Inverse Weibull-Rayleigh Distribution Characterisation with Applications Related Cancer Data Текст научной статьи по специальности «Науки о Земле и смежные экологические науки»

Inverse Weibull-Rayleigh Distribution Characterisation with Applications Related Cancer Data

Aijaz Ahmad 1, S. Qurat ul Ain2, Rajnee Tripathi3 and Afaq Ahmad4

1,2,3 Department of Mathematics, Bhagwant University, Ajmer, Rajasthan, India "Department of Mathematical Sciences, IUST, Awantipora, Kashmir E-mail: 1ahmadaijaz4488@gmail.com E-mail: 2andrabiqurat19@gmail.com E-mail: 3rajneetripathi@hotmail.com E-mail: 4baderaafaq@gmail.com

Abstract

The current study establishes a new three parameter Rayleigh distribution that is based on the inverse Weibull-G family and is an extension of the Rayleigh distribution. The formulation is known as the inverse Weibull-Rayleigh distribution (IWRD). The distinct structural properties of the formulated distribution including moments, moment generating function, order statistics, quantile function, and Renyi entropy have been discussed. In addition expressions for survival function, hazard rate function and reverse hazard rate function are obtained explicitly. The behaviour of probability density function (p.d.f) and cumulative distribution function (c.d.f) are illustrated through different graphs. The estimation of the formulated distribution parameters are performed by maximum likelihood estimation method. A simulation analysis has been carried out to evaluate and compare the effectiveness of estimators in terms of their bias, variance and mean square error (MSE). Eventually, the usefulness of the formulated distribution is illustrated by means of real data sets which are related distinct areas of science.

Keywords: Inverse Weibull-G family, Rayleigh distribution, moments, Renyi entropy, simulation, maximum likelihood estimation.

Mathematics classification: 60E05, 62FXX, 62F10, 62G05

I. Introduction

There is a plethora of univariate distributions in the statistics literature. However, statisticians have found it difficult to find an effective distribution for analysing or modelling complicated real-life data sets. To resolve such challenges, new probability distributions must be formed or fundamental type must be modified. Over recent times, researchers have investigated a plethora of new methods and approaches, and by employing these approaches, generalization or extensions can be accomplished from baseline distributions. The main objective for these modifications is to enhance the accuracy or flexibility of distributions while assessing more complicated real-life data sets.

Waloddi Weibull, a Swedish mathematician, introduced the Weibull distribution in 1951. Because it may be used to analyse real life data with monotone failure rates, this distribution is considered versatile for data sets with bathtub shapes or unimodal. The Weibull distribution, on the other hand,

may not necessarily give a best fit for data sets with a bathtub shape or failure rates that are unimodal.

Let X be a random variable follows the Weibull distribution with parameter b and d. Then its probability density function (pdf) is defined as

y(x,P,d)= (36bxbAe-eqx" ; x > 0,p,6> 0

The inverse of the Weibull distribution is obtained by applying the transformation T = — .

X

Thus the probability density function (pdf) of inverse Weibull distribution takes following form.

h(t,b,e)= pebt' ;t > 0,p,d> 0 (l)

The inverse Weibull distribution is a subclass of the generalised extreme value distribution, which was previously researched by B.V. Gnedenko (1941) and Frechet (1927). In this paper, we develop the inverse Weibull-Rayleigh distribution, which is an extension of the Rayleigh distribution. Rayleigh distributions have a broad array of applications in research to simulate real life data, including reliability analysis, engineering, communication theory, medical science, and applied statistics. Rayleigh distribution has been expanded by researchers to make it more comprehensive and efficient for assessing more diverse factual data, for instance, due to its immense variety of applications. Weibull-Rayleigh distribution by Faton Merovci [11], odd generalized exponential Rayleigh distribution by Albert Luguterah [2], Topp-Leone Rayleigh distribution by Fatoki olayode [12], new generalisation of Rayleigh distribution by A.A Bhat et al [8].The probability density function (pdf) of Rayleigh distribution with scale parameter a is defined by

a 2

g(y,a) = aye 2y ; y > 0,a > 0 (2)

The associated cumulative distribution function (cdf) is given by

a 2

G(y,a) = 1 -;y > 0,a > 0 (3)

In recent past years researcher have focussed to explore new generators from continuous standard distributions. As a result, the obtained distribution enhances the effectiveness and flexibility of data modelling. Some generated families of distribution are as follows: beta-G family of distribution explored by Eugene et al [10], kumaraswamy-G family by Cordeiro et al [9], transformed-transformer(T-X) by Alzaatrh et al [1], Weibull-G by Bourguignon et al [5], Lindley-G by Frank Gomes-silva et al [12], Topp-Leone odd log-logistic family of distributions by Brito et al [7], inverse Weibull-G by Amal S. Hassan et al [3], among others.

II. T-X Transformation

T-X family of distributions defined by Alzaatreh et al [1] is given by

W [G (y)]

F(y)= j r(t )dt (4)

0

Where r(t) be the probability density function of a random variable T and w[<j(y)] be a function of cumulative density function of random variable Y.

Suppose G(y,h) denotes the baseline cumulative distribution function, which depends on parameter vector h. Now using T-X approach, the cumulative distribution function F (y) of inverse Weibull

G( )

generator (IWG) can be derived by replacing r (t) in equation (4) with (l) and W[G(y)j = — , ' I, where

G (y, h)

G (y,h) = 1 - G(y, h) which follows

G( У,h)

G ( y,h)

F (y,ß,0,h)= jßeßrß-1e

ß-i-eßt-ß

= e

dt

G ^ 0 ; y > 0, ß,e> 0

G( yh-

(5)

The corresponding pdf of (5) becomes

f (y, ßM=ßeßg

-в»

G(yh) G ( yn)

[G (yh)V+l

The survival S (y) and hazard rate function h(y) are respectively given by

; y > 0, ß,d> 0

(6)

-8ß\ G(-

S(y) = 1-F(y,ß,e,h) = 1 -/

G( y,h-ß

G\y1njß+l

ßqßg (yh 1 ^ " ■

h(y )=-

-eß

1 - e

G(У h G ( y,n.

III. Useful Expansion Applying Taylor series expansion to the exponential function of the pdf in equation (6) we have

G yh-

G (y,n

Y (- \)elß

Y i!

Gyh

G (y,h).

ß

(7)

Substitute equation (7) in (6), we have

f(v ß e v)- ßs(v v)f (-(G(yh)Yß(MH

Since ß > 0 and |z| < 1, using generalised binomial theorem, we have

(8)

(1 - z Г = X (-1) '

j=o

J

p

p

e

(G (y,h)TMyi = (1 - G{y,hW^ - jj (-l)J [b b +1 ~%(y,h)y (9)

i-n V J 0

J-n

Using equation (9) in equation (8), we have

f (y,M,n) = ±± y,h{b b +1}" yh)

¥ ¥

=YLsu (yhMyhy-p(i+iyi (10)

i=0 J=0

Where Sj >'>"' f"" b+'»-1

The paper is framed as. In section 2, we derive the cumulative distribution function (cdf), probability density function (pdf). In section 3, we study the reliability measures, survival function, hazard rate function and reverse hazard rate function. In section 4, different statistical properties are studied including, moments, moment generating function, quantile function and random number generation. In section 5, Renyi entropy is discussed. In section 6, order statistics is expressed, in section 7, the estimation of parameters are performed by maximum likelihood estimation. In section 8, simulation study is performed. Finally in section 9 the efficiency of the established distribution is examined through data sets.

IV. The Inverse Weibull-Rayleigh Distribution

In this section we explore the inverse Weibull-Rayleigh distribution and studied its different statistical properties. Using equation (3) in equation (5), we obtain the cumulative distribution function (cdf) of the proposed distribution which follows

a 2 TP

a -l|

F(ya,p,e) = e ^ 0 ,y > 0a,p,e> 0 (11)

Figures (1.1) and (1.2) illustrates some of possible shapes of the cdf of IWRD for different values a,b andd

The associated probability density function of inverse Weibull-Rayleigh distribution is given by

(a 2 ö-ß

ö-ß-1 -Ae? -l|

f (y,a,ß,9) = aßOßye 2

a 2

y

a 2

e 2У -1

, y > 0,a, ß,d> 0

(12)

Figures (2.1) and (2.2) illustrates some of possible shapes of the pdf of IWRD for different values a,b andq

V. Reliability Measures

Suppose Y be a continuous random variable with cdf F(y), y ^ 0. Then its reliability function which is also called survival function is defined as

S(y) = pr (7 > y) = J f (yd = 1 - F(y)

The survival function of inverse Weibull-Rayleigh distribution is given as S (y,a,ß,0) = 1-F (y,a,ß,0)

= 1 - e

a 2

ea -1

(13)

e

GO

b

Figures (3.1) and (3.2) illustrates some of possible shapes of the survival function of IWRD for different values a, ¡3 and 6

The hazard rate function of inverse Weibull-Rayleigh distribution is given as

h (ß,e)= f\ya ßß\

Substituting equations (l2)

and (13) in equation (l4), we have

(14)

a

■ßepye

а 2

H ( y,a,p,é)=-

œ а 2 ö-ß-1 -eß

e 2 y -1 è 0

-ß

1-e

а 2

e~ay -1

-ß

e ~2 У -1

2

Figures (4.1) and (4.2) illustrates some of possible shapes of the hazard rate function of IWRD for different values a, b and 0

Reverse hazard rate function of inverse Weibull-Rayleigh distribution is given as

v ; f(ya, ßq)

= aß6ßy<

—y

Yß-i

^ -1

Figures (5.1) and (5.2) illustrates some of possible shapes of the reverse hazard rate function of IWRD for different values a,b andd

VI. Structural properties of inverse Weibull-Rayleigh distribution

Theorem 4.1:- Suppose y denotes a random variable follows IWRD with p.d.f f (y,a, ß,0). Then the rth moment of inverse Weibull-Rayleigh distribution is given by

i,J =0k=0 \K 0 a(k +1)]2

r

22arf- +1

j-b(i+i)-iY "42

Proof:- Let Y denotes a random variable follows inverse Weibull-Rayleigh distribution. Then r f

moment denoted by is given as

¥

Mr' = E (Г )= J yrf (y,a, ß,0)fy

0

Substituting equations (1) and (2) in equation (10), we get

th

¥ ¥ w =zzvj

Hr =

a 2

yr +1e ^ У

r a 2 ö J-ßß+1)-1

1 - e~ 2 У

i=0 J=0 0

dy

We know the formulae of generalized binomial expansion, which follows

œ p-1)

(1 - = X (-1У

j=0

J

Now applying the above formulae, we get

¥ ¥ ¥ a 2 ¥

' -—y

=Z&.ai

Mr =

r+1 2 }

> О £

2=0 j=0

Ve 2 X (- У

k=0

j -b(i +1) -1

ka

e 2 dy

¥ ¥ ¥

^=HI (- 1R

i=0 j=0 k=0

a

œ j-ß(i+1)-i

<k+l)V

^ dy

Making substitution a(k +1)y2 = z so that ydy = —г 2 a(k +1)

dz

¥ ¥ (j-ß(i +1)-10 22 ¥ I-+H-1

^ = Ц(-

i, j=0k=0

a

¥ i Г --+

2 0 „-z.

zè 2 0 e Zdz

[a(k +1)]2+1 о

Vr =

' = XX (- tfd., i j+" - 4

¿-¡¿-G ' ''J k Г / \тГ+1

i ,j=0к=0 \K 0 [a(k +1)]2+1

Theorem 4.2:- Suppose y denotes a random variable follows IWRD with pdf f (y,a,ß,d). Then the moment generating function of inverse Weibull-Rayleigh distribution is given by

¥ ¥ ¥ /. \r

r=0 i, j=0k=0

My (t ) = Ш ^ И

œ j-bit+1)-i

22arlj +1

[a(k +l)}

I-+1 2

2

k

¥

k

r

k

r

r

k

Proof:- Let Y be a random variable follows inverse Weibull-Rayleigh distribution. Then the moment generating function of the distribution denoted by MY (t) is given

¥

MY (t) = E(ety )= Jetyf (y, a, ß,6)dy

0

Using Taylor's series

M+M '

2! 3!

= jj 1+ty + ^ + ^ +...

0 V

f (y,a, p,d)dy

¥ r

X j yf (ya, b,eyiy

r

r=0 0

¥ tr / \ I '-A')

r=0

M,

c )=m s « (k-b(i+,)-■)

r=0 i, j=0k=0 V 0

22aT\ - +1

[a(k +1)]2

+i

The characteristics function of the IWRD denoted as fY (t) can be obtained by replacing t = it, i = is given by

WWW/. \r

f (' )=HI ^ (-u

r=0 i, j=0k=0

f j -b(i+1) -1

22arlj +1

[a(k +1)]2+1

VII. Quantile function of inverse Weibull-Rayleigh distribution

The quantile function of random variable Y, where Y ~ can be obtained by inverting

equation (ll), we have

Q(u )= F ~1(u ) =

— log

a

—j log u 1b +1

In particular, the median of the distribution can be obtained by setting u = 0.5

M =

— loga

>g(Mr +1

VIII. Random number generation of inverse Weibull-Rayleigh distribution

Suppose y denotes a random variable with cdf given in equation (ll). The random number of inverse Weibull-Rayleigh distribution can be generated as

F(y) = u ^ y = F_1(u)

So that

CO

r

T

1

2

1

2

У =

—log

a

q-tog U Y + 1

Where u is the uniform random variable defined in an open interval (0,l).

IX. Renyi entropy of inverse Weibull-Rayleigh distribution

If Y is a continuous random variable having probability density function f (y,a,ß,Q). Then Renyi entropy is defined as

Tr

(^p-logH/г(у)ф1, where g > 0 and g^ 1

Using equation (6), we have

TR W = T—log"

1 -g

» htm*

dy

1 -g

log

fa* (g MY ll^l

(ß+i )g -ß+i)r

G y h)

G ( y,h)

dy

Now using the power series expansion for exponential function, we have

G( yh)1 b _G ( yh) _

Substituting equation (16) into (17), we obtain

= у (-\)geßi у i\

i=0

G(yh)1 b

_G (yh)_

(g) = (- 1j ßg!q ' j(g(yh)Y(G(yh))-{ß+l]g-ß(1 -G(yh)f--1g+ßdy\

(15)

(16)

.. I ¥ ¥

'g I '=0 j=0 i!

(-1)'+J ßrg1eß[r+i ) Çb(g+i ) -g j

J (g (yh))r(G(yh)Y-ß(r+i

Thus, the Renyi entropy for inverse Weibull-Rayleigh distribution, is given by

, + , , 2 ( a , öJ-b,r+

v ']-rW У

ч У

1

T-r

log

ss

i=0 j=0

0

1 - e 2

dy

\ /

j - ß(g + i) - g^jyre~a(r+k)у i, j=0k=0 V k 0 0

.. I ¥ ¥

jyre 2 " dy)

2

¥

7

¥

1-ß

1

e

e

R

Where

d =

(-1)' 1 ßggeß{g+l ) (ß(g +i ) -g

Making the substitution — (g + k )y2 = z, we have

1 -g

log

II (-i)agd

i, j=0 к=0

œ j -ß(r+i ) -g

i, j

1 I 2 ö 2

g+1

2 èa(g + к )

f

I z 2 e zdz

After solving above integral, we obtain

tr м = т"^1о&

1 -g

i, j=0k=0

g+1

(-1 a,'*-ß(r+ihrg 1 2 12 1

2 {a(g+ k)

X. Order statistics of inverse Weibull-Rayleigh distribution

Let us suppose Y1,Y2,...,Yn be random samples of size n from IWRD distribution with pdf f (y) and cdf F (y). Then the probability density function of kth order statistics is given as

4 )(y)=

n!

(k-l)(n - k )!

f (yp (y )]k-1[l-F (y)]

\n-k

(17)

Now using the equation (ll) and (12) in (17). The probability of kth order statistics of inverse Weibull-Rayleigh distribution is given as

h )(y )=

n!

(k - l)(n - k )

,2 У

\-ß-1 -efi\e2

—y

e 2 -1

« 2

—y

V /

eb e2 -1

k-1

1 - e

в" e^ -1

Then, the pdf of first order statistics Yx inverse Weibull-Rayleigh distribution is given as

fijy)=

napq ye

a 2 ~2 У

f a , \-ß-1

a 2

ay -1

^y2

e2 -1

V /

1-e

q\ e~2y -1

Then, the pdf of nth order statistics Yn inverse Weibull-Rayleigh distribution is given as

1

к

b

e

e

n—1

ß

1

n

ß

ß

a. 2

e

f%)(y) = naßebye 2

a 2 У

f a 2 \-ß-1

a2

a -1

^y2

e2 -1

V /

qß\ e>' -1

-ß

-1

XI. Maximum likelihood estimation and Fisher's information matrix of inverse

Weibull- Rayleigh distribution

Suppose Y1,Y2,...,Yn denotes random sample of size n from inverse Weibull-Rayleigh distribution then its likelihood function is given by

i = П f

( a 2 Vß-1 -eß\ e*™ -1

e 2 -1 è 0

i \ n \aßoß)n П:

ni a 2

fa 2 Vß-1 2 -eß^e^-i

i=0

a 2 12yi 1 e 2 -1

\

e i=1 e

The log likelihood function is given by

logl = nloga + nlogß + nßlog0 + ^logyi + ^yf -(ß +l)^log

i=i

i=i

i=i

fa 2 ö-ß -1

i=1

fa f ^

e-1

è 0

(18)

Differentiating equation (18) with respect each parameter a, b and 0, we have

5logl _ n +1Yy 2 _b+D + q

a ^ ' 2 /Li

da a 2'

a 2 _1

n a 2

Y У'е 2

( a 2 ea _ 1

e2 _ 1

\ /

(19)

д log l n -— = — + n

dß ß

= — + nlogq-£log e2 ' -1

fa 2 ^ i

e2 -1

fa 2 ö-ß

V /

i

e2 -1

è 0

log

a2 e 2 -1

v /

-qbiog(q)ï

fa 2 Yb -1

è 0

(20)

д logl nß

дв ~ ~в'

i=l

fa 2 ö-ß 1

e 2 -1

\ 0

ß

a . 2

e

e

1

ß

2

2

P

a

i=1

i=1

By setting equations (19) (20)and (2l) to zero the MLE of parameters can be obtained. However the

above equations are non-linear which cannot be expressed in closed form. So numerical techniques such as Newton-Raphson, Regula-Falsi and bisection methods must be applied to obtain MLE of parameters denoted by ç{â, p,d) of ç(a,P,0).

Since the MLE of Ç follows asymptotically normal distribution which is given as

(ç-ç) ® N (o, i-Ç

Where Iis the limiting variance-covariance matrix Ç and I(ç) is a 3x 3 Fisher information matrix

i,e

/ (V)=-i

E 'd2 log l E ^ d2 lOg l ^ E œ d 2 log l

I da2 , dadp j dadd V

1 E œ d2 log lN E ' d 2 log l ^ E ' d2 logl

n { dpda 1 dP2 0 { dpdd

E œ d2 log l ^ E ^ d 2 log l ^ E ^ d2 logl

ddda \ / { dddp 0 { dd2

Where

52 logl = -n + (b +1) y yfe 2 da2 a2 4 ^

a 2

—yi

1 (

e 2 yi -1

n a 2

4e "

z y

4-2 y

œ a 2 Vb-1 i

e 2 -1

- 1)

œ a 2 vb-2 i

e2 -1

è

d logl - n

dp2 p2

(

i=1

-yt

yPf

e 2" -1 è /

jlog

e 2 y -1

qp(iogq)2 Z

Í a 2 VP

eay -1

i=1

(

a 2

e 2 -1

Yb

log

a2 e2 -1

d2 logl _ - nb

d02 _ 02

Pp-i)qp-2 Z

fa 2 e2* -1

i_1

\~P

d2 logl _ d2 logl

dadp dpda

n 2 2 y' 2

y,e'

pep

i_1

2y¡ 1 e 2 -1

n a 2

Z2 ~2yi y,e 2

i_1

f

a2

xP-1

e 2 -1 è

f

log

a2 7y¡

\

e 2 -1 è

2

+

4

2

a

a

a

+biogq)! yi

n a 2

2e 2

i=1

e2 -1

52 logl _ d2 logl _ p2e13

2a b-1 n a 2

dadd ddda

2e 2

i=l

e** -1

Vb-i

d2 log l _ d2 log l _ n

dfidd ~ dddp ~ 9 +

1Z

i _1

a 2

1

e 2 -1

f

log

a2 e 2 -1

, , n i a 2

ie+b-b)Yje2 y'-i i=l

\-p

J

Hence the approximate 100(1 - y/%> confidence interval for a, b and d are respectively given by

a±z^raV) ±ZyfibV) And q±ZrJiqV)

2

y 2

y 2

Where Zy denotes the yth percentile of the standard normal distribution.

XII. Results

I.

Simulation Analysis

In this section we demonstrate the simulation analysis which examines the effectiveness of the M L estimators. The inverse cdf method is employed to generate random samples of size n = 30,50,75,100and150 which is discussed in section (4). This procedure is repeated N = 500 times for calculation of bias, variance and MSE. Four separate combinations of parameters are selected and it is observed that bias, variance and MSE decrease significantly, when we increase sample size. The efficiency of ML estimators is therefore relatively strong, consistent in case of IWRD.

Table 1: Average bias, variance and MSEs of 5,00 simulations of IWRD for different parameters values.

2

a

2

II. Applications

This section is dedicated to demonstrate the effectiveness of the established distribution by taking into account real data sets taken from medical science. The established distribution is compared with power Erlang distribution (PED), Weighted Gumbel-II distribution (WG-IID), power Gompertz distribution (PGD), inverse Weibull distribution (IWD), Rayleigh distribution (RD), inverse Rayleigh distribution (IRD) and inverse Lindley distribution (ILD). It is revealed that the developed distribution offers an appropriate fit.

To compare the versatility of the explored distribution, we consider the criteria like AIC (Akaike

Sample Parameters a = 0.4 , b = 0.7, q = 0.8 a = 0.4 , b = 0.5, q = 0.7

Size n Bias variance MSE Bias variance MSE

30 a 0.10667 0.09549 0.10687 0.03396 0.04469 0.04584

b 0.01601 0.03820 0.03846 0.03240 0.01454 0.01559

e 0.99840 6.86551 7.86231 0.46101 2.29695 2.50948

50 a 0.03325 0.04242 0.04353 0.00960 0.02461 0.02471

b 0.02828 0.01952 0.02032 0.02701 0.00881 0.00954

e 0.29418 1.33982 1.42637 0.22597 0.96048 1.01155

75 a 0.03266 0.02648 0.02755 0.01118 0.01587 0.01599

b 0.01992 0.01395 0.01434 0.01276 0.006403 0.00656

e 0.23048 0.54670 0.59982 0.14271 0.26261 0.28298

100 a 0.01522 0.01673 0.01696 0.00267 0.01220 0.01221

b -0.0008 0.00980 0.00980 0.01304 0.00462 0.00479

e 0.15424 0.34874 0.37253 0.06209 0.12997 0.13383

150 a 0.01240 0.01161 0.01177 0.00537 0.00995 0.00998

b 0.01080 0.00754 0.00765 0.01761 0.00403 0.00434

e 0.09099 0.16775 0.17603 0.06589 0.13020 0.13255

a = 0.6 , b = 0.5, 0 = 0.7 a = 0.6 , b = 0.8, 0 = 0.9

30 a 0.01382 0.01359 0.01379 -0.0018 0.01282 0.01282

b 0.00500 0.00398 0.00400 0.01203 0.00447 0.00462

e 0.10007 0.19036 0.20038 0.07110 0.19553 0.20058

50 a 0.01956 0.01135 0.01174 0.00314 0.01279 0.01280

b 0.00397 0.00422 0.00423 0.00645 0.00402 0.00406

e 0.11939 0.15628 0.17053 0.07557 0.19296 0.19867

75 a 0.01085 0.01139 0.01151 -0.0052 0.00959 0.00962

b 0.00533 0.00384 0.00386 0.01512 0.00440 0.00403

e 0.08410 0.13345 0.14052 0.03145 0.09593 0.09691

100 a 0.00531 0.01086 0.01089 0.00303 0.00873 0.00874

b 0.00616 0.00401 0.00305 0.01137 0.00396 0.00402

e 0.06256 0.12807 0.13199 0.04911 0.10033 0.00275

150 a 0.01534 0.010668 0.01070 0.02240 0.01451 0.00502

b 0.00422 0.00389 0.00301 0.00551 0.00436 0.00400

e 0.12016 0.17372 0.12816 0.14134 0.10026 0.00084

information criterion), CAIC (Consistent Akaike information criterion), BIC (Bayesian information criterion) and HQIC. Distribution having lesser AIC, CAIC, BIC, HQIC and KS values is considered better also having higher probability value (p-value).

2kn

AIC = 2k - 2 In l; CAIC =—^1--2 ln l; BIC = k ln n - 2 ln l

n - k-1

And HQIC = 2k ln(ln(n))-21nl

The descriptive statistics of the data set 1 and 2 are presented in Table land 4.The estimates of the parameters are shown in Table 2 and 5 for data set land 2 respectively. Log-likelihood, Akaike information criteria (AIC) etc for the data set 1 and 2 are generated and presented in Table 3 and 6 respectively.

Data set 1:- The data was collected from a group of 46 patients, per years, upon the recurrence of leukemia whom received autologous marrow. The data repoted by Jhon H Kersey [14], as follows

0.0301,0.0384,0.063, 0.0849, 0.0877, 0.0959, 0.1397, 0.1616, 0.1699, 0.2137,0.2137, 0.2164, 0.2384, 0.2712, 0.274, 0.3863, 0.4384, 0.4548, 0.5918, 0.6,0.6438, 0.6849, 0.7397, 0.8575, 0.9096, 0.9644, 1.0082, 1.2822, 1.3452, 1.4,1.526, 1.7205, 1.989, 2.2438, 2.5068, 2.6466, 3.0384, 3.1726, 3.4411, 4.4219,4.4356, 4.5863, 4.6904, 4.7808, 4.9863, 5

Table 2: Descriptive statistics of data set first

Min Qi Median Mean Q3 Skew Kurt. Max

0.0301 0.221 0.798 1.517 ' 2.441 1.036 2.655 5

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

Model IWRD PED WG-IID PGD IWD RD IRD ILD

a 0.6737 1.2685 0.7023 0.6483 0.4075 0.0343 0.4333

ß 0.2764 0.6654 0.4651 0.2006 0.7017

q 0.0641 1.4619 0.0010 0.7515 0.3371

s.e a 0.2252 2.6342 0.3287 0.1832 0.1154 0.0600 0.0050 0.0462

b 0.0399 0.6548 0.6414 0.3050

q 0.0491 2.3917 0.5574 0.1553

Table 4: Performance of distributions for data set first

Model IWRD PED WG- PGD IWD RD IRD ILD

IID

-2 log l 118.90 127.91 138.90 127.36 138.89 207.42 303.76 176.43

AIC 124.90 133.91 144.90 133.36 142.89 209.42 305.76 178.43

CAIC 125.47 134.48 145.47 133.93 143.17 209.51 305.85 178.52

BIC 130.39 139.40 150.38 138.84 146.55 211.25 307.59 180.25

HQIC 126.96 135.97 146.95 135.41 144.26 210.10 306.45 179.11

K-S Value 0.079 0.0980 0.1399 0.1013 0.139 0.3998 0.566 1.342

P Value 0.935 0.7686 0.3286 0.7325 0.328 8.1e-07 2.9e-13 2.2e-16

The asymptotic variance-covariance matrix of maximum likelihood estimates under IWRD for data set first is computed as

/ » =

( 0.0507 -0.0053 0.0075 ^ -0.0053 0.0015 -0.0011

è 0.0075 -0.0011 0.0024 0

Therefore, the 95%confidence interval fora,band£ are given as(0.2322,1.1152) (0.1982, 0.3546) and (-0.0321,0.1604), respectively.

Fig 9.1 Estimated pdfs of the fitted models for data set 1 F¡g 9 2 Emp¡r¡ca| cdf versus fitted cdfs

Data set 2:- The data set represents the survival times(in years) of a group of patients given chemotherapy treatment reported by Bekker et al.[4]. The data follows

0.047, 0.115, 0.121, 0.132, 0.164, 0.197, 0.203, 0.260, 0.282, 0.296, 0.334, 0.395, 0.458, 0.466, 0.501, 0.507, 0.529, 0.534, 0.540, 0.641, 0.644, 0.696, 0.841, 0.863, 1.099, 1.219, 1.271, 1.326, 1.447, 1.485, 1.553, 1.581, 1.589, 2.178, 2.343, 2.416, 2.444, 2.825, 2.830, 3.578, 3.658, 3.743, 3.978, 4.003, 4.033

Table 5: Descriptive statistics of data set second

Min Qi Median Mean Q3 Skew Kurt. Max 0.047 0.39 0.84 1.34 2.17 0.972 2.663 4.03

Table 6: The ML Estimates of the unknown parameters for data set second

Model IWRD PED WG-IID PGD IWD RD IRD ILD

CO 0.9335 1.9113 0.8677 0.6504 0.6026 0.1072 0.6584

b 0.3361 0.6756 0.4977 0.1215 0.8671

ê 0.1524 2.1173 0.0010 0.9762 0.4482

S.E OO 0.3257 3.6555 0.3392 0.1719 0.0898 0.0159 0.0723

b 0.0505 0.6238 0.5701 0.2587 0.0927

ê) 0.1078 3.4214 0.5884 0.1949 0.0818

Table 7: Performance of distributions for data set second

Model IWRD PED WG-IID PGD IWD RD IRD ILD

-2 log l 110.24 115.97 127.64 115.96 127.63 155.83 230.17 138.88

AIC 116.24 121.97 133.64 121.96 131.63 157.83 232.17 140.88

CAIC 116.82 122.56 134.23 122.54 131.92 157.92 232.26 140.97

BIC 121.66 127.39 139.06 127.38 135.25 159.63 233.98 142.69

HQIC 118.26 123.99 135.66 123.98 132.98 158.50 232.84 141.56

K-S Value 0.0660 0.9811 0.1383 0.1139 0.138 0.353 0.507 1.252

P Value 0.982 2.2e-16 0.3251 0.5638 0.325 1.5e-05 2.8e-11 2.2e-16

The asymptotic variance-covariance matrix of maximum likelihood estimates under IWRD for data set first is computed a

/ » =

œ 0.1061

■0.0105 0.0263

-0.0105 0.0263 ö 0.0025 -0.0034 -0.0034 0.0116

Therefore, the 95% confidence interval for a,bandd are given as(0.2950,1.5720) (0.2370, 0.4353) and (-0.0590, 0.3638), respectively.

it is evident from Table (4) and (7) that IWRD has lesser values of AIC, CAIC, BIC, HQIC and K-S statistics along with higher p-value. When it is compared with IWD, RD, IRD and ILD models. Hence we conclude that IWRD provides an adequate fit than compare ones

XIII. Discussion

This paper deals with a new generalisation of Rayleigh distribution called inverse Weibull-Rayleigh distribution. We have added extra two parameters to the Rayleigh distribution by inverse Weibull-G generator, the main purpose for such modification is that the formulated distribution become more richer and flexible in modelling datasets. Several distinct properties of formulated distribution has been studied and discussed. The model parameters of the distribution are estimated by the known method of maximum likelihood estimation. Eventually, the efficiency of the explored distribution is examined through real data sets which reveals that the formulated distribution provides an adequate model fit than competing ones.

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