Weibull Inverse Power Rayleigh Distribution with Applications Related to Distinct Fields of Science Текст научной статьи по специальности «Науки о Земле и смежные экологические науки»

Weibull Inverse Power Rayleigh Distribution with Applications Related to Distinct Fields of Science

Muzamil Jallal1, Aijaz Ahmad2, Rajnee Tripathi3

1,2,3 Department of Mathematics, Bhagwant University, Ajmer Email: muzamiljallal@gmail.com Email: ahmadaijaz4488@gmail.com Email: rajneetripathi8@gmail.com

Abstract

In this paper an extension of Weibull Power Rayleigh Distribution has been introduced, and named it is as Weibull Inverse Power Rayleigh Distribution. This distribution is obtained by adopting T-X family technique. Various Structural properties, Reliability measures and Characteristics have been calculated and discussed. The behaviour of Probability density function, Cumulative distribution function, Survival function, Hazard rate function and mean residual function are illustrated through different graphs. Various parameters are estimated through the technique of MLE. The versatility and flexibility of the new distribution is done by using real life data sets. To evaluate and compare the out effectiveness of estimators, a simulation analysis has also been carried out.

Keywords:- Weibull distribution, Inverse Rayleigh distribution, Renyi entropy, maximum likelihood estimation, Order statistics.

I. INTRODUCTION

Weibull distribution, although being first identified by Frechet [8] and first applied by Rosin and Rammler [9] to describe particle size distribution, was named after Swedish Mathematician Waloddi Weibull, who in 1951 described the distribution in detail in his paper "A Statistical Distribution Function of Wide Applicability". This distribution is now commonly used to assess product reliability, analyze life data and model failure times, see inverse Weibull-G by Amal S. Hassan et al [3]. This distribution has found its importance in the fields of biology, economics, engineering sciences and hydrology.

The pdf of Weibull distribution is given by

f (t;a,3) =aptp"le"at" a,3> 0,t > 0 (^

Where a, 3 and t are Location, Shape and Scale parameters respectively.

Rayleigh being a one-parameter continuous and simplest velocity probability distribution has a diverse range of applications. Various researchers have developed several extensions and modifications of the distribution, resulting in some flexible and more effective distributions by adding some more parameters or by compounding and thus showing its importance in various fields like

Social, Medical and Engineering Sciences. For instance transmuted Rayleigh distribution by Faton Merovci [6], Weibull-Rayleigh distribution by Faton Merovci et al [7], odd generalized exponential Rayleigh distribution by Albert Luguterah [1], Topp-Leone Rayleigh distribution by Fatoki Olayode [5], odd Lindley-Rayleigh distribution by Terna Godfrey Ieren [10], new generalisation of Rayleigh distribution by A.A Bhat et al [4].

The Inverse Rayleigh distribution finds its application in the field of reliability studies. Voda [11] worked out that the lifetime distributions of various types of experimental units can be approximated by Inverse Rayleigh distribution. In this article we use Weibull and Inverse Power Rayleigh Distributions to define a new model which generalizes the Inverse power Rayleigh distribution.

The cdf of Power inverse Rayleigh Distribution is given by

e

G(x,e,l) = e 2x" x > 0,e,l> 0 (2)

e

G (x,e,X) = 1 - e 2x" (3)

And its associated pdf is given by

e

--

g (x,e,X) = e 21" x > 0,e,X> 0

x + (4)

II. Materials and Methods

Transformed-transformer (T-X) family of distributions (Alzaatreh et al [2]) is given by

W [G( x)]

F(x) = J f (t)dt

(5)

Where f(t) is the probability density function of a random variable T and W[G(x)] be a function of cumulative density function of random variable X .

Suppose [G,g] denotes the baseline cumulative distribution function, which depends on parameter vector g. Now using T-X approach, the cumulative distribution function of Weibull Inverse Power Rayleigh distribution can be derived by replacing f(t) in equation(5) by equation (1) and

W [G(x)] = G(xg) ,where G (x, g) = 1 - G(x, g) which follows

G (x,g)

G(x.g)

G (x,g)

1 -at?

e

F(x,a,P,g) = Japtp-1e-^ dt

G( x,g) f

F (x,a,fi,g) = 1 - e The corresponding pdf of (6) becomes

G(xg J (6)

0

0

a

\G( xc)]ß-1 f (x,a,ß,c) = aßg(x,c) _ 'Ç)\ß1 e

\G ( x,c)]ß 1

G(x,c)

G (x,c)

(7)

Where G(x,0,X) = e 2x" is known as Power Inverse Rayleigh Distribution.

G (x,6U) = 1 -G{x,e,X) = 1-e 2x2

(8) (9)

And g(x,9,X) e 21 ,9 > 0,1 > 0

x

III. Linear Transformation

Apply Taylor series expansion to the pdf in equation (7) we have

(10)

G(x,c) T

G(x,c)_ = V (-1)

-r

i=0

G( x, c)

G ( x,c)

On substituting equation (11) in (7), we get the following expression

" G(x, c) "

f ( ) ß ( ) \G(x, c)]ß-1 V (-1)' f ( x, c) =aßg( -ßr L —

\G ( x, c)]'

G ( x, c)

(11)

f ( x, c) =L ^f^ßg ( x, c) \G( x, c)]ß(i+1)-1 \g ( x, c)]-

i=o

\ß(i+1)+1]

(12)

Using generalized binomial expansion, we have

" - =L ( ' + j -1) "

f (x,c) =LL (ß(i + jV^ßg(x,c)\G(x,c)]J'+ß(i+1)-1

1=0j=0 I j J

(13)

Using (8) and (9) in (13) we get

f ( x, c) =

LL i-n iß(i+1)+

i=0 j=0

ex

,2 x 2

,2 x2'

j+ß(i+1)-1

f ( x,c) =LL5i

i=0 j=0

ex

y x2X+1

j+ß(i+1)-1

(14)

(15)

ß

e

e

-a

e

a

i=0

e

e

x

e

e

e

e

Where = +VV

IV. Weibull Inverse Power Rayleigh Distribution

Let X be a random variablefollowing Weibull Inverse Power Rayleigh Distribution with its cdf given

by "

( 0 \-ß

■J r 2 1

-a e2x -1

F ( x;a, ß,0,1) = 1 - e

, x > 0,a, ß,0,1> 0

(16)

And its pdf is given by

0(0 Yß-1

( о Xß

r, a n i\ aß0X 2x2

f( xa,ßß,X) = -2j+r e

e -1

v /

, x > 0,a,ß,0,1 > 0

(17)

V. Reliability measures

The Survival function, Hazard function, Cumulative Hazard function, Reverse Hazard function is given by:

( Vß 2 x2i , -a e 2x -1

)

sx (x) = e V

h = арШ 2 x hx (x) x 21+1 e

о ( о vß-1

e 2x21 - 1

Hx (x) =a

0

e 2 x -1

v /

-ß

aß01

P. 1 ( о VP

0(0 xß-1 T721

e 2 * -1

v /

1-e

2x

e -1

a

e

21

2 x

a

e

e

x

Г.. =

x

p

0

a I e-^—1

21

2x

i .1

! \

\

[ I alpha=1.2,beta=0 6,theta=2 5,lambda=3.7

n aipha-0.3,beta-0 9,thela-2 3Jambda-3.2

□ a!pha-0.4,beta-0 9,theta-2 5Jambda-3.5

□ a!pha-0.6,beta-1.0,theia-2.7:lambda-3.B

□ a!pha=0.02,beta=1 7,theta=2.6Jlambda=3.9

Vi \

v \ S I . \ * \ \

: |. 1 ■li !

li \ ■ 1 '1

: ; !i □ aipha-0.5,beta-0.6,theta-2.5,iambda-3.7

" li \ s alpha-0 2,beta-0.5,theta-2.3,lambda-3.2

alpha-0 2,beta-0.7,theta-2.5,1a mbda-3.5

alpha=0.4,bBla=0.4,theta=2.7,lambda=3.8

p ll □ a[pfia-0.3,beta-0.6,theia-2.6,la[nbda-3.1

■ t '. 11

■A- ' \

\ \ \

w 1 \ \ -

Fig c HRF of WPIRD under different values to parameters

Fig d HRF of WPIRD under different values to parameters

Theorem 1: Show that the Quantile Function of Weibull Inverse Power Rayleigh Distribution is given by

1

12A

e

" 1 "

2log [ -log(1- «)] P +1

)

Proof: Let x be a random variable following Weibull Inverse Power Rayleigh Distribution with parameters a,P,e and X, then we derive its Quantile Function from the corresponding cdf as given below

We know that cdf of WIPRD is given by

-a\ e—jj-1

1 - e ^ 2x

Put 1 - e ^ 2 x

Apply log on both sides

( e 1

2x2X - 1

= log(1 - u)

After solving we get

x(u) =

2 log

- 1

— log(1 - u ) 1 P + 1

a )

(18)

Median:

The median for the new WIPRD can be derived from the quantile function in equation (18) by putting u=0.5 as below

Me = Qx (0.5) =

2 log

-W) 1 p +1

a 2

Theorem 2: If X ~ WIPRD(a,p,e,2) then its rth moment is given by.

E (-X )' = *'=££ (f- (^ + 1 + J ja^'+'P

i=0 j=o

J

r r|1--

el 2X \ 2X

\j + P(i +1)]1-2X

Proof: We know that rth moment about origin is given by

-p

= u

-a

e

e

»

E( X )r = ¡Lir ' = J xr/ ( x;a, Pf,X)dx

0

Using (14) we get the following equation

_!)!iPi ++« + jja<^

»;-] x' LL P ?+j Y"^ ' ^

Vr -

0 i-0 j-0

f -[]+P(i+1)f

-r f „ 2x"

-g

j+P(i+i)-i

dx

i»

' -IE* J-

x 21+1 i-0 j-0 0 x

Where - tf |P(i ++1) + jWOp

ii

i

. - -[j+p(i+i)f

/ur'-LLSy ÛÀJ xr-21-1 e ^ dx i-0 j-0 0

„ -[ j + P(i + 1)f

Put —^—^—- z, we get

2x 21

E( X )-,,-n* if].

i-0 j-0 v 2 y [j + P(i +1)]1-21

r|1 - r

On substituting r=1,2,3,4 we get first four moments about origin.

Theorem 3. Show that the Moment generating function of Weibull Inverse Power Rayleigh Distribution is given by

r r|1 - —

M «) =1 f I r j + J Wf) V^,'

r=0 r i=0 J=0 v J ) v2) \j + p(i + 1)]1-^x

Proof: We know that Moment generating function is given by

»

Mx (/ ) - E (e* ) -J e*/( x;a, Pf,X)dx

0

Mx (/) - E(e* ) - J jl + tx + &2L + — + "j/(x; a, P'f ^

Mx(/) - E(e*)-jL^y-/(x;a,Pf,X)dx

0 r-0 r!

r

GO

-- tr f

c (t) = E (e*) = 1 — J xrf (x; a, p,e, 7)dx

Mx (t) = E(eK ) = 1 — | xr f (x; a, P, e,

r=0 r! 0

After solving the above equation we get,

r rl 1 - r

w r w w / - \ — -1- i 1

^^ V'w e ( e\2A y 27

Mx(t)=¿711% I —-r-r

r=0 r! i=0 j=0 V 2 1 [j + P(i + 1)]1-

Theorem 4. Show that the Characteristic function of Weibull Inverse Power Rayleigh Distribution is given by

r r|1 - r

4 <«)=1 (P<i j1'+j Wf)r^,

r=0 r! i=0 j =0 i! v j J y21 [j + P(i +1)]1-^

Proof: We know that Characteristic function is given by

4x (it) = E(eitr) = J eitrf ( x;a, P,e,7)dx

0

w w (TvY

4 (it) = E(eto) = JX^f (x;a, P,e,7)dx

0 r = 0 r w /. y w

4x (it) = E(eiir) = J xrf (x;a, P,e,7)dx

r = 0 r! 0

After solving the above equation we get

r rf1 - —

4 (t) v(it)r 27 1V1 27

4(it) = 1iTT¿¿A12J --r

r=0 ¡=0 j=0 y/-J [j + P(i +1)]1

VI. Incomplete Moments

We know that

T. (S) = J x f (x; a, P, e, 7)dx

Put (14) in the above equation, we get

-|j+p<i+l)-l

s w ^ .A e7 ~e

T. (S) = J x-1 S^f (P + 'j + j YOP^

0 i=0 j=0 '! v j J x

-e

dx

s

0

GO G ^

(S) =II

i=0 j=0 0 -

-(j+p(i+1))f

2x2'

dx

Tq(S) = II 2Sj

i=0 j=0

(J+P(i+1))f

2

s-2X

r| 1--, M

2X

Where M =

(j+P(' + 1)f

2 x

2X

VII. Renyi Entropy

If X is a continuous random variable following WIPRD with pdf f (x;a, P,f,X), then

Tr (p) = y^log-jf / P( x,c)dxj

tr (p;>=-—log 1 - p

aPg (feF

G( x,c)

lP]P

G (x,g) J

dx

Tr (p) = t:1" log|I I "j f \g( x, C)]P \G( x, ?)]•

1 P ' i=0 J=0 0

X,C)\ |G(x,c)p'+(p+i)P-pdxi

(19)

Where UJ =i=il (pa)' (ap)P(P(P +-) + P + J -1

Now put (10) and (8) in equation (19) we get

_ I GG GG G

Tr (P) = -U>g \IIoj (ex)Pf

1 P 1 i=0 j=0 0

-\j+(p+Qp]e

x-p(2X+1)e 2 x- dxi

Put \J + (P+ = z ^ dx =

2 x

2X

2X+1

\j + (p + ')p]e 1 2z J \j + (p + ')p]ex

dz

tr (P) = "-log

1 -P

i=0 J=0

p-1

\j + (p + op]

j + (p + ope

2

-p(2X+1)+2X+1

2X i p(2X + 1) - 1

2X

VIII. Tsallis Entropy of WIPRD

We know that Tsallis Entropy is defined as

s

x

e

G f

0

S—=—1 j1 -J f P( x,g)dx[

S— =-7

P- 1

1 -

( r G( ) 1P^P

aPg (x,) G^Gg aPg([G(x,,)]P+1 6

dx

S—=—1 i1 -

¿¿u, J[g (x,,)—G( x,g)]

x, C) p'+(p+i)P-Pdx

i=0 j=0 0

Now put (10) and (8) in equation (20) we get

Sp= —

P-1

1-

ww

¿¿U, J

i=0 j =0 0

ei

,2 x2'

P ( -e ) J+(P+i)P-P

2x2

V /

dx

After solving the above equation we get

P-1

1 -

w w ¿¿«

i=0 j=0

(i—1

y [j + (P + i)P]

j+(p+i)Pe 2

-p(27+1)+2l+1

27 r( p(27 + 1) - 1 27

IX. Mean Deviation from Mean

We know that

D(j) = E(| x -j\)

w

D(M) = J1 x -M|f (x)dx

0

M

D(ju) = 2jF(m) - 2 J xf (x)dr

Now we have J xf (x)dx = J x^ |P(i + 1) + jla(i+1)P" 0 0 i=0 j=0 i! ( j J ;

Mxfi „(IJ-rft(1 -

i=0 j = 0

7 .21+1

j+P(i+1)-1

dx

(20)

(21)

Substituting (22) in (21) we get

X)

GO

-e

x

0

e

e

e

e

D(p) =

1 - e

- 251 , (I)<1 - j; 1)1

21

X. Mean Deviation from Median

We know that

D(M) = E(| x - M |)

D(M) = J | x - M | f (x)dx

M

D(M) =>u- 2 Jxf (x)dx

(23)

Now we have

Jxf(x)dx = J .,55 Mß -1) - jy^

0 0 ¿=0 j=0 ( j J x

,2 x2

3 2 x 2

j ß +!)-!

dx

1

J xf (x)dx = 55,, (I )-JI(1 - 2-

i=0 j=0

1 ^ (j + ß('" + 1))I

2-J 2M

2-

(24)

Substituting (24) in (23) we get

D(M) =,-2:5:5 *(I)^(1 -j-1

XI. Maximum Likelihood Estimation

Let x-, x2 , x3 ,•••, x^ be n random samples from Weibull Inverse Power Rayleigh Distribution, and then its likelihood function is given by

l = ^ f (x;a,ß,|-)

i=1

^^^x-2-r e Vß-1 (-I- Vß

l = (aßd—)n n x-(2-+1)e 2 "

i=1

,2 x 2 x - 1

Its Log Likelihood function is given by

ß

0

e

e

M

0

e 2 -1

a

e

n £> n

ZC^ _2X

log x, H— Z x,

n ( 0 r n ( 0 \

(P + 1)Z log i =1 e2x2X _1 V y _aZ i =1 e2x2X _1 V y

Differentiating the above w.r.t a, P, 0 and 1, we will get the following equations

n f 0 \~P

d log l _ n

da a

d log l _ n

Z e 2x2X _ 1

,=1 V y

n ( 0 n ( 0 r _P ( 0 \

Z log e 2 x2X _ 1 +aZ e2x2X _ 1 log e2x2X _ 1

,=1 V y i =1 V y V y

3 l0g 1 = 0 H 2 Z x _2X_ (P + 1)Z

1 1

30 0 2

¿=1

¿=1

( 0 \ 2x¿2X

2 x,

e ' +

e2x" _1

v y

apZ

1 2 x 2X

0(0 VP_

i=1 2xi

e2x" _1

v /

3 log 1 n

9X X

n n n -, -

= XX - 2£ log x, _ 0Z x,_2X log x, + (P + 1)0Z T~6-2x'" X, _2X log x

¿=1

,=1

,=1

0

e 2?X_ 1

v

0(0 vP_

_aP0Z

2 x,

ei

0

e 1

v

_2X , x, logx,

The above equations are non-linear equations which cannot be expressed in compact form and it is difficult to solve these equations explicitly for a, P, 0 and X . By applying the iterative methods such as Newton-Raphson method, secant method, Regula-Falsi method etc. the MLE of the parameters denoted as rj (a, P,0,X) of rj{a, can be obtained by using the above methods.

Since the MLE of rj follows asymptotically normal distribution as given as follows

4ñ (r _

r)^ N (0,1 (r))

Where 1 1 (r) is the limiting variance covariance matrix rj and 1 (r) is a 4x4 Fisher Information matrix

,=1

,=1

0

e

0

i =1

I })= —1

E

E

E

E

( 5 2 log / ] da2

( 5 2 log / ] dftda

( 52 log / ] 5O5a

(52 log / ]

did a

E

E

E

( 5 2 log / ] dadft

( 52 log / ] 5p2

( 52 log / ]

E

( 52 log / ]

dade

E

( 5 2 log / ] 5a5X

E

E

(5 2 log / ] 5p5O

( 52 log / ]

5e2

E

E

(52 log/] 5P5X

( 52 log / ] 5O5X

(*2, ,] (a2,„„ ,]

E

5 2 log /

5X5p

E

52 log /

5X5O

E

(52 log/] 5X2

Hence the approximate 100(l-y)% confidence interval for a,p,O and X are respectively given by

a ±

± ZyJ a (v) p± z^ i pP (}) e± z^fifff}} x± zj ixX (})

Where Zw is the iyth percentile of the standard distribution.

XII. Order Statistics

Let x(T),x(2),x(3),...,x(n) denotes the order statistics of n random samples drawn from Weibull Inverse Power Rayleigh Distribution, then the pdf of X (k) is given by

fx(k)(xf) =

n!

(k - 1)!(n - k)

fx (x)\Fx (x)]k-1\l - Fx (x)J

n-k

e ( e ]-p-1

( e ]-p

fx(k)(xf) =

n!

apex 2x2

-e

(k - 1)!(n - k)! x2

2 x2

e -1

-a\ e——-1

1 - e V 2x

k-1 ,

-p^

n-k

-a\ e 2^-1J

Then the pdf of first order X (T) Weibull Inverse Power Rayleigh Distribution is given by

fx(1)( x,fl) =

napOX

e ( e ]—p—1

e -1

-a\ e2x2X-1

A n

and the pdf of nth order X (n) Weibull Inverse Power Rayleigh Distribution is given by

fx(n)(x;e):

napex

( o Vp—1

1 - e

-a\ e—-¡—1

p

2x

e -1

a

e

e

p

2x

e -1

2x

oc

e

x

p

n—1

p

e

2x

e —1

2x

2x

a

1

e

e

XIII. Simulation Study

In this section, we study the performance of ML estimators for different sample sizes (n=, 50,150, 250,500). We have employed the inverse CDF technique for data simulation for WIPRD distribution using R software. The process was repeated 1000 times for calculation of bias, variance and MSE.

Table 1: The Mean values, Average bias and MSEs of 1,000 simulations of WIPRD for parameter values

Sample Size n parameters a = 1.0 P- = 0.9 e = 2.3 and X = 0.3

Average Bias Variance MSE

50 a 0.9493 -0.050 0.0001 0.0027

P 0.7717 -0.128 0.0001 0.0166

e 2.0815 -0.218 0.0246 0.0723

X 2.0510 1.7510 0.0768 3.1429

150 a 0.9537 -0.046 8.41e-05 0.0022

P 0.7760 -0.123 8.11e-05 0.0154

e 2.0854 -0.214 7.07e-03 0.0531

X 1.9675 1.6675 2.55e-02 2.8063

250 a 0.9545 -0.045 6.23e-05 0.0021

P 0.7767 -0.123 5.81e-05 0.0152

e 2.0866 -0.213 4.24e-05 0.0497

X 1.9479 1.6479 1.56e-05 2.7313

500 a 0.9556 -0.044 3.43e-05 0.0020

P 0.7778 -0.122 3.11e-05 0.0149

e 2.0905 -0.209 2.15e-05 0.0460

X 1.9261 1.6261 8.36e-05 2.6528

a = 1.1 P = 0.8 e = 2.0 and X = 0.5

50 a 0.9465 -0.153 0.0001 0.0236

P 0.7688 -0.031 0.0002 0.0011

e 1.7951 -0.204 0.0216 0.0635

X 1.1839 0.6839 0.0248 0.4926

150 a 0.9496 -0.150 6.99e-o5 0.0226

P 0.7716 -0.028 7.04e-05 0.0008

e 1.8038 -0.196 6.59e-03 0.0450

X 1.1476 0.6476 8.11e-03 0.4275

250 a 0.9508 -0.149 5.19e-05 0.0223

P 0.7728 -0.027 5.04e-05 0.0007

e 1.8116 -0.188 4.33e-03 0.0398

X 1.1355 0.6355 4.83e-03 0.4087

500 a 0.9513 -0.148 2.83e-05 0.0221

P 0.7733 -0.026 2.69e-05 0.0007

e 1.8128 -0.187 2.08e-03 0.0371

X 1.1277 0.6277 2.59e-03 0.3967

As is clear from table 1, decreasing trend is being observed in average bias, variance and MSE as we increase the sample size. Hence, the performance of ML estimators is quite well, consistent in case of Weibull Inverse Power Rayleigh Distribution.

XIV. Application

In this segment, the efficacy of the developed distribution has been assessed using two realistic sets of data. As the new distribution is compared to New Modified Weibull distribution (NMWD), Additive Weibull distribution (AWD), Power Gompertz distribution (PGD), Inverse power Rayleigh distribution (IPRD), Weibull distribution (WD), Lindley distribution (LD) and Hamza distribution (HD). It is revealed that the new developed distribution offers an appropriate fit.

Various criterion including the AIC (Akaike information criterion), CAIC (Consistent Akaike information criterion), BIC (Bayesian information criterion), HQIC (Hannan-Quinn information criteria) and KS (Kolmogorov-Smirnov) are used to compare the fitted models. The p-value of each model is also recorded. A distribution having lesser AIC, CAIC, BIC, HQIC and KS values and with large p-value is considered better one.

2k

AIC = 2k- 2ln / CAIC =--2ln /

n-k-1

BIC = k ln n — 2 ln / HQIC = 2k ln(ln(n)) - 2ln /

Data set 1:- The following represents the dataset of 63 Observations of the tensile strength measurements on 1000 carbon fiber-impregnated tows at four different gauge lengths. The data reported by Bader and Priest (1982) as follows:

1.901, 2.132, 2.203, 2.228, 2.257,2.350, 2.361, 2.396, 2.397, 2.445, 2.454, 2.474, 2.518, 2.522, 2.525, 2.532, 2.575, 2.614, 2.616, 2.618, 2.624, 2.659, 2.675, 2.738, 2.740, 2.856, 2.917, 2.928, 2.937, 2.937, 2.977, 2.996, 3.030, 3.125, 3.139, 3.145, 3.220, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346, 3.377, 3.408, 3.435, 3.493, 3.501, 3.537, 3.554, 3.562, 3.628, 3.852, 3.871, 3.886 ,3.971, 4.024, 4.027, 4.225, 4.395, 5.020.

Data set 2: The second data represents COVID-19 mortality rates data belongs to Italy of 59 days that is recorded from 27 February to 27 April 2020. The data is as follows:

4.571, 7.201, 3.606, 8.479, 11.410, 8.961, 10.919, 10.908, 6.503, 18.474, 11.010 ,17.337, 16.561, 13.226, 15.137, 8.697, 15.787,13.333, 11.822, 14.242, 11.273, 14.330, 16.046, 11.950, 10.282, 11.775, 10.138, 9.037, 12.396, 10.644, 8.646, 8.905, 8.906, 7.407, 7.445 ,7.214, 6.194, 4.640, 5.452, 5.073 ,4.416 ,4.859 ,4.408, 4.639, 3.148, 4.040, 4.253, 4.011, 3.564, 3.827, 3.134, 2.780, 2.881, 3.341, 2.686, 2.814, 2.508, 2.450, 1.518.

The fitted models are compared using empirical goodness of fit measures such as the AIC (Akaike information criterion), CAIC (Consistent Akaike information criterion), BIC (Bayesian information criterion), HQIC (Hannan-Quinn information criteria), and KS (Kolmogorov- Smirnov). Each model's p-value is also displayed. A distribution with a lower AIC, CAIC, BIC, and HQIC together with a higher p value is rated as the top distribution

Table 2 and 5 shows the descriptive statistics for data set 1 and data set 2 respectively. Table 3 and 6 displays the parameter estimates for the data set 1 and data set 2 respectively. Table 4 and 7 displays

the log-likelihood, Akaike information criteria (AIC), BIC (Bayesian information criterion) etc. details and some other statistics for the data set 1 and data set 2.

Table 2. Descriptive Statistics for data set 1

Mean Min. Max. Qi Q3 Median S.D Skew. Kurt. 3.059 1.901 5.020 2.554 3.421 2.996 0.620 0.632 3.286

Table 3. The ML Estimates and standard error of the unknown parameters

Model WIPRD NMWD AWD

â 11.4370 0.01469 0.00292

ß 0.7586 0.00100 0.00100

e 60.8587 3.14576 4.68182

i 0.91032 0.43441 4.54990

Stan- â 5.880 0.0063 0.00081

dard ß 0.5707 0.00034 0.00058

Error e 53.0940 0.29681 0.21350

i 0.0810 0.26257 0.26577

PGD

IPRD

WD

LD

HD

0.00105 0.22866

0.00438 ------------------0.003775 ---------------------------

2.72725 ------------------4.69909 ------------------0.00100

0.71363 15.1809 ----------------------0.53923 2.28724

0.91031 ----------- ---------------------------

--------- 0.62231

0.04958 0.05084

0.08109

Table 4. Performance of distributions

Model WIPRD NMWD AWD PGD IPRD WD LD HD

-2logl 111.612 138.816 124.704 134.466 185.528 124.546 242.714 150.514

AIC 119.612 146.816 132.705 140.466 189.528 128.546 244.715 154.515

AICC 120.301 147.505 133.395 146.896 193.814 128.746 244.780 154.715

HQIC 122.983 150.187 136.077 140.873 189.728 130.232 245.558 156.201

BIC 128.184 155.388 141.278 142.995 191.214 132.832 246.858 158.802

K-S Value 0.08215 0.1448 0.11151 0 .143 0.35439 0.11139 0.4308 0.21545

P Value 0.7888 0.1424 0.4137 0.152 2.68e-07 0.4151 1.39e-10 0.00576

Table 5. Descriptive Statistics for data set 2

Mean Min. Max. Q1 Q3 Median S.D Skew. Kurt.

8.156 1.518 18.474 4.146 11.341 7.445 4.5267 0.4523 2.1281

Table 6. The ML Estimates and standard error of the unknown parameters

Model

WIPRD NMWD AWD

PGD

IPRD

WD

LD

HD

a ß e i

Standard Error

a ß e i

0.2065 0.82124

11.26308

0.51936

0.16217 0.12048

7.0582 0.04781

0.02612 0.00100

1.434910 0.293231 0.01111

0.02612 0.00100

1.43491 0.29323 0.01111

0.013344 0.01334 0.15826 0.15826 0.23093 0.230931

0.01746 0.00387

1.75839

0.01006 0.00864

0.31366

------------------0.014105 ---------

------------------1.918049 ---------

22.8479 ----------------------0.222869

0.51936 -------------------------------

------------------0.006603 ---------

------------------0.18601 ---------

3.6430 ----------------------0.02068

0.04781 -------------------------------

21.5900 0.81384

28.7503 0.05698

Table 7. Performance of distributions

Model WIPRD NMWD AWD PGD IPRD WD LD HD

-2logl 267.932 336.232 336.232 334.882 289.302 335.404 346.598 362.508

AIC 275.932 344.232 344.232 340.882 293.302 339.404 348.599 366.508

AICC 276.673 344.973 344.973 341.319 293.516 339.618 348.669 366.722

HQIC 279.176 347.476 347.476 343.315 294.924 341.026 349.410 368.129

BIC 284.242 352.543 352.543 347.115 297.457 343.559 350.677 370.663

K-S Value 0.10235 0.12345 0.21187 0.12154 0.22795 0.121 0.14516 0.21409

P Value 0.5331 0.3041 0.00839 0.3216 0.00354 0.3268 0.1506 0.00748

As it is obvious from table 4 and table 7 that the Weibull inverse power Rayleigh distribution has smaller values for AIC, AICC, BIC, HQIC and K-S statistics as compared with its sub models. Accordingly we arrive at the conclusion that Weibull inverse power Rayleigh distribution provides an adequate fit than the compared ones.

Figure e, f, g and h represents the estimated densities and cdfs of the fitted distributions to data set 1st and 2nd.

XV. Conclusion

This newly introduced distribution "Weibull inverse power Rayleigh distribution" which is obtained by T-X method. Several mathematical quantities for the newly developed distribution are derived including moments, moment generating function, incomplete moments, order statistics, different measure of entropies etc. To show the behavior of p.d.f, c.d.f and other related measures different plots have been drawn. The parameters are obtained by MLE technique. Lastly by carrying out through two real life data sets to show that the formulated distribution leads an improved fit than the compared ones.

References

[1] Albert L (2016).Odd Generalized Exponential Rayleigh Distribution. Advances and Applications in Statistics, Vol 48(1) , 33-48.

[2] Alzaatreh A, Lee C, Famoye F (2013).A New Method for Generating Families of Distributions. Metron, 71, 63-79.

[3] Amal H and Said G.N (2018).The Inverse Weibull-G Family. Journal of Data Science, 723-742.

[4] Bhat A.A and Ahmad S.P (2020).A New Generalization of Rayleigh Distribution: Properties and Applications. Pakistan Journal of Statistics, 36(3), 225-250.

[5] Fatoki O (2019).The Topp-Leone Rayleigh Distribution with Application. American Journal of Mathematics and Statistics, 9(6), 215-220.

[6] Faton M (2013).Transmuted Rayleigh Distribution. Austrian Journal of Statistics, 42(1), 21-31.

[7] Faton M and Ibrahim E (2015).Weibull-Rayleigh Distribution Theory and Applications. Applied Mathematics and Information Sciences, an International Journal, Vol 9(5), 1-11.

[8] Frechet M (1927).Sur La Loi De Probabilite De Lecart Maximum. Ann. Soc. Polon. De Math. Cracovie. 6:93-116.

[9] Rosin, P; Rammler, E (1933)."The laws Governing the Fineness of Powdered Coal". Journal of the Institute of Fuel, 7:29-36.

[10] Terna G.I, Sauta A, Issa A.A (2020).Odd Lindley-Rayleigh Distribution, Its Properties and Applications to Simulated and Real Life Datasets. Journal of Advances in Mathematics and Computer Science, 35(1), 63-88.

[11] Voda, R (1972). "On the Inverse Rayleigh Variable", Rep. Stat. Res. Ju, Vol. 19(4), 15-21.