality Parameter Data Sets For Study The Usefulness Of The Model.

A Random Variable X is Said to Have a Cubic Transmuted Distribution if its Cumulative Distribution Function (CDF) is Given By:

Where G(x) is the CDF of The Cubic Transmuted Distribution and F(x) is the CDF OF The base Distribution. differentiating (1) it Gives The Probability Density Function (PDF) of The Transmuted Distribution as Zhao& Lu,. (2008) [17]:

Let X be the random variable having Kappa distribution with parameters (�,??????,�), then the probability density function and cumulative distribution function of the distribution is given by":

2. Cubic Transmuted Kappa Distribution

"The cubic transmuted Kappa distribution is useful to analyze complex data arising in the distribution of wealth. The cubic transmuted Kappa distribution is defined as follows; The CDF of a cubic transmuted Kappa distribution is obtained by using " (4) in (1) Zhao& Lu,. (2008) [17].

The reasonable shapes of CDF cubic transmuted Kappa distribution Fig 1: The Shape for the CDF cubic transmuted Kappa distribution Suppose a continuous random variable X follows a Kappa distribution with parameters, then the density function of the cubic transmuted Kappa distribution is given by:- �(x,α,θ,β,γ1,γ2)= ( −α−2α −α−3α )

The reasonable shapes of PDF cubic transmuted Kappa distribution Fig 2: The Shape for the PDF cubic transmuted Kappa distribution proof the room ∫ ∞ 0�(x,α,θ,β,γ1,γ2).dx=∫ ∞ ( −α−2α −α−3α )

=1 The Survival function of the cubic transmuted Kappa distribution given is:

The reasonable shapes Survival cubic transmuted Kappa distribution given is:

Fig 3: The Shape for the Survival cubic transmuted Kappa distribution The hazard function of the cubic transmuted Kappa distribution given is: For any random variable X which follows cubic transmuted Kappa distribution, its hazard function is given as:

3. Statistical Properties

In this section, some of the properties of the cubic transmuted Kappa distribution are discussed:

3.1. Moments

Let X mean the Random variable Follows cubic transmuted Kappa distribution then ????????????ℎ order moment near origin of µ?????? is: (afify et al.,( 2015) [1] "

3.2. Moments About the Mean

Let X mean The random variable Follows cubic transmuted Kappa distribution Then Moments About The Mean order Moment about origin of µ?????? is:"

3.3. Coefficients of Variation (Cv)

Coefficient of cubic transmuted Kappa distribution is given by(ashour Aand eltehiwy, (2013) [5],:

3.4. Coefficient of Skewedness (C.S)

COEFFICIENT OF SKEWEDNESS for cubic transmuted Kappa distribution is given by:

3.5. Coefficient of Kurtosis (KS)

COEFFICIENT OF KURTOSIS (KS) for cubic transmuted Kappa distribution (ashour Aand eltehiwy, (2013) [6], is given by":

5. Parameter Estimation

The Method of Maximum Likelihood Estimate is Used for Estimating The Parameters of The Newly Proposed Distribution Known as of cubic transmuted Kappa distribution. Let x1, x2,…,xn be a Random Sample of Ssize n From of The cubic transmuted Kappa distribution, Fhen the Corresponding likelihood Function is Given By(Ahmad et al., (2014) [2]: " Let x1,x2,x3, x4,....xn be a random sample of size n from cubic transmuted Kappa distribution The likelihood function, L cubic transmuted Kappa distribution. is given by:

4. Application of Cubic Transmuted Kappa Distribution

"We compare our proposed distribution with the new cubic transmuted Kappa distribution, Kappa distribution. (khan and king, 2013) [9] Here. By utilizing the maximum likelihood method, we estimate distribution parameters. The goodness of fit measures including the minus log-likelihood function (-LL), Akaike information criterion (AIC), Bayesian information criterion (BIC) and consistent Akaike information criterion (CAIC) are calculated to make a comparison among the fitted distributions. Generally, the lesser values of the above-mentioned measures, the better and the most appropriate fit to the information data set. "

Data Set Implementing the applied aspect on the real data that represents the working times until the malfunction of the axial spraying device measured in months, which was obtained with the help of the team of the Modern Irrigation and Mechanization Technologies Project, which amounts to (n=100) devices, and the following TABLE 1shows the real data under study".

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The maximum likelihood estimates and the goodness of fit measures are computed by by the Wolfram Mathematical program software which is presented in the table below.

Table 3: Parameters "Estimates and Goodness" – of – fits by Akaka Information Criterion (AIC), Akaka Information Criterion Amended (AICC) Distributions Estimate of Parameters -2log AIC AICc BIC Cubic Transmuted Kappa Distribution 524 202.778677 210.7776849574 211.447869498 — Kappa Distribution 465.321582 332.246868 336.24686495 336.442870494 — Note: The BIC values were not clearly identifiable in the provided text, so I have left them blank rather than guessing.

It is also clear that the distribution is preferable by cubic transmuted Kappa distribution representing the real data and its Applications:

Fig 4: The Shape for the PDF cubic transmuted Kappa distribution in data and its Applications

5. Conclusion

We proposed and studied a new lifetime cubic transmuted Kappa distribution. The derived model is flexible with variable hazard rate. The Shapes of hazard rate function indicates that the new developed model is a competitive" model for other probability models having "constant, increasing and decreasing failure rate. Moreover, cubic transmuted Kappa distribution has special cases, which style it of divergent statistical significance from other models. We used real data for application purpose. The derived "model is more flexible and give better fit than "well-known existing models. So, we can conclude that our new distribution can be used an alternative models for muddling in various areas. For example, engineering, failure analysis, survival analysis, weather forecasting, extreme value theory, economics (income inequality) and others.