Generalized Rank Mapped Transmuted Distribution for Generating Families of Continuous Distributions

M. A. Ali; Haseeb Athar

doi:10.2991/jsta.d.210129.001

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Volume 20, Issue 1, March 2021, Pages 132 - 148

Generalized Rank Mapped Transmuted Distribution for Generating Families of Continuous Distributions

Authors

M. A. Ali¹, Haseeb Athar²^{, *}

¹Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia

²Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, India

^*Corresponding author. Email: haseebathar68@gmail.com

Corresponding Author

Haseeb Athar

Received 7 October 2018, Accepted 5 January 2021, Available Online 5 February 2021.

DOI: 10.2991/jsta.d.210129.001 How to use a DOI?
Keywords: Transmuted map; Order Statistics; Weibull distribution; Beta distribution; Hazard function; Continuous distributions
Abstract: This study introduces generalized transmuted family of distributions. We investigate the special cases of our generalized transmuted distribution to match with some other generalization available in literature. The transmuted distributions are applied to Weibull distribution to find generalized rank map transmuted Weibull distribution. The distributional characteristics such as probability curve, mean, variance, skewness, kurtosis, distribution of largest order statistics, and their characteristics studied to compare with ordinary Weibull distribution. Hazard rate functions and distributional characteristics of largest order statistics of transmuted distributions are also studied. It is observed that the transmuted distributions are more flexible to model real data, since the data can present a high degree of skewness and kurtosis. If someone is interested to locate more flexible and higher degree of skewed distribution can explore this generalized transmuted family of distributions for future use.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The ideas of developing new distributions are important issues in recent literatures. Numerous families of distributions have proposed by several authors for modeling data in several areas such as engineering, economics, finance and actuarial science, medical and life sciences. However, in many applied areas like lifetime analysis, insurance analysis we need extended distributions, that is, new distributions which are more flexible to model real data, since the data can present a high degree of skewness and kurtosis.

Lee et al. [1] provided an overview of most method used to generate family of continuous distributions earlier in 1980. For more details about these methods can be referred to Pearson [2], Johnson [3], and Tukey [4]. Recently several literatures were discussed generalized method to generate extended generalized family of distributions. For more details about the recent development may refer to Johnson et al. [5], Eugene et al. [6], Jones [7], Alzaatreh et al. [8], Bourguignon et al. [9], Afify et al. [10], Granzotto et al. [11], Al-Kadim and Mohammed [12], Jayakumar and Babu [13], Mahdavi and Kundu [14], Alizadeh et al. [15], Al-Kadim [16], Pobocikova et al. [17], Elgarhy et al. [18], Afify et al. [19], and references therein. Apart from the above, a more extended generalized nth degree transmuted method suggested in this study to generate transmuted distributions. An application of the generated transmuted map is extended to the Weibull distribution. The distributional characteristics of the generated transmuted distributions are also simulated to compare with traditional Weibull distribution.

This paper is organized in the following way: In Section 2, we developed nth degree generalized transmutation map based on continuous family of distribution. Particular cases are also discussed. Some other members of generalized transmutation map are identified. In Section 3, the survival function, hazard rate function, and reserved hazard rate function of newly generated generalized transmuted distribution are discussed. In Section 4, we developed the nth degree generalized transmutation map based on Weibull distribution. Also, probability density function (pdf) graphs are simulated and presented in figure to compare each other. In Section 5, some distributional characteristics such as mean variance, skewness, and kurtosis are simulated and presented in tabular form to compare each other for different parametric set of values. The Section 6 is based on discussion and proof of some theorems related to order statistics (OS) of generalized transmuted Weibull distribution (TWD). Simulated distributional characteristics of largest OS of quadratic TWD are also presented in section 6. The Section 7, states conclusion and then inserted references.

2. GENERALIZED nth RANK TRANSMUTATION MAP

The construction of the generalized n-degree transmutation map considered here is simple and intuitive. Let X1,X2,…,Xn be a random sample from an absolutely continuous population with pdf gx,x∈a,b corresponding to cumulative distribution function (cdf) Gx with ∫abfxdx=∫φψfh(t)h′(t)dt;x=ht. This is valid under the following conditions:

fx is continuous on some interval A≤x≤B containing the original limits of integration, a,b.
The equalities a=hφ and b=hψ holds.
ht and its derivative h′t are continuous on the interval φ≤t≤ψ.
As t varies from φ to ψ, the function ht always varies in the same direction from hφ=a to hψ=b.

Let X1:n≤X2:n≤…≤Xn:n be the OS obtained by arranging the preceding random sample in increasing order of magnitude. The cdf of Xr:n1≤r≤n is given by

Gr:nx=PXr:n≤x=∑i=rnniGxi1−Gxn−i=IGxr,n−r+1,(1)

where, the beta function with Bp,q=B1p,q, and Ixp,q=Bxp,qBp,q is incomplete beta function ratio. The corresponding pdf is given by

gr:nx=∂∂xFr:nx=n!r−1!n−r!Gxr−11−Gxn−rgx=gxbGx;r,n−r+1;−∞<x<∞,1≤r≤n,(2)

where bt;p,q=1Bp,qtp−11−tq−1,0≤t≤1.

Now, consider the random variable, Y as

Y~gr:nx with probability πr:nr=1,2,3,…,n; n=1,2,3,…,

where 0≤πr:n≤1 and ∑r=1nπr:n=1.

Hence, generalized transmuted cdf of nth rank mapped (n = 1, 2, 3, …) distribution is given by

FYx=∑r=1nπr:nIGxr,n−r+1=∑r=1nmr:nx,(3)

and corresponding generalized transmuted pdf of nth rank mapped distribution is

fYx=∂∂xFYx=∑r=1n∂∂xmr:nx=∑r=1nπr:n∂∂xIGxr,n−r+1(4)

=gx∑r=1nπr:n bGx;r,n−r+1,(5)

where mr:nx=πr:nIGxr,n−r+1 and b.;.,. is same as before.

Now,

bt;p,q=1Bp,qtp−11−tq−1=1Bp,qtp−1∑i=0q−1−1q−1−initq−1−i=1Bp,q∑i=0q−1−1q−1−initp+q−2−i(6)

Using (6) in (5) to get

fYx=gx∑r=1n∑j=0n−r−1n−r−jπr:nBr,n−r+1n−rjGxn−1−j=gxGxn−1∑r=1n∑j=0n−rkrjx=gxGxn−1∑r=1nkr,(7)

where, ki=∑j=0n−ikij and kij=−1n−i−jπi:nBi,n−i+1n−ijGx−j.

Expression (3) and (7) will also be helpful for simulation study for the cdf and pdf respectively.

2.1. Particular Cases

Put n=2,π1:n=π,i.e.,π2:2=1−π and λ=2π in (3) and (7) to get the quadratic transmutation map of Shaw and Buckley [20]
Fx=λGx+1−λG2x
and the corresponding pdf of quadratic transmutation map of Shaw and Buckley [20] is
fx=gxGx+λ1−Gx.
Put n=3,π3:3=1−π1:3−π2:3 and λ1=3π1:3,λ2=3π2:3 in (3) and (7) to get the cubic ranking transmutation map of Granzotto et al. ([11], Eq. (3), pp. 2761)
Fx=λ1Gx+λ2−λ1G2x+1−λ2G3x
and the pdf of cubic ranking transmutation map of Granzotto et al. [11] is
fx=gxλ1+2λ2−λ1Gx+31−λ2G2x.
Put π1:n=1 and πi:n=0∀2≤i≤n in (3) and (7) to develop the simple transmutation map of Eugene et al. [6] on using beta distribution as a generator.
Fx=IGxp,q
and the pdf of simple transmutation map of Eugene et al. [6] is
fx=gxbGx;p,q.
Put n=3,λ=λ1=3π1:3=−λ2, λ2=3π2:3 in (3) and (7) to develop the cubic ranking transmutation map of Al-Kadim and Mohammed [12] with cdf
Fx=1+λGx−2λG2x−λG3x
and the corresponding pdf of the cubic ranking transmutation map of Al-Kadim and Mohammed [12] is
fx=1+λgx−4λGxgx−3λG2xgx.

It may be noted that the general formula of the transmuted distribution of Al-Kadim [16] can also be deduced from (3) as a particular case for odd and even n respectively.
Put n=4,λi=2πi:4;1≤i≤4 in (3) and (7) to develop the generalized Quartic ranking transmutation map which is new cdf as suggested.
Fx=2λ1Gx+3λ2−λ1G2x+2λ1−2λ2+λ3G3x+1−λ1+λ2−2λ3G4x(8)
and the corresponding generalized Quartic ranking transmutation map with new pdf as suggested is given by
fx=gx2λ1+6λ2−λ1Gx+6λ1−2λ2+λ3G2x+41−λ1+λ2−2λ3G4x.(9)
Put n=5,λi=5πi:5;1≤i≤5 in (3) and (7) to develop the generalized Quintic ranking transmutation map with a new cdf suggested as
Fx=λ1Gx+2λ2−λ1G2x+2λ1−2λ2+λ3G3x+3λ2−λ1−3λ3+λ4G4x+1−λ2−λ3−λ4G5x(10)
and the corresponding pdf of generalized Quintic ranking transmutation map which is a new pdf as suggested is
fx=gx[λ1+4λ2−λ1Gx+6λ1−2λ2+λ3G2x+43λ2−λ1−3λ3+λ4G3x+51−λ2−λ3−λ4G4x].(11)
Put λi=nπi:n;1≤i≤n, and n=6,7,8,… in (3) and (7) to develop the cdf of suggested generalized desired nth (n≥6) ranked transmutation map as well as the corresponding pdf of generalized desired nth ranked transmutation map.
If one put, πi:n=1n for 1≤i≤n, then Fx=Gx
If we put, πi:n=ni/2n−1, for all 1≤i≤n and n=2,3,4,…, then from (3) and (7) we get another new generalized nth ranked transmuted map with cdf for generating families of distributions is
FTDnx=∑r=1nπr:nIGxr,n−r+1=∑r=1nmr:nx,(12)
where mr:nx=nr/2n−1IGxr,n−r+1 and the corresponding generalized n^th ranked transmuted map of pdf for generating families of distribution are given by
fTDnx=gxGxn−1∑r=1nkr,(13)
where, ki=∑j=0n−ikij and kij=−1n−i−j2n−1Bi,n−i+1nin−ijGx−j.

Some Specific Cases of (12) and (13)

2.1.1. Quadratic rank transmuted distribution (TD2)

For n = 2 in (12) and (13), the new form of quadratic map ranked transmuted cdf is given by

FTD2x=134Gx−G2x=13Gx3+1–Gx(14)

and the corresponding new form of quadratic map ranked transmuted pdf is given by

fTD2x=gx34−2Gx=2gx31+1−Gx.(15)

2.1.2. Cubic rank transmuted distribution (TD3)

For n = 3 in (12) and (13), the new form of cubic ranked transmuted cdf is given by

FTD3x=179Gx−2G3x(16)

and the corresponding new form of cubic ranked transmuted pdf is given by

fTD3x=gx79−6G2x.(17)

2.1.3. Quartic rank transmuted distribution (TD4)

For n = 4 in (12) and (13), the new form of quartic ranked transmuted cdf is given by

FTD4x=11516Gx+12G2x−16G3x+3G4x(18)

and the corresponding new form quartic ranked transmuted pdf is given by

fTD4x=gx1516+24Gx−48G2x+12G3x.(19)

2.1.4. Quintic rank transmuted distribution (TD5)

For n = 5 in (12) and (13), the new form of quintic ranked transmuted cdf is given by

FTD5x=13125Gx+50G2x−50G3x+6G5x(20)

and the corresponding new form of quintic ranked transmuted pdf is given by

fTD5x=gx3125+100Gx−150G2x+30G4x.(21)

In a similar manner, one can generate any desired higher order n≥6 rank transmuted map from (12) and (13).

3. HAZARD FUNCTION

The survival function Sx, hazard rate function hx, and reserved hazard rate function rx of newly generated generalized transmuted cdf FTDn(x) (Eq. 12) corresponding to pdf fTDn(x) (13) are respectively given by

STDnx=1−FTDnx(22)

hTDnx=fTDnx/STDnx=fTDnx/1−FTDnx(23)

rTDnx=fTDnx/FTDnx,(24)

where, fTDnx and FTDnx are as before.

Use (3) in (22) to get survival function for newly derived nth ranked transmuted distribution as

STDnx=1−∑r=1nmr:nx.(25)

Use (7) and (22) in (23) to get hazard rate function for newly derived nth ranked transmuted distribution as

hTDnx=gxGxn−1∑r=1nkr1−∑r=1nmr:nx.(26)

Use (3) and (7) in (24) to get reserved hazard rate function for newly derived nth ranked transmuted distribution as

rTDnx=gxGxn−1∑r=1nkr∑r=1nmr:nx,(27)

where, mr:nx and kr are as before.

For more about the hazard function one can refer to Zubair et al. [21] and references therein.

Theorem 3.1.

The quadratic generalized transmuted hazard rate function is given by

hTD2x=2gx2−Gx3−Gx4−Gx.

Proof.

From (23) and for n = 2, we get the generalized transmuted hazard rate function as

hTD2x=fTD2x1−FTD2x.(28)

Now using (14) and (15) in (28) and on algebraic simplification gives the proof of the theorem.

Theorem 3.2.

The quadratic generalized transmuted Weibull hazard rate function is given by

hTWD2x=2αxα−1e−x/βα+e−x/β2α3βα1−23e−x/βα+13e−x/β2α.

Proof.

From (28) and on using (29) and (30) gives the proof of the theorem.

Simulated hazard function (WD, TWD2 to TWD5) for some specific sets of parameters (α, β) values are presented in Figures 1–4 to compare among themselves. The hazard function is one of the most important quantities to character life phenomenon. Compare with many other modified Weibull distributions, the shape of the hazard function is easy to decide. It can be derived from (23) and it is flexible. As we know, it is very common for a bathtub-shaped hazard function of a system or component to have a long useful lifetime with low constant rate portion in the middle and sharp change in the initial and wear-out of phase, so a distribution which can fit this kind of hazard rate would be very useful in reliability studies.

4. TRANSMUTED WEIBULL DISTRIBUTION

The Weibull distribution which was proposed by Weibull [22] is a very important lifetime distribution and is widely used in many fields. However, the hazard function of the traditional Weibull distribution can only be increasing, decreasing, or constant. To meet the need of fitting complex modes and the bathtub-shaped hazard rate, researchers have proposed many improved flexible models based on the traditional Weibull distribution. To know more about modified or improved models based on the traditional Weibull distribution, one may refer to Johnson et al. [5], Xie et al. [23], Bebbington et al. [24], Nassar et al. [25], Afify et al. [19], and references therein. Still even available modified Weibull model are not enough to represent or fit the data obtain all cases such as engineering, economics, finance and actuarial science, medical and life sciences. Our proposed transmuted model will be more flexible and will cover such limitation for which data present a higher degree of skewness and kurtosis.

A random variable X is said to have traditional Weibull distribution (WD) with parameters (α>0,β>0,θ≥0) if its cdf is given by

Gx=1−exp−x−θβα; x≥θ(29)

and corresponding pdf is given by

gx=αβx−θβα−1exp−x−θβα; x≥θ,(30)

where, α is the shape parameter, β is the scale parameter, and θ is the location parameter. Throughout the paper take θ=0 without loss of generality, then (29) and (30) transform into cdf and pdf of 2-parameters traditional Weibull distribution, respectively.

Quadratic ranked map Transmuted Weibull distribution (TWD2)

Using (29) in (14); (29) and (30) into (15) to get the new Quadratic map ranked transmuted Weibull cdf as
FTWD2x=1−exp−xβα1+13exp−xβα(31)
and the corresponding new Quadratic map ranked transmuted Weibull pdf is given by
fTWD2x=2α3βxβα−1exp−xβα1+exp−xβα.(32)
Cubic ranked map Transmuted Weibull distribution (TWD3)

Using (29) in to (16); (29) and (30) into (17) to get the new Cubic map ranked transmuted Weibull cdf as
FTWD3x=1791−exp−xβα−21−exp−xβα3(33)
and the corresponding new cubic map ranked transmuted Weibull pdf is given by
fTWD3x=α7βxβα−1exp−xβα9−61−exp−xβα2.(34)
Quartic ranked map Transmuted Weibull distribution (TWD4)

Using (29) in to (18); (29) and (30) into (19) to get the new Quartic map ranked transmuted Weibull cdf as
FTWD4x=115161−exp−xβα+121−exp−xβα2−161−exp−xβα3+31−exp−xβα4(35)
and the corresponding new Quartic ranked transmuted Weibull pdf is given by
fTWD4x=α15βxβα−1exp−xβα16+241−exp−xβα−481−exp−xβα2+121−exp−xβα3.(36)
Quintic ranked map Transmuted Weibull distribution (TWD5)

Using (29) in to (20); (29) and (30) into (21) to get new the Quartic map ranked transmuted Weibull cdf as
FTWD5x=131251−exp−xβα+501−exp−xβα2−501−exp−xβα3+61−exp−xβα5(37)
and the corresponding new quartic ranked transmuted Weibull pdf is given by
fTWD5x=α15βxβα−1exp−xβα25+1001−exp−xβα−1501−exp−xβα2+181−exp−xβα4.(38)

In a similar way one can find new any desired rank map transmuted cdf of Weibull distribution by using (29) in to (12); corresponding pdf by using (29) and (30) into (13).

The simulated pdf curves of different ranked TWD for some specific sets of parametric values (α, β) were plotted in Figures 5–8 to observe and compare the change of skewness and the pdf curve shapes with the change of transmutation rank.

It is observed from the above Figures 5–8 that the TWDs are more skewed compare to ordinary Weibull distribution. The degree of skewness of TWDn curves increases if the rank of transmutation map increases. So, the newly generated TWDs have advantages to fit if the data sets are more skewed.

5. DISTRIBUTIONAL CHARACTERISTICS

The k^th (k=1,2,3,…) raw moment corresponding to generalized n^th rank transmuted map of Weibull pdf (30) are given from (13) as

μk′=EXk=∫0∞xkfxdx=∑r=1nkr,(39)

where, ki=∑j=0n−ikij, kij=−1n−i−j2n−1Bi,n−i+1nin−ijMk,n−1−j and Mk,m=∫0∞xkgxGmxdx.

Now on using (29) and (30), we have

Mk,m=αβ∫0∞xkxβα−1exp−xβα1−exp−xβαmdx=αβα∫0∞xk+α−1e−xβα∑i=0m−1imiexp−xβαidx=αβα∑i=0m−1imi∫0∞xk+α−1exp−xβαi+1dx.

We know that ∫0∞xnexp−qxmdx=Γn+1mmqn+1, m,n>0 and real. For more about it refer to Jeffrey and Dai ([26], p. 272). So that

Mk,m=αβk∑mm−1imiΓk+ααi+1αi+1=αβk∑i=0mδim,(40)

where, δim=−1imiΓk+ααi+1αi+1.

5.1. Moments for Quadratic TWD2

For n = 2 in (39), the kth raw moment of the quadratic TWD is given by

μk′=EXk=∫0∞xkfxdx=βk32Γk+αα+Γk+α2α(41)

and for k = 1, 2, 3, 4, the first four moments of the quadratic TWD are given by

Mean=μ1′=β32Γ1+αα+Γ1+α2α

μ2′=β232Γ2+αα+Γ2+α2α

μ3′=β332Γ3+αα+Γ3+α2α

and μ4′=β432Γ4+αα+Γ4+α2α.

The central moments are given by

VarX=μ2=β232Γ2+αα+Γ2+α2α−β292Γ1+αα+Γ1+α2α2μ3=μ3′−3μ2′μ1′+2μ1′3and μ4=μ4′−4μ3′μ1′+6μ2′μ1′2−3μ′4.

Pearson's four coefficients, based upon the first four central moments are

β1=μ32μ23,γ1=+β1 and β2=μ4μ22,γ2=β2−3.

It may be noted that these coefficients are true numbers independent of units of measurement.

The pthp∈0,1 percentile point of quadratic TWD (32) is given by

xp=β−ln1−21−1−3p/41/α

and random observation can be generated from the following inverse function X=β−ln1−21−1−3U/41/α, where U~U0,1.

5.2. Moments for Cubic TWD3

For n = 3 in (39), the kth raw moment of the Cubic TWD is given by

μk′=EXk=∑r=13kr=k1+k2+k3,

where, k1=∑j=02k1j=k10+k11+k12, k2=∑j=01k1j=k20+k21, k3=k30.

k10=97Mk,2, k11=187Mk,1, k12=97Mk,0, k20=−187Mk,2, k21=187Mk,1, k30=37Mk,2.

Mk,0=αβkδ00, Mk,1=αβkδ01+δ11, Mk,2=αβkδ02+δ12+δ22.

δ00=Γk+α, δ01=Γk+α, δ02=Γk+α, δ11=−11+αΓk+α1+α, δ12=−21+αΓk+α1+α, δ22=−11+2αΓk+α1+2α.

From (34), the r^th raw moment of TWD3 is given by

μk′=EXk=βk73Γα+kα+6Γα+k2α−2Γα+k3α.(42)

Mean=μ1′=β73Γα+1α+6Γα+12α−2Γα+13α,

μ2′=β273Γα+2α+6Γα+22α−2Γα+23α

μ3′=β373Γα+3α+6Γα+32α−2Γα+33α,

μ4′=β473Γα+4α+6Γα+42α−2Γα+43α.

and VarX=β273Γα+2α+6Γα+22α−2Γα+23α−β2493Γα+1α+6Γα+12α−2Γα+13α2.

Other central moments and Pearson's four coefficients can be obtained from the above by simple algebraic manipulation.

5.3. Moments for Quartic TWD4

For n = 4 in (39), the kth raw moment of the TWD4 is given by

μk′=EXk=βk154Γα+kα+18Γα+k2α−4Γα+k3α−3Γα+k4α.(43)

5.4. Moments for Quintic TWD5

For n = 5 in (39), the kth raw moment of the TWD5 is given by

μk′=EXk=βk315Γα+kα+40Γα+k2α+10Γα+k3α−30Γα+k4α+6Γα+k5α.(44)

For example, some distributional properties like mean, variance, skewness, and kurtosis are simulated and presented below in Table 1 for some specific values of the parameters (α, β) to observe and compare differentiation of traditional Weibull distribution (30) along with some other different ranked map TWD (TWDn, n = 2,3, 4, 5), where, n indicates nth rank (31–38) map. It is observed that skewness of transmuted distribution is more flexible as rank of transmutation increases. So, one can use flexible desired rank map transmuted distribution to fit desired skewed data set. For all simulation work MATLAB R2015a version is used.

Distributional Characteristics		Different Combination of Parameter (α, β) Values
Distributional Characteristics		α=0.4β=0.3	α=1.0β=1.0	α=1.3β=1.2	α=1.4β=1.2	α=1.5β=1.3	α=1.5β=1.2	α=0.5β=0.3	α=2.0β=0.3	α=0.6β=1.3	α=0.5β=0.1
Means	WD	0.9970	1.0000	1.1083	1.0937	1.1736	1.0833	0.6000	0.2659	1.9559	0.2000
	TWD2	0.7566	1.0000	1.1714	1.1714	1.2715	1.1737	0.4886	0.2998	1.6909	0.1629
	TWD3	0.5841	0.8988	1.0684	1.0717	1.1662	1.0765	0.3993	0.2771	1.4330	0.1331
	TWD4	0.4571	0.7511	0.8879	0.8890	0.9658	0.8915	0.3255	0.2280	1.1889	0.1085
	TWD5	0.3654	0.6025	0.6980	0.6951	0.7515	0.6936	0.2673	0.1743	0.9806	0.0891
Variances	WD	9.8060	1.0000	0.7391	0.6266	0.6349	0.5410	1.8000	0.0193	11.8246	0.2000
	TWD2	6.6876	0.6287	0.3727	0.2819	0.2473	0.2107	1.2411	0.0001	8.1840	0.1379
	TWD3	4.4160	0.5232	0.3566	0.2867	0.2715	0.2314	0.8456	0.0041	5.7940	0.0940
	TWD4	2.8471	0.5209	0.4601	0.4087	0.4322	0.3683	0.5756	0.0156	4.2317	0.0640
	TWD5	1.7331	0.3269	0.2397	0.1943	0.1842	0.1570	0.3615	0.0022	2.7453	0.0402
Skewness	WD	142644.2686	35.0000	21.4433	15.4247	19.0513	11.7856	372.0816	0.0013	67852.7372	0.5104
	TWD2	63555.1105	18.5292	13.1605	9.9132	12.7894	7.9119	166.8259	0.0010	30860.4307	0.2288
	TWD3	26345.0289	9.9275	8.1805	6.4000	8.5316	5.2779	69.9412	0.0008	13291.0082	0.0959
	TWD4	10238.2132	5.4371	5.1186	4.1148	5.6021	3.4656	27.6680	0.0005	5489.6068	0.0380
	TWD5	3764.5469	3.2185	3.4063	2.7832	3.8259	2.3668	10.4650	0.0004	2226.7486	0.0144
Kurtosis	WD	305.6793	24.0000	25.1054	26.5616	28.4265	28.4265	100.8000	43.4273	52.9786	100.8000
	TWD2	438.1475	41.5948	70.8843	95.9613	139.5531	139.5531	141.3656	741.4459	73.7997	141.3656
	TWD3	645.9979	40.8002	56.3043	68.9036	87.7369	87.7369	195.8565	642.8510	94.8898	195.8565
	TWD4	967.0436	27.9066	24.9716	25.6054	26.6692	26.6692	263.1634	36.1091	111.1517	263.1634
	TWD5	1578.6245	48.6008	69.6797	87.7467	115.7325	115.7325	404.0340	455.9642	160.7934	404.0340

Table 1

Distributional characteristics of nth,n=2,3,4,… degree mapof transmuted Weibull distribution (TWDn, n=2,3,4,…).

6. ORDER STATISTICS

OS and functions of OS play an important role in statistical theory and methodology. Floods and droughts, longevity, breaking strength, aeronautics, oceanography, duration of humans, organisms, components, and devices of various kinds can be studied by the theory extreme values. Life tests provide an ideal illustration of the advantage of OS in censored data. Since such an experiment may take a long time to complete, it is often advantageous to stop after failure of the first r out of n similar items under test. For more details and development of OS one may refer to Sarhan and Greenberg [27], Arnold and Balakrishnan [28], Balakrishnan and Cohen [29], Arnold et al. [30], Ali [31], and David and Nagaraja [32].

The pdf of rth OS for the TD2 (15) is given by

fr:nTD2=Br,n−r+1FTD2xr−11−FTD2xn−rfTD2x.(45)

The pdf of extreme OS follows from (45) at r = 1 and r = n respectively given by

f1:nTD2=n1−FTD2xn−1fTD2x.(46)

fn:nTD2=nFTD2xn−1fTD2x.(47)

Theorem 6.1.

For n=2,3,…, the recurrence relation between the pdf of largest OS of quadratic rank transmuted distribution given in (47) and any ordinary distribution is given by

fn:nTD2x=n4/3n∑i=0n−1−1/4in−1ign+i:n+in+i−gn+i+1:n+i+12n+i+1,(48)

where, gm:m=mGm−1xgx. Gx and gx are cdf and pdf of any continuous distribution, respectively.

Proof.

From (47) and on using (14) and (15), we have

fn:nTD2x=n134Gx−G2xn−1gx34−2Gx=n4/3nGn−1x1−14Gxn−11−12Gxgx=n4/3n∑i=0n−1−1/4in−1iGn+i−1x−12Gn+ixgx.

Now using gm:m=mGm−1xgx to get the required result.

Theorem 6.2.

For n=2,3,…, the recurrence relation between kth order moment of largest OS for the pdf (47) of quadratic rank transmuted distribution and kth order moment of largest OS for the pdf of ordinary continuous distribution is given by

μn:nkTD2=n4/3n∑i=0n−1−1/4in−1iμn+i:n+ikn+i−μn+i+1:n+i+1k2n+i+1.(49)

Proof.

Multiplying both sides of (48) by Xk and then take expectation to get the result of the theorem.

Theorem 6.3.

For n=2,3,…, the recurrence relation between largest OS pdf of (47) of quadratic rank transmuted distribution and largest OS pdf of Weibull distribution (30) is given by

fn:nTWD2x=n43nαβxβα−1∑i=0n−1∑j=0n+i−1−1i+j4in−1in+i−1j−12n+ije−x/βj+1α−∑i=0n−1−1n+2i4i12n−1ie−x/βn+i+1α.(50)

Proof.

We know that

gm:m=mGm−1xgx=mαβxβα−1e−x/βα1−e−x/βαm−1 Using 29 and 30=mαβxβα−1∑j=0m−1−1jm−1je−x/βj+1α on binomial expansion.

Now using this result in (48) and on algebraic manipulation, we get the required result.

Theorem 6.4.

For n=2,3,…, the k^th order moment of largest OS for the pdf (32) of quadratic rank TWD is given by

μn:nkTWD2=n4/3nβk∑i=on−1∑j=0n+i−1−1i+jj+14in−1in+i−1j−12n+ijΓk+αj+1α+∑i=0n−1−1n+i+1n−1i2n+i+14iΓk+αn+i+1α.(51)

Proof.

For Weibull distribution defined in (29) and (30), we have

gm:mx=mGm−1xgx.

By expanding above expression binomially, we get

gm:m(x)=m1−exp−xβαm−1αβxβα−1exp−xβα=mαβαxα−1∑j=0m−1−1jm−1jexp−xβj+1α.(52)

Now kth order moment of largest OS of Weibull distribution (30) is given by

μm:mk=∫0∞xk gm:mxdx=mαβα∑j=0m−1−1jm−1j∫0∞xk+α−1exp−xβj+1αdx on using 52.

Since ∫0∞xn exp−qxmdx=Γn+1mmqn+1, m,n>0.

Therefore,

μm:m(k)=mαβα∑j=0m−1−1jm−1jβk+αj+1αΓk+αj+1α,=mβk∑j=0m−1−1jm−1j1j+1Γk+αj+1α.(53)

Now using (53) in (49) for m=n+i and m=n+i+1 then the kth order moment of largest OS for the pdf (32) of quadratic rank TWD is given by

μn:nkTWD2=n4/3n∑i=0n−1−1/4in−1iβk∑j=0n+i−1−1jn+i−1j1j+1Γk+αj+1α−βk12∑j=0n+i−1jn+ij1j+1Γk+αj+1α=n4/3nβk∑i=0n−1−1/4in−1i∑j=0n+i−1−1jn+i−1j−12n+ij1j+1Γk+αj+1α+−1n+i+12n+i+1Γk+αn+i+1α.

Now on algebraic manipulation, we have

=n4/3nβk∑i=on−1∑j=0n+i−1−1i+jj+14in−1in+i−1j−12n+ijΓk+αj+1α+∑i=0n−1−1n+i+1n−1i2n+i+14iΓk+αn+i+1α.

Hence the theorem.

7. CONCLUSIONS

In this paper we have generated new generalized transmuted family of distributions (TDn). Some generalized transmuted distributions available in literature are found as particular cases of our transmuted family of distributions. These new generalized transmuted families of distributions are applied to Weibull distribution to find generalized rank map transmuted Weibull distribution (TWDn). Simulated hazard function, pdf curves, and some distributional characteristics such as mean, variance, skewness, and kurtosis for some specific parametric values of generalized transmuted families of Weibull distribution are presented in Figures 1–8 and in Table 1 respectively to make a comparative study among changes of rank maps. Also simulated quadratic ranked transmuted largest os's distributional characteristics are studied and presented in Table 2. These new distributions are more flexible and skewed compare to ordinary Weibull distribution. Flexibility prominently increases as degree of rank of transmutation map increases. These are observed in pdf curves (Figure 4–8) plotting as well as in distributional characteristics presented in Table 1. It is observed that the transmuted distributions are more flexible to model real data, since the data can present a high degree of skewness and kurtosis. If someone is interested to locate more flexible and higher degree of skewed distribution can explore this generalized transmuted family of distributions for future use.

n	α=1.5, β=0.5				α=0.5, β=0.9				α=2.0, β=2.0
n	Mean	Var	Skew	Kurto	Mean	Var	Skew	Kurto	Mean	Var	Skew	Kurto
2	0.2533	0.3362	1.2027	1.8339	2.0547	23.3801	25.5977	56.0054	0.8056	3.9894	1.4560	1.8314
3	0.3605	0.4288	0.3614	1.0651	2.8803	32.3545	18.0286	41.2666	1.1767	5.1747	0.4353	0.9362
4	0.4428	0.4923	0.0961	0.8081	3.6262	40.2543	14.0814	33.6843	1.4542	5.9506	0.1244	0.6501
5	0.5112	0.5401	0.0097	0.7112	4.3217	47.2606	11.6460	29.0873	1.6804	6.5160	0.0184	0.5415
6	0.5708	0.5779	0.0053	0.6877	4.9808	53.4868	9.9987	26.0429	1.8741	6.9525	0.0013	0.5097
7	0.6242	0.6084	0.0482	0.7061	5.6114	59.0147	8.8196	23.9196	2.0454	7.3006	0.0325	0.5195
8	0.6731	0.6334	0.1233	0.7526	6.2186	63.9072	7.9448	22.3947	2.2003	7.5835	0.0946	0.5553
9	0.7184	0.6539	0.2231	0.8201	6.8060	68.2150	7.2812	21.2858	2.3425	7.8156	0.1789	0.6095
10	0.7608	0.6706	0.3435	0.9048	7.3763	71.9799	6.7727	20.4830	2.4746	8.0067	0.2807	0.6779

Table 2

Distributional characteristics of largest OS of TWD2 (48) for different parametric values and sample size.

CONFLICT OF INTEREST

The authors declare that there are no conflicts of interest regarding the publication of this paper.

AUTHORS' CONTRIBUTIONS

All authors have read and agreed to the published version of the manuscript.

ACKNOWLEDGMENTS

The author would like to thank the Editor-in-Chief, and the anonymous referees for their careful reading and constructive comments and suggestions which greatly improved the presentation of the paper.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 20 - 1
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AU  - M. A. Ali
AU  - Haseeb Athar
PY  - 2021
DA  - 2021/02/05
TI  - Generalized Rank Mapped Transmuted Distribution for Generating Families of Continuous Distributions
JO  - Journal of Statistical Theory and Applications
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