A NEW GENERALIZED EXPONENTIATED FAMILY OF CONTINUOUS DISTRIBUTIONS WITH APPLICATIONS TO ENVIRONMENTAL DATA SETS Текст научной статьи по специальности «Математика»

A NEW GENERALIZED EXPONENTIATED FAMILY OF CONTINUOUS DISTRIBUTIONS WITH APPLICATIONS TO ENVIRONMENTAL DATA SETS

Ibrahim, Sule1 Olalekan Akanji, Bello2 Ismail Adekunle, Kolawole3

Department of Mathematical Sciences, Faculty of Pure and Applied Sciences, Kaduna State

University, Kaduna, Nigeria1.

Department of Statistics, Faculty of Basic and Applied Sciences, Osun State University, Osogbo,

Osun State, Nigeria2.

Department of Mathematics and Statistics, Kaduna Polytechnic, Kaduna, Kaduna State, Nigeria3.

[email protected] [email protected] [email protected]

ABSTRACT

Different researchers in the field of distribution theory have derived new models for generalizing the classical ones to make them more flexible and to aid their application in various fields. This generalization and extension of the classical models is mostly done using families of distributions. This article presents a new family of distributions called the Exponentiated Pareto-G family of distributions with two positive shape parameters. Some statistical properties of the new family of distributions, such as explicit expressions for the quantile function, probability-weighted moments, moments, generating function, Reliability function, hazard function, and order statistics are discussed. A maximum likelihood estimation technique is employed to estimate the model parameters. Two submodels such as Weibull and Frechet distributions are employed to check the fit of the family of distributions with the aid of their pdf and hazard function plots. Also, a simulation study is presented to assess the performance of the maximum likelihood estimator. Furthermore, two real-life applications are carried out to assess the fit and flexibility of the new family using the Weibull model as the baseline. The results showed that the new distribution fits better in the two real data sets considered among the range of distributions considered.

Keywords: Exponentiated Pareto-G, maximum flood level, precipitation, consistent, flexibility

I. INTRODUCTION

Research in the field of statistical distribution theory has increased tremendously in the past few years and still growing rapidly. Different researchers in the field of distribution theory have

derived new models for generalizing the classical ones to make them more flexible and to aid their application in various fields. This generalization and extension of the classical models is mostly done using families of distributions. These families of distributions developed have aided the fit of many classical distributions with the addition of extra parameters to the baseline distributions.

Several considerations motivate the development of new generalized families of distributions. More adaptable, flexible, and robust models are required since current distributions frequently fall short of capturing the variability and patterns found in modern data. New distributions that can explain high-dimensional data and adjust to different contexts become crucial as data dimensionality and complexity rise. By offering a more accurate representation of the underlying processes, such distributions improve the resilience and accuracy of statistical analysis. Some of the well-known recently proposed modified families of distributions in the literature by different researchers to improve the standard theoretical distribution and also add flexibility to the classical distributions are: the new generalized family of distributions by [2], Topp-Leone Kumaraswamy-G family of distributions by [12], Topp-Leone Exponentiated-G family of distributions by[ 11], Rayleigh-exponentiated odd generalized-X family of distributions by [17], Type I half-logistic exponentiated-G family of distributions by [6], new generalized family of distributions by [15], Exponentiated type II generalized Topp-Leone-G family of distributions by [1].

II. THE EXPONENTIATED PARETO-G FAMILY (ETP-G) OF DISTRIBUTIONS

A new two-parameter distribution, called the exponentiated Pareto distribution introduced by [9] with cdf and pdf given as

F (x;r, p) = 1 -(1 + x )

f ( x;r, p) = zp(l + x )_T_1 1 -(1 + x )

p-i

According to [3], the cdf of the T-X family of distribution is given as

W [G ( x )]

F (x )= J r (t )dt = R\W [G (x )]"

(1)

(2)

(3)

Where W [G(x)] satisfies the following conditions

(i) (ii)

(iii)

W [G (x)]e[a, b]

W [G( x)]is differentiable and monotonically non-decreasing, and

(4)

W [G(x)] a as x ^ -<» and W [G(x)] b as x ^ ^

Let r (t) be the pdf of a random variable T G [c, d ]for c < d and W [G( x)] be a

function of the cdf of a random variable X .

Then the pdf corresponding to equation (3) is given by;

f ( x) = { dw [G( x) ]j r{W [G( x) ]}

Proposition 1:

(4)

Let G(x;<^)be the cdf of any arbitrary random variable X . Also, let T e (c,d) be a random variable with a pdf, r(t) . Furthermore, let our proposed link function be given as G(x), using the expoenentiated Pareto distribution as the generator, then the cdf of Exponentiated Pareto-G family of distributions is given as:

F ( x;r, p,Ç) = \ 1 -(1 + G( x;CY

(5)

Proof:

G ( x;Ç)

F(x; г, p,C) = тр J (1 +tp1 1 -(1 +t)

p-1

dt

Let y = 1 +1, when t = 0, y = 1 and when t = G(x; y = 1 + G(x; Ç)

So, dt = ôy Now,

1+G(x;Ç)

F(x; r, p,0 = rp J y[1 - yày

(6)

From equation (6),

Let k = 1 -y~T, when y = 1,k = 0 and when y = 1 + G(x;^),k = 1 — (l + G(x;)

So,

1-(1+G ( x ;£))"

F ( x; r, = rp j yT-lhp-

1u P-1 дк

py

,-p-1

1-(1+G ( x;Ç))~

F ( x; r, p,C) = p j к p-1dk

1-(1+G(x;£))"r кP

F ( x; r, = p f — dk

0 p

F ( а О = p

kL

P

1-(1+G ( x;£))"

n1-(1+G ( X))-'

Jo

F ( x; r, aO = [ k ']„ F(x; r, p, Ç) = [ 1 - (1 + G(x; £)) T J, 0 < x < «>

(7)

Where T, p> 0 are the shape parameters and ^ > 0 is a vector of parameters depending on the baseline distribution used.

The pdf to equation (7) is given as

f ( x; t, p, O = rpg ( x; Ç) (1 + G( x; Ç) p1 1 - (1 + G( x; Ç) )

p-1

(8)

III. EXPANSION OF DENSITY

This section presents the densities expansion which will be used to estimate some of the distributions properties.

(1 + Z )-=£ (-1)

1 -1

j=0

fa + i -

For | z |< 1 and (D is a positive real non integer.

Applying equation (9) and equation (10) on the last term in equation (8), we have

1 -(1 + G( x;C)) [1 + G ( x;Oj

p-1

K-0

i (p-1

[1 + G( x; Ç) J"

T<i+1)-1

= i(-1Y [I{'+1)+ - 21[G( x;Ol '

V J

J=0

f(x;r,p,Q = zpg(x;*Q£ ("1)

i, J =0

'+1

(p- 1Yr(i +1) + J - 2^ V J

V ' y

[G ( x;0 ]'

In this vain, using equation (9) and equation (10) on equation (6), we have

[F (x; r, p,C) ]" =[ 1 - (1 + G( x;0)"'J* [ 1 -(1 + G(x;0)^f =±(-lf [Phk ![1 + G(x;0fk

k=0

[1 + G( x;Ork=:£(-1)'

d=0

(zk + d -1 d

[G( x;C)J

(9)

(10)

(11)

i=0

ад h

[F(x;r,AOh S (- 1fk

( ptiVrk + d -1

d=0 к = 0

V k У

d

[G( x;0 f

(12)

IV. PROPERTIES OF ETP-G

I. PROBABILITY WEIGHTED MOMENTS (PWMS)

= E [ XrF (X)s ] = J" xrf (x )(F (x))s dx

(13)

The PWMs of EtP-G is derive by substituting equation (11) and equation (12) into equation (13) by replacing h with s, we have

<p-1-i ,

0 i,j,d=0 k=0 II. MOMENTS

i+j+d+k

r(i + 1 + j-2](psYTk + d-1]

J

V k У

d

xrg(x;0[G(x;0fJdx (14)

/•œ

E(Xr ) = J xrf (x)dx

(15)

The rth moments for EtP-G distribution is derive by substituting equation (11) into equation (15) to obtain

^ ад

E(Xr) = J x-тр^ (-1)i+J

(p- 1Yr(i +1) + j - 2 ^

i, j=0

V '

g(x;0 [G(x; £)) dx

(16)

III. MOMENT GENERATING FUNCTION (MGF)

The Moment Generating Function of x is given as

Mx (t) = f etXf (x)dx (17)

The MGF for EtP-G distribution is derive by substituting equation (11) into equation (17) we obtain

Mx (t ) =

f V ( rt+J (P~ 1lir(i+1) + J - 21 * ( ^rG( r)1

J H) . . e g(x;0lG(x;0\

0 ', J=0

V ' У

œ tzxz

dx

(18)

tx V * I -V

where the expansion of £ = > - and following the process of moments above, we have the

z=0 Z !

MGF for EtP-G distribution in equation (18) given as

œ œ t\ j (p~ +1) + j - 2^

Mx(t)=J rp-z s (-1Г

0 i, J =0 q=0 z !

V ' J

zg ( x; Ç) [G( x;C)]dx

(19)

IV. RELIABILITY FUNCTION

R ( x;r, p,Q = 1 - 1 "(1 + G ( x;0

V. HAZARD FUNCTION

T( x; r, p,Ç) =

zpg ( x; O (1 + G ( x; Ç) p1 1 - (1 + G( x; Q)

p-i

1 -

1 -(1 + G ( x;Ç)-

VI. QUANTILE FUNCTION [ 1 - (1 + G( x;0)U

1 - (1 + G ( x;C)) T = U p 1 - U ^=[1 + G ( x-C) f

(20)

(21)

1 - Uf

= 1 + G( x,C)

G( x;0 =

1 - Uf

-1

x

= Q (u ) = G 1

1 - Uf

-1

VII. ORDER STATISTICS

f ( x)

fn (x) =

B( r, n - r +1)

K-1)'

C n - r^

v=0

V ' y

F ( x )

v+ r-1

(22)

(23)

The pdf of rlh order statistic for distribution is obtained also replacing h with v+r-1 in cdf expansion, we have

1 œ v+ r-1 n - r

fn (x) = B( 1 ^ 1) I H (-1)

B(r, n r + 1) i, j=0 k=0 v= 0

v+i+j+d+k ( p{' + r -1) ^ k

\

J

'rk + d - 1Yp- 1Yz(i +1) + j -2 Yn -r

V d J

V ' J

[G ( x;C)]

j+d+v+r-1

The pdf of the minimum order statistic of the EtP-G distribution is obtained by setting r=1 in equation (24)

f:n (x) = n I S S (" 1)

i, j ,d=0 k=0 v=0

v+i+j+d+k

P' k

Yk + d - 1Y p-1 Yr(z +1) + j - 2 Y n -1

(25)

d

V ' J

j

[G ( x;0]

j+d+v

Also, the pdf of the maximum order statistic of the distribution is obtained by setting r = n in equation (24)

œ v+ n-1

fn:n (x) = n I S (" 1)v

i, j ,d=0 k=0

¥j+d+k ( p(v + n -1)N k

frk + d -1 Yp-1 Yr(z +1) + j - 2

(26)

d

V ' J

[G ( x;0]

j+d+v + n-1

VIII.

MAXIMUM LIKELIHOOD ESTIMATION

This section explores the maximum likelihood estimation (mle) technique to estimate the unknown parameters of the EtP-G distribution. Let Xj,x2,...,Xn be a random sample of size n from the EtP-G distribution. Then, the likelihood function based on observed sample for the vector of parameter (T, P, C)T is given by

log L = wlog r +»log p +£logg(x,.;0- r-1 ¿log l+G(x;C) + p-1 ¿logil- l+G(x;CT

i=i

(27)

The components of score vector U = UT, U, Uq are given as

UT=-~Ë logl + G(x;C)+ p-1 £

/=1

7=1

l + G(x;C)'Tlogl + G(x;C) l-l + G(x;C)"r"

(28)

£/,=- + £ logl-l + G(x;C)

P 7=1

= 0

(29)

f/C=E

7=1

t-

•Ë

7=1

G(x;Oc

l + G(x;C)

+ hE

7=1

rl + G(x;C) G(x;C)c

l-l + G(x;C)

(30)

Equations (28), (29) and (30) cannot be solved analytically, so we have to resort to numerical method to estimate the unknown parameters.

V. SUB MODELS

I. EXPONENTIATED PARETO-WEIBULL (ETPW) DISTRIBUTION

The cdf and pdf of the Weibull distribution are given as

G( x;0, p) = 1 - e-iexf (31)

p-\a - (dx)

fi

g(x;0,P) = 0P°x" e Where x > 0,0, P> 0.

The cdf for ETPW distribution is obtained by inserting equation (31) into equation (7) as

(32)

F (x; t, p,0, P) =

1 -

2 - e

-(dxf

(33)

And the pdf for ETPW distribution is obtained by differentiating equation (33) with respect to x as

f (x; t, p,0, P) = zp0pexpie-

2 - e"

-T-l

1 -

2 - e~

p-i

(34)

Where x > 0,T, p,0, P> 0

- x — 2 8 a = 1 2 p 1 .5 P = 1 .S

— - T = 2 2 e = 1.5 p = 3.5 P — 2 8

---- X 1 .5 e — 2 5 p = 1 8 P — 3 8

--- X = 2.5 0=1.2 p = 1 .5 P = 4.8

x = 3.3 0=1.9 p = 2.5 P = 1 .5

/ V

/ N

/ \

/ V

i \ __

1 ----

f / \

/ ■ " - \

^ \

/ ■■ j

t / ^^

/ / ■ -

/ -- /

—

s - — .......... ---- ---"r-- -

i i O.O 0.5 1 o I 1 .5

X

Figure 1: Plots of pdf of ETPW distribution with different parameter values Reliability function for the ETPW distribution is given as

R( x; t, p,0, P) = 1 -

1 -

2 - e

(0x)p

(35)

Hazard function for the ETPW distribution is given as

zpdßäxp e

ß-\0 - (вх )

T (x; t, p,0, ß) = ■

2 - e

-(вх f

1 -

2 - e

-(вх f

p-1

1 -

1 -

2 - e

<exf

Quantile function for the ETPW distribution is given as

= б (u ) =

--log

e

i -

i - w-

-1

II. EXPONENTIATED PARETO-FRECHET (ETPFr) DISTRIBUTION

The Frechet distribution's cdf and pdf are provided as

G(x;6,S) = e {x} , x > 0,0,8 > 0

g(x;0,S) = S6sx"tfV(x} , x > 0,0,8 > 0 The cdf for ETPFr distribution is given as

F (x;r, p,6,S) =

1 -

1 + e

,x > 0,т,p,6,ö> 0

The pdf for ETPFr distribution is given as

lS-8-1,

-Г-1 -(-)s -X

e x 1 + e x 1- 1 + e x

p-1

,x>0,т,р,в,5 >0

(36)

(37)

(38)

(39)

(40)

(41)

Figure 2: Plots of pdf of ETPFr distribution with different parameter values Reliability function of the ETPFr distribution is given as

R( x; r, p,Q, P) = 1 -

1 -

1 + e

Hazard function of the ETPFr distribution is given as

zppdf!x-p-1e

<-r

T (x;r, p,Q, P) =

1 + e

<-r

-T-1

1 -

1 + e

<-r

p-1

1 -

1 -

1 + e

Quantile function of the ETPFr distribution is given as

e

= 0 (u ) =

- log

1 - Uf

-1

p

(42)

(43)

(44)

VI. SIMULATION STUDY

This section addresses a numerical analysis to evaluate the performance of MLE for ETPW Distribution.

Table 1: MLEs, biases and RMSE for some values of the parameters of ETPW distribution (25,2,6,5) (27,4,8,6)

N Parameters Estimated Values Bias RMSE Estimated Values Bias RMSE

T 25.7129 0.7129 3.7808 27.7166 0.7166 4.1678

20 p 2.7524 0.7524 2.0073 5.5891 1.5891 4.0788

6 6.7789 0.7789 2.2489 9.1209 1.1209 2.8826

p 5.0041 0.0041 0.4382 6.0493 0.0493 0.4292

T 25.5242 0.5242 3.1134 27.4919 0.4919 3.2601

50 p 2.2234 0.2234 0.9125 4.5111 0.5111 1.8794

d 6.4049 0.4049 1.5416 8.5580 0.5580 1.7238

P 5.0226 0.0226 0.2663 6.0445 0.0445 0.2728

100 T 25.3816 0.3816 2.3289 27.4562 0.4562 2.2990

p 2.0852 0.0852 0.4054 4.2018 0.2018 0.9604

d 6.2106 0.2106 1.0451 8.2503 0.2503 1.1375

P 5.0193 0.0193 0.1648 6.0314 0.0314 0.1609

250 T 25.5657 0.5657 1.6176 27.3854 0.3854 1.4242

p 2.0123 0.0123 0.2027 4.0360 0.0360 0.4619

e 5.9821 -0.0179 0.5937 8.0288 0.0288 0.6385

P 5.0174 0.0174 0.0796 6.0192 0.0192 0.0843

500 T 25.4954 0.4954 1.2777 27.2583 0.2583 0.8795

p 1.9991 -0.0009 - 0.1293 4.0074 0.0074 0.2968

d 5.9229 0.0771 0.4329 7.9697 -0.0303 0.4076

P 5.0078 0.0078 0.0553 6.0076 0.0076 0.0495

1000 T 25.4001 0.4001 0.9152 27.1578 0.1578 0.5850

p 1.9956 -0.0044 - 0.0896 4.0003 0.0003 0.1969

d 5.9244 0.0756 0.2979 7.9761 -0.0239 0.2774

P 5.0045 0.0045 0.0323 6.0031 0.0031 0.0342

Table 1 displays the values of biases, estimated values and RMSEs It is noticed from the table that the RMESs approach zero and the estimates tend to the true parameter values as the sample increases. This is an indication that that the maximum likelihood estimates are efficient and consistent.

VII. APPLICATION

The fit of ETPW distribution is tested with applications to environmental data sets to assess its flexibility and robustness. The fit of the new model is compared with some existing distributions having Weibul distribution as the baseline. The comparators are: the Type I Half-Logistic Exponentiated Weibull (TIHLEtW) Distribution by [7], Type II Exponentiated Half Logistic Weibull (TIIEHLW) distribution by [4], Half-Logistic Generalized Weibull (HLGW) Distribution by [13], Exponentiated Weibull (EW) by [14]vand Weibull Distribution by [16].

The data set 1 consists of 20 observations with respect to maximum flood level data to see how the new model works in practice. The data has been obtained from [8] and is given as: 0.654,

0.613, 0.315, 0.449, 0.297, 0.402, 0.379, 0.423, 0.379, 0.3235, 0.269, 0.740, 0.418, 0.412, 0.494, 0.416, 0.338, 0.392, 0.484, 0.265.

The data set 2 is obtained from [10] and also reported in [5]. It consists of thirty successive values of March precipitation (in inches) in Minneapolis/St Paul. The data are:

0.77, 1.74, 0.81, 1.20, 1.95, 1.20, 0.47, 1.43, 3.37, 2.20, 3.00, 3.09, 1.51, 2.10, 0.52, 1.62, 1.31, 0.32, 0.59, 0.81, 2.81, 1.87, 1.18, 1.35, 4.75, 2.48, 0.96, 1.89, 0.90, 2.05.

Table 2: The models' MLEs and performance requirements based on data set 1

Models r 6 P P ll AIC BIC

EtPW 38.9937 0.0354 206.5286 0.5620 16.3103 -24.6205 -20.6376

TIHLEtW 13.8158 2.3621 37.6306 0.5298 13.9359 -19.8717 -15.8888

TIIEtHLW 4.2059 0.6008 2.7424 6.0126 13.3035 -18.6071 -14.6242

HLGW - 0.2951 6.5128 6888.7174 14.9716 -23.9432 -20.0560

EtW - 1.4919 3.0333 2.3652 13.9497 -21.8993 -18.9121

W - 3.5083 - 14.2303 13.2633 -22.5261 -20.5352

0.2 0.4 0.6 0.8 Data Theoretical probabilities

Figure 3: Fitted cdf, pdf, Q-Q, and P-P plots for data set 1

Table 3: The models' MLEs and performance requirements based on data set 1

Models r e P P ll AIC BIC

EtPW 48.1901 0.0022 2.9823 1.0738 -38.0910 84.1820 89.7868

TIHLEtW 4.1437 0.6170 13.5135 0.5563 -38.4067 84.8135 90.4183

I. , Sule, O. A., Bello, I. A., Kolawole RT&A, No 1 (82)

GENERALIZED EXPONENTIATED-G FAMILY OF DISTRIBUTIONS_Volume 2°, March 2025

TIIEtHLW 0.7691 1.6782 0.4909 1.5904 -38.1060 84.2121 89.8168

HLGW - 0.3416 2.7617 2.7351 -40.1181 86.2362 90.4398

EtW - 2.4241 1.1680 0.8941 -39.8193 85.6386 89.8422

W - 1.8088 - 0.3154 -41.6433 87.2866 90.0891

Figure 4: Fitted cdf, pdf, Q-Q, and P-P plots for data set 2

Tables 2 and 3 outline the results of the mle of the parameters of the EtPW distribution together with the comparator distributions. Based on the goodness of fit statistic AIC and BIC, the new probability model recorded the lowest AIC as well the lowest BIC value suggesting that the EtPW is best fits the two data sets. Figures 3 and 4 also buttress and reaffirm the fit of the EtPW distribution as it follows the pattern and shape of the data.

VIII. CONCLUSION

This research article proposed and studied a new family of distributions called the Exponentiated Pareto-G family of distributions. The family was derived from the exponentiated Pareto distribution using the T-X methodology proposed by [3]. The properties of the new family such as quantile function, probability-weighted moments, moments, generating function, reliability

function, hazard function, and order statistics were examined as statistical components of the newly proposed family of distributions. The parameters of the family are estimated using the method of maximum likelihood technique. Two submodels such as Weibull and Frechet are used to show the shape of the family as baseline distributions. A simulation results to evaluate the new distribution's performance is carried out using Weibull as the baseline distribution. This is to assess the efficiency of the estimation method used. Two real data sets are applied to ascertain the importance and flexibility of the new family of distributions. The results reveal that the new exponentiated Pareto Weibull distribution appears to be superior to the existing models considered. This implies that the new family has added flexibility to the baseline distribution and it can be used to model data in a variety of fields.

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