Transmuted Exponentiated Gumbel Distribution (TEGD) and its Application to Water Quality Data

The Transmuted Exponentiated Gumbel Distribution (TEGD) has been derived using Exponentiated Gumbel Distribution (EGD) and the Quadratic Rank Transmutation Map (QRTM). The analytical expressions and shapes of the distribution function, probability density function, hazard rate function and reliability function are studied. The parameters of the TEGD are estimated by the method of maximum likelihood. Finally the TEGD is applied to real data set of water quality parameter and found to be better fit than Exponentiated Gumbel Distribution (EGD) and Gumbel Distribution (GD).


Introduction
Transmuted distributions have been discussed dynamically in frequently occurring large scale experimental statistical data for model selection and related issues. In applied sciences such as environmental, medicine, engineering etc. modeling and analyzing experimental data are essential. There are several distributions which can be used to model such kind of experimental data. The procedures used in such a statistical analysis depend heavily on the assumed probability model or distributions. That is why the development of large classes of standard probability distributions along with relevant statistical methodologies has been expanded. However, there still remain many important problems where the real data does not follow any of the classical or standard probability models.
The Gumbel Distribution (GD) is a very popular statistical distribution due to its extensive applicability in several areas and its wide applications has been reported by Kotz and Nadarajah (2000). The applicability of GD in the field of flood frequency analysis, network, space, software reliability, structural and wind engineering are reported by Cardeiro et al., (2012). Nadarajah (2006) introduced Exponentiated Gumbel Distribution (EGD) based on Gumbel Distribution (GD) and illustrated its applicability in the area of global warming modeling, rainfall modeling, wind speed modeling etc. Due to its wide applicability in different fields of science, the generalization of Gumbel Distribution has become important. Now a days transmuted distributions and their mathematical properties are widely studied for applied sciences experimental data sets. Transmuted Rayleigh Distribution (Merovci, 2013) 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) 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 transmuted distribution if its cumulative distribution function (cdf) is given by Where ( ) is the cdf of the transmuted distribution and ( ) is the cdf of the base distribution. Differentiating (1) w.r.t. X, it gives the probability density function (pdf) of the transmuted distribution as Where ( ) and ( ) are the corresponding pdf of ( ) and ( ) respectively. It is observed that at , we have the base distribution of the random variable X.

cdf & pdf of TEGD:
For and the pdf and cdf of Exponentiated Gumbel Distribution (EGD) can be expressed as equation (3)  Using series representation as The expression (5) & (6) can be written in mixture form by using (*) as (7) & (8) respectively

Graphical Representation of pdf & cdf of TEGD:
For known values of μ, σ, α and the possible shapes of cdf and pdf of the TEGD are represented in Fig. 1 & Fig. 2.

Moments of TEGD:
The n th moment ( ) of a TEGD for random variable X can be obtained as By combining ( ) ( ) ( ) ( ) and ( ), the n th moment of X can be expressed as ( ) ( Putting and in the equation (19), the expression for moment (mean) and moment are obtained as follows Where C is Euler's constant. In particular ( ) ( ) ( ) and .∑ / And ( ) ( ) * ( )+

Moment Generating Function of TEGD:
If X has a TEGD then the moment generating function of X, say ( ) is, obtained as

Random Number Generation and Parameter Estimation of TEGD:
Using the method of inversion we have generated random numbers for the TEGD as  We set the sample size at , the parameter α at and at = 0.6. The location and scale parameter were fixed at and respectively. The Monte Carlo simulation experiments are performed using the R-Programming language. We get the estimated value of the parameter and standard error as follows:

Reliability Analysis of TEGD:
The Reliability Function ( ), is defined by ( ) ( ) and for TEGD it is given as Using ( ) we can expressed Reliability Function of TEGD can be written in mixture form as

Hazard Rate Function:
The hazard rate function or instantaneous failure rate, which is an important quality characterizing life phenomenon defined by ( ) ( )

( )
The hazard rate function for TEGD is given by (

Application of TEGD to Experimental Data
Survivability of a theoretical statistical model depends on its practical application. That is why, we provide an application to real data set to illustrate the importance of TEGD.
Here we work with water quality data using some water quality parameters. Data for water quality parameters have been taken from the Department of Chemistry, Gauhati University. Various water quality parameters were estimated for the project entitled Since the probability distribution of estimated data of iron (Fe) follows the theoretical TEG distribution. Therefore, to test the goodness of fit of this distribution, estimated values of iron (Fe) have been applied in TEG distribution. Here we compute the MLE's of the parameters and the goodness-of-fit statistics for this distribution are compared with EGD and GD, to show the importance of new model. All computations have been carried out using R-Programming language.
The graphical representations of Reliability Function & Hazard Rate Function of TEGD for known values of different parameters are presented in Fig. 3 & Fig. 4 respectively.
To compare TEGD with EGD & GD, we consider some criteria like ( ): where is the maximum value of log-likelihood function, AIC (Akaike Information Criterion), CAIC (Corrected Akaike Information Criterion) and BIC (Bayesian Information Criterion) for the data set. In general the better fit of the distribution corresponds to the smaller value of the statistics ( ) , AIC, CAIC and BIC.   The observed values of ( ), AIC, CAIC and BIC for TEGD, EGD & GD from Table  3, it is clear that Transmuted Exponentiated Gumbel Distribution (TEGD) gives lower values than that of Exponentiated Gumbel Distribution (EGD) and Gumbel Distribution (GD). Hence TEGD leads to a better fit than the other two distributions.

Conclusion
The TEGD has been generated and their parameters are estimated. The TEGD is applied to water quality data and it is compared with EGD & GD and leads to better fit than the other two distributions. Hence the new TEGD can be applied to environmental science for better modeling.