The Kumaraswamy transmuted Pareto distribution
 Sher B. Chhetri^{1},
 Alfred A. Akinsete^{2}Email authorView ORCID ID profile,
 Gokarna Aryal^{3} and
 Hongwei Long^{1}
https://doi.org/10.1186/s4048801700654
© The Author(s) 2017
Received: 22 March 2017
Accepted: 6 July 2017
Published: 15 August 2017
Abstract
In this work, a new fiveparameter Kumaraswamy transmuted Pareto (KwTP) distribution is introduced and studied. We discuss various mathematical and statistical properties of the distribution including obtaining expressions for the moments, quantiles, mean deviations, skewness, kurtosis, reliability and order statistics. The estimation of the model parameters is performed by the method of maximum likelihood. We compare the distribution with few other distributions to show its versatility in modeling data with heavy tail.
Keywords
AMS Subject Classification
Introduction
A closely related generalized distribution similar to using the beta random variable as the baseline distribution as defined in Eqs. (1) and (2), is the work due to Cordeiro and de Castro (2011), where the authors combined the works of Eugene et al. (2002) and Jones (2009) by replacing the baseline distribution with the Kumaraswamy (Kw) distribution to construct a new class of KwG distributions. See also Bourguignon et al. (2013), Cordeiro et al. (2010), and Elbatal (2013). The work by Shams (2013) also introduced and studied another type of generalization of the Kumaraswamy distribution.
Note that when λ=0, the above relation reduces to the distribution (G(x)) of the baseline random variable.
Following the work by Aryal and Tsokos (2009) on the transmuted extreme value distribution, a number of transmuted family of distributions have been proposed and discussed by many authors in the literature. A few of them are the work by Aryal and Tsokos (2011): transmuted Weibull distribution; Khan and King (2013): transmuted modified Weibull distribution; Ashour and Eltehiwy (2013): transmuted Lomax distribution; Hady and Ebraheim (2014): exponentiated transmuted Weibull distribution; Hussian (2014): transmuted exponentiated gamma distribution; Merovci and Puka (2014): transmuted Pareto distribution; Owokolo et al. (2015): transmuted exponential distribution, and Khan et al. (2016): transmuted Kumaraswamy distribution; Afify et al. (2014): transmuted complementary Weibull geometric distribution; Yousof et al. (2015): the transmuted exponentiated generalizedG family of distributions, and a host of many others. Tahir and Cordeiro (2016) provided a comprehensive list of contributed works on transmuted distributions. This paper is outlined as follows. In Section 2, we introduce the Kumaraswamy transmuted Pareto distribution. Some of its mathematical properties are discussed in Sections 3, 4, 5, 6, 7, 8 and 9. Section 10 discusses the estimation of the parameters of the distribution by the method of maximum likelihood. We provide the application of the distribution to two real life data in Section 11. In Section 12 we have simulation, and concluding remarks in Section 13.
The Kumaraswamy transmuted Pareto (KwTP) distribution
Following the transmutation map by Shaw and Buckley (2007) as defined in Eq. (3), we combine the Kumaraswamy distribution and the transmuted Pareto to form what we refer to as the Kumaraswamy transmuted Pareto (KwTP) distribution. Without loss of generality, we give a brief description of these component distributions in what follows.
2.1 Component distributions
2.1.1 The KumaraswamyKw distribution
where both a>0 and b>0 are shape parameters. A generalized form of this distribution is obtained when t is replaced by the CDF G(t) of another random variable, to have what is referred to as the KumaraswamyG (KwG) distribution. The beta and Kumaraswamy distributions share similar properties. For example, the Kumaraswamy distribution, also referred to as the minimax distribution, is unimodal, uniantimodal, increasing, decreasing or constant depending on the values of its parameters. A more detailed description, background, genesis, and properties of Kumaraswamy distribution are outlined in Jones (2009), where the author highlighted several advantages of the Kumaraswamy distribution over the beta distribution, namely; its simple normalizing constant, simple explicit formulas for the distribution and quantile functions, and simple random variate generation procedure. There have been many contributions to the theory and applications of the Kumaraswamy distribution in the literature. See for example the works by Cordeiro et al. (2010): Kumaraswamy Weibull; Cordeiro et al. (2012): Kumaraswamy Gumbel; Pascoa et al. (2011): Kumaraswamy generalized gamma; Saulo (2012): Kumaraswamy BirnbaumSaunders; Akinsete et al. (2014): Kumaraswamy geometric, and a host of many others.
2.1.2 The Pareto and the transmuted Pareto (TP) distributions
The Pareto distribution was named after a sociologist Vilfredo Pareto. It is used in modelling the distribution of incomes and other financial variables, and in the description of social and other phenomena. Many distributions have been derived using the Pareto distribution. A few examples are the gammaPareto distribution by Alzaatreh et al. (2012), the beta exponentiated Pareto distribution by Zea et al. (2012), and the betaPareto distribution by Akinsete et al. (2008).
The distribution in Eq. (5) becomes the Pareto when λ=0.
2.2 The KwTP distribution
Special cases of KwTP

When λ=0, KwTP reduces to KwP distribution by Bourguignon et al. (2013) with PDF$$\begin{array}{*{20}l} f(x;\alpha, \theta, a, b)= \frac{ab\alpha \theta^{\alpha}}{x^{\alpha+1}} \times \Bigg\{\bigg[1\left(\frac{\theta}{x}\right)^{\alpha}\bigg] \Bigg\}^{a1} \Bigg\{1\bigg[1\left(\frac{\theta}{x}\right)^{\alpha}\bigg]^{a}\Bigg\}^{b1}. \end{array} $$

When a=b=1, KwTP reduces to the transmuted Pareto distribution by Merovci and Puka (2014) with PDF$$\begin{array}{*{20}l} f(x;\alpha, \theta, \lambda)=\frac{\alpha \theta^{\alpha}}{x^{\alpha+1}}\bigg[1\lambda+2\lambda \left(\frac{\theta}{x}\right)^{\alpha}\bigg]. \end{array} $$

By setting b=1 and λ=0, the KwTP reduces to the exponentiated Pareto distribution, defined by Nadarajah (2005), with PDF$$\begin{array}{*{20}l} f(x;\alpha, \theta, a) = \frac{a\alpha \theta^{\alpha}}{x^{\alpha+1}} \times \Bigg\{\bigg[1\left(\frac{\theta}{x}\right)^{\alpha}\bigg] \Bigg\}^{a1}. \end{array} $$

When a=1=b and λ=0, KwTP reduces to the classical Pareto distribution with PDF$$\begin{array}{*{20}l} f(x;\alpha, \theta) = \frac{\alpha \theta^{\alpha}}{x^{\alpha+1}}. \end{array} $$
Mixture representation
In this section, we provide alternative and useful expressions for the PDF and the CDF of KwTP distribution. Let X be a random variable having the KwTP density (7).
and g(x) and G(x) are the PDF and CDF of the transmuted Pareto distribution respectively.
By setting i+j=m, and \(w_{k,m}^{*}=\sum \limits _{i,j:i+j=m}^{} w^{k}_{ij}= \sum \limits _{i=0}^{m}\sum \limits _{j=0}^{mi} w^{k}_{ij}\), we can further express the PDF of the KwTP in the form
That is, m=0 ⇒ i=0,j=0. From \(w_{ij}^{k}={(k+1)a1\choose i}{(k+1)a1\choose j}(1)^{i}\lambda ^{j}\) and i=0,j=0, we have w _{0,0}=1. Thus, \(w_{k,0}^{*}=1\), and hence, \(u_{0}=\sum \limits _{k=0}^{\infty }{b1\choose k}(1)^{k}.\)
Let \(v^{k}_{ij}={(k+1)a\choose i}{(k+1)a\choose j}(1)^{i}\lambda ^{j}\), where in particular \(v^{k}_{00}=1\).
we can express the CDF of the KwTP as,
Quantile function
Moment functions
where Γ(.,.) denotes the upper incomplete gamma function. That is, \(\Gamma (s,\theta)=\int _{\theta }^{\infty } e^{t} t^{s1} dt\).
Median, mean, variance, skewness, kurtosis for selected values of the parameters
α  λ  a  b  Median  Mean  Variance  Skewness  Kurtosis 

5  0.2  1  1  0.5636  0.6111  0.0224  4.9196  81.8250 
6  0.2  1  1  0.5525  0.5891  0.0130  4.0458  42.9488 
7  0.2  1  1  0.5446  0.5744  0.0084  3.5980  30.9806 
8  0.2  1  1  0.5388  0.5638  0.0059  3.3243  25.2955 
10  0.2  1  1  0.5309  0.5497  0.0034  3.0056  19.8668 
5  1  1  1  0.6392  0.6944  0.0386  4.1113  60.5271 
5  0.5  1  1  0.6061  0.6997  0.0335  4.2366  63.3030 
5  0.1  1  1  0.5688  0.6181  0.0243  4.7743  77.4267 
5  0.5  1  1  0.5505  0.5903  0.0162  5.4825  101.8249 
5  0.8  1  1  0.5408  0.5694  0.0091  6.0430  135.7943 
5  1  1  1  0.5359  0.5556  0.0039  2.8111  17.8286 
5  0.5  1  1  0.5505  0.5903  0.0162  5.4825  101.8249 
5  0.5  2  1  0.5964  0.6422  0.0249  4.7467  78.6519 
5  0.5  3  1  0.6304  0.6802  0.0313  4.4534  70.3107 
5  0.5  4  1  0.6578  0.7106  0.0366  4.2881  65.8432 
5  0.5  5  1  0.6810  0.7362  0.0411  4.1793  63.0006 
5  0.2  1  1  0.5636  0.6111  0.0224  4.9196  81.8250 
5  0.2  1  2  0.5303  0.5480  0.0031  3.0122  20.2613 
5  0.2  1  3  0.5199  0.5305  0.0011  2.6351  15.2360 
5  0.2  1  4  0.5150  0.5224  0.0006  2.4635  13.3142 
5  0.2  1  5  0.5118  0.5176  0.0003  2.3646  12.2905 

Bowley skewness (B _{ sk }): By definition, the Bowley’s measure of skewness is expressed as$$ B_{sk} =\frac{Q_{3}+Q_{1}2Q_{2}}{Q_{3}Q_{1}}= \frac{Q_{0.75}2Q_{0.5}+Q_{0.25}}{Q_{0.75}Q_{0.25}}. $$

Moors kurtosis M _{ kur }: This is defined as$$M_{kur}=\frac{(E_{7}E_{5})+(E_{3}E_{1})}{E_{6}  E_{2}}= \frac{Q_{0.875}Q_{0.625}+Q_{0.375}Q_{0.125}}{Q_{0.75}Q_{0.25}}, $$
where Q _{ i } is the i ^{ t h } quartile for i=1,2,3, and E _{ i }=F ^{−1}(i/8), i=1,2,⋯,7 represents the i ^{ t h } octile.
Mean deviation
where F(.) is the CDF of the KwTP distribution, and \(J(t)=\int _{\theta }^{t}x f(x)dx.\)
Using Eq. (21), we can write appropriate expressions for J(μ) and J(M). Combining these with the expression for F(μ), it is easy to obtain the expressions for D(μ) and D(M) in Eqs. (19) and (20) respectively.
Reliability
The following lemma shows the limiting behavior of the hazard function:
Lemma 1
Proof
Order statistics
and \(d_{r,0}=(\rho _{0}^{*})^{r}\).
We can obtain d _{ r,m } from d _{ r,0}, ⋯, d _{ r,m−1} and, therefore, from ρ _{0},ρ _{1}, ⋯, ρ _{ m }.
and d _{ i+s−1,0}=(ρ _{0})^{ i+s−1}, where ρ _{0} is defined in Eq. (16).
Parameter estimation
In this section we estimate the parameters of the Kumaraswamy transmuted Pareto distribution by the method of maximum likelihood estimation.
where G(x _{ i };α,θ,λ) is the PDF of the transmuted Pareto distribution.
The maximum likelihood estimates \(\hat {\alpha }, \hat {\lambda }, \hat {a},\hat {b}\) of the unknown parameters α,λ,a,b respectively, can be obtained by setting Eqs. (25)  (28) equal zero and solving for the parameters. We can use numerical methods such as the quasiNewton algorithm to numerically optimize the loglikelihood function given in Eq. (24), to get the maximum likelihood estimates of the parameters α,λ,a,b. To compute the standard error and the asymptotic confidence interval, we use the usual large sample approximation in which the maximum likelihood estimators for γ can be treated as being approximately normal. This will require the computations of the second order derivatives of Eq. (24) with respect to the vector of parameters.
Applications of KwTP
In this section we apply the KwTP to two data sets. These are exceedances of flood peaks (in m ^{3}/s) of the Wheaton River near Carcross in Yukon Territory, Canada, and the Norwegian fire insurance claims data.
11.1 The exceedances of flood peaks (in m ^{3}/s) of the Wheaton River near Carcross in Yukon Territory, Canada
Descriptive statistics for the Wheaton river data
Min.  Q1  Median  Mean  Q3  Max.  Skewness  Kurtosis 

0.100  2.125  9.500  12.200  20.12  64.00  1.4725  5.8895 
Estimated parameters and their standard errors for the Wheaton river data
Model  \(\hat {a}\)  \(\hat {b}\)  \(\hat {\lambda }\)  \(\hat {\alpha }\)  \(\hat {\theta }\) 

KwTP  4.2684  17.0139  0.3687  0.2003  0.1 
(1.5669)  (12.6727)  (0.5308)  (0.0609)    
KwP  2.8553  85.8468    0.0528  0.1 
(0.3371)  (60.4213)    (0.0185)    
BTP  3.9118  17.3874  0.8518  0.1159  0.1 
(1.8159)  (11.7365)  (0.2588)  (0.0509)    
BP  3.1473  85.7508    0.0088  0.1 
(0.4993)  (0.0001)    (0.0015)    
TP  1  1  0.952  0.3490  0.1 
    (0.089)  (0.072)    
EP  2.8797  1    0.4241  0.1 
(0.4911)      (0.0463)    
P  1  1    0.2438  0.1 
      (0.0287)   
The AIC, CAIC, BIC, HQIC and KS test statistic of the Wheaton river data
Model  Statistics  

−ℓ(.,x)  AIC  CAIC  BIC  HQIC  KS  
KwTP  254.017  516.034  516.641  525.085  519.634  0.147 
BTP  256.577  521.154  521.760  530.204  524.753  0.160 
KwP  271.200  548.400  548.753  555.230  551.119  0.170 
BP  283.700  573.400  573.753  580.230  576.119  0.175 
TP  286.201  576.402  576.575  580.954  578.214  0.287 
EP  287.300  578.600  578.774  583.153  580.413  0.199 
P  303.100  608.200  608.257  610.477  609.106  0.332 
11.2 The Norwegian fire insurance data (in kr)
We apply the KwTP to the Norwegian fire insurance claims data for the years 1988 and 1990. Several authors have analyzed these data. See for example, Mdziniso and Cooray (2017) and related references in the paper. The 1988 Norwegian fire claims data consist of 827 fire insurance losses in thousand Norwegian krones, ranging from 500 to 465,365 thousand Norwegian krones. The 1990 Norwegian fire data consist of 628 fire insurance losses in thousand Norwegian krones, ranging from 500 to 78,537 thousand Norwegian krones. Data for both years are highly positively skewed. Using these data, Mdziniso and Cooray (2017) compared the performances of the Pareto (P), the 3paramater generalized Pareto (GP), the oddPareto (OP) and its extension (OP ^{∗}), the 3parameter Burr, the exponentiated Pareto (EP), the exponentiated odd Pareto (EOP) and the odd generalized Pareto (OGP) distributions.
Norwegian fire insurance claims: Estimated values for KwTP
Parameters  −ℓ  AIC  BIC  AD  KS  pvalue  

(Year = 1988,n=827)  
\(\hat {\alpha }\)  \(\hat {\lambda }\)  \(\hat {a}\)  \(\hat {b}\)  
1.0197  0.7034  0.9698  1.2808  6751.24  13510.49  13529.29  0.46  0.03  0.62 
(0.4104)  (0.2148)  (0.1609)  (0.6186)  
(Year = 1990,n=628)  
\(\hat {\alpha }\)  \(\hat {\lambda }\)  \(\hat {a}\)  \(\hat {b}\)  
0.6310  0.7285  1.0873  3.1417  5039.91  10087.82  10105.57  0.96  0.04  0.40 
(0.2739)  (0.1873)  (0.1669)  (1.8428) 
Simulation study
Simulation from KwTP of the MLE for α=2,λ=0.1,a=3, and b=2.5
Sample size  Parameter  Estimate  RMSE  MAE 

(n)  
5  \(\hat {\alpha }\)  2.0490  0.7727  0.6466 
\(\hat {\lambda }\)  0.1342  0.6446  0.5640  
\(\hat {a}\)  2.7096  1.0869  0.8522  
\(\hat {b}\)  2.4806  1.1296  0.9987  
10  \(\hat {\alpha }\)  2.0722  0.6983  0.5810 
\(\hat {\lambda }\)  0.1046  0.6089  0.5274  
\(\hat {a}\)  2.7421  0.9994  0.7871  
\(\hat {b}\)  2.5427  1.0715  0.9389  
20  \(\hat {\alpha }\)  2.0429  0.6157  0.5030 
\(\hat {\lambda }\)  0.0228  0.5786  0.5022  
\(\hat {a}\)  2.8104  0.9036  0.7274  
\(\hat {b}\)  2.5201  0.9400  0.8048  
50  \(\hat {\alpha }\)  2.0078  0.5674  0.4486 
\(\hat {\lambda }\)  0.0330  0.5507  0.4692  
\(\hat {a}\)  2.8363  0.8251  0.6755  
\(\hat {b}\)  2.4863  0.8682  0.7382  
100  \(\hat {\alpha }\)  1.9644  0.5361  0.4302 
\(\hat {\lambda }\)  0.0590  0.5288  0.4445  
\(\hat {a}\)  2.8337  0.7392  0.5914  
\(\hat {b}\)  2.4907  0.7856  0.6617  
200  \(\hat {\alpha }\)  1.9783  0.5214  0.4246 
\(\hat {\lambda }\)  0.0628  0.5082  0.4210  
\(\hat {a}\)  2.8550  0.6567  0.5191  
\(\hat {b}\)  2.5186  0.7352  0.6141  
500  \(\hat {\alpha }\)  1.9691  0.5515  0.4607 
\(\hat {\lambda }\)  0.0846  0.4972  0.4057  
\(\hat {a}\)  2.8452  0.6056  0.4562  
\(\hat {b}\)  2.5001  0.7121  0.5980  
1000  \(\hat {\alpha }\)  1.9802  0.5694  0.4841 
\(\hat {\lambda }\)  0.1073  0.4818  0.3830  
\(\hat {a}\)  2.8629  0.5301  0.3895  
\(\hat {b}\)  2.4704  0.6748  0.5693 
Conclusion
We have proposed in this article, a new distribution that is being referred to as the Kumaraswamy transmuted Pareto (KwTP). The transmuted Pareto distribution is used as a baseline distribution in the Kumaraswamy distribution to construct the KwTP distribution. Many mathematical and statistical properties and special cases of the KwTP are obtained. The estimation of the model parameters is performed by the maximum likelihood method. We compare the distribution with few other distributions in modeling two real datasets. Various statistics indicate that KwTP better fits the Wheaton river data set than other comparative distributions, and majority of its competitive distributions for the Norwegian insurance claims data. We conducted a simulation study to assess the performance of the maximum likelihood estimation procedure for estimating the parameters of the KwTP distribution. It is expected that the KwTP distribution will serve as a better alternative in modeling data sets exhibiting the extreme value properties.
Declarations
Acknowledgements
The authors would like to thank the Editor and the anonymous reviewers for their valuable comments and suggestions which have greatly improved the quality of the paper.
Authors’ contributions
The authors, SBC, AAA, GA and HL with the consultation of each other carried out this work and drafted the manuscript together. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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