Om Distribution With Properties And Applications Текст научной статьи по специальности «Науки о Земле и смежные экологические науки»

Om Distribution With Properties And Applications

Rama Shanker & Kamlesh Kumar Shukla*

•

Department of Statistics College of Science, EIT, Asmara, Eritrea [email protected] , [email protected] * Corresponding Author

Abstract

A new one parameter lifetime distribution named, 'Om distribution' has been proposed and studied. Its various statistical properties including shapes for probability density, moments based measures, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, distribution of order statistics, and stress-strength reliability have been discussed. Estimation of parameter has been discussed with the method of maximum likelihood. Applications of the distribution have been explained through two examples of real lifetime data from engineering and the goodness of fit found to be quite satisfactory over several one parameter lifetime distributions.

Keywords: Lifetime distributions, Statistical Properties, Maximum likelihood estimation, Applications

I. Introduction

In the present world, the time to the occurrence of some event is of interest for some populations of individuals in almost every field of knowledge. The event may be death of a person or any living creature, failure of a piece of electronic equipment, development (or remission) of symptoms. In reliability analysis, the time to the occurrences of events are known as "lifetimes" or "survival times" or "failure times" according to the event of interest in the fields of study. The modeling and statistical analysis of lifetime data has been a topic of considerable interest to statisticians and research workers in engineering, biomedical science, insurance, finance, amongst others. Applications of lifetime distributions range from investigations into the endurance of manufactured items in engineering to research involving human diseases in biomedical sciences.

During recent decades, a number of one parameter and two-parameter lifetime distributions for modeling lifetime data have been introduced by different researchers in statistics. The popular one parameter lifetime distributions available in statistics literature are exponential distribution and Lindley distribution introduced by Lindley (1958). Recently Shanker (2015 a, 2015 b, 2016 a, 2016 b, 2017 a, 2017 b, 2017 c, 2017 d) has proposed several one parameter lifetime distributions, namely Shanker, Akash, Sujatha, Aradhana, Rama, Akshaya, Amarendra and Devya, and it has been showed by Shanker that these distributions have advantages and disadvantages over the others. The probability density function (pdf) and the cumulative distribution function (cdf) of exponential, Lindley, Shanker, Akash, Sujatha, Aradhana, Rama, Akshaya, Amarendra and Devya distributions along with their introducers and year have been presented in table 1.

Table 1: pdf and cdf of one parameter lifetime distributions

Distributio ns pdf and cdf Introducer(Year)

Devya ns fix; 6) = , 6 (1 + x + x2 + x3 + x4) e6 V ' 64 + 63 + 262 + 66 + 24 V ' Shanker (2016 d)

F ( x,6) = 1 - 1+ 6" (x4 + x3 + x2 + x) + 63 (4x3 + 3x2 + 2x) + 662 (2x2 + x)" +246x e-6x

6" + 63 + 262 + 66 + 24

Amarendra f)4 f ( x; 6)= 6 (1 + x + x2 + x3) e6 V ' 63 + 62 + 26 + 6V ' Shanker (2016 c)

F (x,6) = 1 - 6V +62(6 + 3)x2 +6(62 + 26 + 6)x] _gx + 63 +62 + 26 + 6 e

Akshaya 64 . f (x;6)= 6 (1 + x)3e6 ;x>0, 6>0 V ' 63 + 362 + 66 + 6V ' Shanker (2017 b)

F (x;6) = 1 - 63x3 + 362 (6 +1)x2 + 36(62 + 26 + 2)x 1 + 63 + 362 + 66+6 e

Rama Q4 f ('•6)=6h(1+xJ) Shanker (2017 a)

F (x,6) = 1 - , 63x3 + 362x2 + 66 x 1 +-5- 63 + 6 e6

Aradhana 63 9 f(x;6)=026+2(1+x)2e6 ;x>06>0 Shanker (2016 b)

F (x;6) = 1 - " 6x (6x + 26 + 2 )" 62 + 26 + 2 eex ;x>0,6>0

Sujatha 63 f (x;6)= , (1 + x + x2) e-6x ; x > 0, 6> 0 V ' 62 + 6+ 2V ' Shanker (2016 a)

F (x,6) = 1 - ^ 6x (6x + 6 + 2)" + 62 + 6 + 2 e~6x

Akash 63 f (X;6) = 0^2 (1 + x2)e-x ;x > 0, 6> 0 Shanker (2015 b)

F (x;6) = 1 - r 6x (6x + 2)" „ 1 + \ ' e ;x > 0,6 > 0 62 + 2 J

Shanker 62 f (x;6) = ^(6 + x)e-6x ;x > 0, 6> 0 Shanker (2015 a)

F (x, 6) = 1 - T 6x " L1 +62 + 1_ e~6x ;x > 0,6 > 0

Lindley 62 f (x;6) = ——-(1 + x) e-6x ; x > 0, 6> 0 6 +1 Lindley (1958)

F ( x;6) = 1 - " 6x " 1 +- _ 6+1_ e-6x ;x > 0,6 > 0

Exponentia l f (x;6) = 6e-x ; x > 0, 6> 0

F(x;6) = 1 -e-6x ;x>0,6>0

Ghitany et al (2008) have discussed various statistical properties, estimation of parameter and application of Lindley distribution for modeling waiting time data in a bank. It has been observed that these lifetime distributions are not always suitable for modeling lifetime data from biomedical sciences and engineering. In the present paper an attempt has been made to propose a one parameter lifetime distribution named 'Om distribution' which gives better fit than all one parameter lifetime distributions. Its various statistical properties, estimations of parameter and applications for modeling two real lifetime data from engineering have been discussed.

II. Om Distribution

A new one parameter lifetime distribution named Om distribution can be defined by its pdf and cdf

e5

f (*e) =

f (x-e) = i -

e4+4e3 + i2e2+24e+24

4 , \3

(1 + x )4

; x > 0, e> 0

(1+x) e4+4(1+x) e3+12(1+x) e2+24(1+x)e+24

e4+4e3 + 12e2+24e+24

-ex

; x > o,e> 0

(2.1)

(2.2)

The nature of the pdf and cdf of Om distribution for varying values of parameter e have been shown graphically in figures 1 and 2 respectively.

/ \ — theta=0.3 -- theta=0.5

_ - ' L theta=0.7

I Z / \\ " . l ^ „W", ^ theta=0.9

theta=1

■' ' ^—^—

1 1 1 0 5 10 i i 15 20

X

— theta=1.2 -- theta=1.5

— theta=1.7

1 ' theta=1.9

theta=2

* T. —. -

Fig. 1: Nature of the pdf of Om distribution for varying values of parameter e

0 5 10 15 20

x

— theta=1.2

■ ■ / s >.' / / -- theta=1.5

/■"' s — theta=1.7

/./ / theta=1.9

theta=2

0 2 4 6 8

Fig. 2: Nature of the cdf of Om distribution for varying values of parameter 6

3. Moments and Associated measures

The r th moment about origin of Om distributon (2.1) can be obtained as

6 + 4 (r +1)63 + 6 (r +1)( r + 2 )62 + 4 ( r +1)( r + 2)( r + 3)6] r !<

[+(r +1)(r + 2)(r + 3)(r + 4)_^ ^ = 123

r 6r (64 + 463 +1262 + 246 + 24) ; , , ,

(3.1)

The first four moments about origin of Om distribution can be given as

64 + 863 + 3662 + 966 +120

M =

6 (64 + 463 +1262 + 246 + 24) 14 +1263 + 7262 + 2406 + 3

, 2 (64 +1263 + 7262 + 2406 + 360)

M2 =

62 (64 + 463 +1262 + 246 + 24) 14 +1663 +12062 + 4806 + <

, 6 (64 +1663 +12062 + 4806 + 840) M " 63 (64 + 463 +1262 + 246 + 24) , 24 (64 + 2063 +18062 + 8406 +1680)

LI =---—

4 64 (64 + 463 +1262 + 246 + 24)

Thus the moments about mean of the Om distribution (2.1) are obtained as

08 +1607 +1280 + 6240 +19200 + 38400 + 576002 + 57600 + 2880

M ='

02 (04 + 403 +1202 + 240 + 24)

f012 + 24011 + 276010 +192809 + 88560s + 2851207 + 7084806 +14169605 + 23328004^

M =

_ V

+31104003 + 31104002 + 2073600 + 69120

03 (04 + 403 +1202 + 240 + 24)

3016 + 960 +14720 +140480 + 92672012 + 454656011 +1767936010 + 56409600 +1503475208 + 3367526407 + 6314803206 + 9803980805 +12386304004 +12386304003 + 9289728002 + 464486400 +11612160

39A

M4 =

04 (04 + 403 +1202 + 240 + 24)

The coefficient of variation (CV ), coefficient of skewness ( V^), coefficient of kurtosis (fî2 ) and Index of dispersion (y) of Om distribution (2.1) are thus obtained as

_ V08 +1607 +12806 + 62405 +192004 + 384003 + 576002 + 57600 + 2880

f

M

CV = ( =

04 + 80 + 360" + 960 +120

M

M

3/2

f012 + 24011 + 276010 +192809 + 88560s + 2851207 + 7084806 +14169605 + 23328004^ +31104003 + 31104002 + 2073600 + 69120

(08 +1607 +12806 + 62405 +192004 + 384003 + 576002 + 57600 + 2880)3/2

A = M = v

M

f3016 + 96015 +1472014 +14048013 + 92672012 + 454656011 +1767936010 + 564096009^ +1503475208 + 3367526407 + 6314803206 + 9803980805 +12386304004 +12386304003 + 9289728002 + 464486400 +11612160

(08 +1607 +12806 + 62405 +192004 + 384003 + 576002 + 57600 + 2880)

y = ■

(j2 08 +1607 +12806 + 62405 +192004 + 384003 + 576002 + 57600 + 2880

M 0 (04 + 403 +1202 + 240 + 24) (04 + 803 + 3602 + 960 +120)

The behaviors of coefficient of variation, skewness, kurtosis and index of dispersion of Om distribution have been shown graphically for varying values of parameter 0 in figure 3.

2

2

3

3

4

2

3

2

0.5 1.0 15 2.0

theta

Fig. 3: Behavior of coefficient of variation, skewness, kurtosis and index of dispersion of Om distribution for

varying values of parameter d

The conditions of dispersion of Om distribution along with other one parameter lifetime distribution for values of the parameter 6 have been presented in table 2.

Table 2. Over-dispersion, equi-dispersion and under-dispersion of Om distribution and other one parameter lifetime distributions for varying values of their parameter e

Distributions Over-dispersion Equi-dispersion Under-dispersion

(<u<*2) (, = a-) (<u>*2)

Om e< 1.306113562 e = 1.306113562 e> 1.306113562

Devya e< 1.451669994 e = 1.451669994 e> 1.451669994

Amarendra e< 1.525763580 e = 1.525763580 e> 1.525763580

Akshaya e< 1.327527885 e = 1.327527885 e> 1.327527885

Rama e< 1.950164618 e = 1.950164618 e> 1.950164618

Aradhana e< 1.283826505 e = 1.283826505 e> 1.283826505

Sujatha e< 1.364271174 e = 1.364271174 e> 1.364271174

Akash e< 1.515400063 e = 1.515400063 e> 1.515400063

Shanker e< 1.171535555 e = 1.171535555 e> 1.171535555

Lindley e< 1.170086487 e = 1.170086487 e> 1.170086487

Exponential e< 1 e=1 e> 1

IV. Statistical Properties

I. Survival function, Hazard rate function and Mean Residual life function

Suppose f (x) and F (x) be the pdf and cdf of a continuous random variable X . The survival function, S (x), hazard rate function h (x) (also known as the failure rate function) and the mean residual life function m (x) of X are respectively defined as

S (x ) = P ( X > x ) = 1 - F (x ) P (X < x + Ax|X > x) f (x)

h ( x) = lim

^ ' Av if

Ax^0

Ax

1 - F ( x )

and m (x) = E [X - x|X > x] = * J"[1 - F (t)] dt

- F ( x )■

The corresponding survival function S (x), hazard rate function, h (x) and the mean residual life function, m (x) of Om distribution are thus obtained as

(1+x)4 e4+4 (1+x)3e3 +12 (1+x)2 e2 + 24 (1+x)e+24

S (x ) = h ( x ) =

e4 + 4e3 + 12e2 + 24e+24

; x > 0,e> 0

e5 (1+x)

(1+x)4 e4+4(1+x)3 e3+12(1+x)2 e2+24(1+x)e+24 '

and m(x)=

x3/?3+12 (1+x )2 e2 e4+4e3+12e2+24e+24

; x > 0,e> 0

(1+x )4 e4+4 (1+x )3 e3+12 (1+x )2 e2 + 24 (1+x) e+24

\3 n3

2 /}2

J[(1+1 )4 e4+4 (1+1 )3 e3+12 (1+1 )2 e2+24 (1+1 )e+24

e-etdt

X

(1 + x)4 64 + 8(1 + x)3 63 + 36(1 + x)2 62 + 96(1 + x)6 +120

6

(1 + x) 64 + 4 (1 + x) 63 +12 (1 + x) 62 + 24 (1 + x) 6 + 24

6

It can be easily verified that 64 + 863 + 3662 + 966 +120

h ( 0 ) = ■

= f 0)

and

m

( 0 ) =

6 (64 + 463 +1262 + 246 + 24) distribution have been shown in figures 4 and 5 respectively.

64 + 463 +1262 + 246 + 24 = M. The behaviors of h (x) and m (x) of Om

— theta=0.3

- - theta=0 5

--- theta=0.7

theta=0.9

theta=1

-1"

...............

0 2 4 6 8 10

x

— theta=1.2

- - theta=1.5

\ \ "" theta=1.7

— theta=1.9

theta=2

N. ■ - . **■ _ ---_ ---

0 5 10 15

x

Fig. 5: Behavior of m (x) of Om distribution for varying values of parameter 6

II. Mean deviations from the mean and the Median

The amount of scatter in a population is measured to some extent by the totality of deviations usually from their mean and median. These are known as the mean deviation about the mean and the mean deviation about the median and are defined as

S(X) = J|x-M f (x) dx and S2(X) = J|x-M\f (x) dx, respectively, where M = E (X)

0 0

and M = Median (X)

. The measures Sx (X) and S2 (x) can be computed using the following

simplified relationships

M

S(X) = 2mF (m)-21 xf (x) dx (4.2.1)

0

M

and S( X ) = M- 2 J xf ( x ) dx (4.2.2)

0

Using pdf (2.1) and expression for the mean of Om distribution, we get

r

J x f ( x;9) dx = /-

(/5 + 4/4 + 6/3 + 4/2 + /)95 + (5/4 +16/3 +18/2 + 8/ +1)94 +(20/3 + 48/2 + 36/ + 8) 93 + (60/2 + 96/ + 36)92 + (120/ + 96)9 +120

-«u

9 (94 + 493 +1292 + 249 + 24)

(4.2.3)

J x f (x;9) dx = /-

|(m 5 + 4M4 + 6M3 + 4M2 + M )95 + (5M4 + 16M3 + 18M2 + 8M +1)94 - (20M3 + 48M2 + 36M + 8) 93 + (60M2 + 96M + 36) 92 + (120M + 96) 9 +1201

9 (94 + 493 +1292 + 249 + 24)

0

0

(4.2.4)

Using expressions (4.2.1), (4.2.2), (4.2.3) and (4.2.4), the mean deviation about mean, Sx (X) and the mean deviation about median, S2 (x) of Om distribution (2.1),, after tedious algebraic simplification are obtained as

(1+/)4 e4+8 (1+/)3 e3+36 (1+n)2 e2+96 (1+/) e+120

S (X ) = 2

(4.2.5)

9 (94 + 493 +1292 + 249 + 24)

-9/

d 2 ( X ) =

(m 5 + 4M4 + 6M3 + 4M2 + M ) 95 + (5M4 + 16M3 +18M2 + 8M +1) 94 (20M3 + 48M2 + 36M + 8) 93 + (60M2 + 96M + 36) 92 + (120M + 96) 9 +120

9 (94 + 493 +1292 + 249 + 24)

III. Bonferroni and Lorenz Curves

The Bonferroni and Lorenz curves ( Bonferroni, 1930) and Bonferroni and Gini indices have applications not only in economics to study income and poverty, but also in other fields like reliability, demography, insurance and medicine. The Bonferroni and Lorenz curves are defined as

i q i

B (p ) =-J x f (x ) dx =-

P/ o P/

J x f ( x ) dx-J x f (x ) dx

1

P/

/-J x f (x ) dx

and

J x f ( x ) dx-J x f ( x ) dx

iq i

L (p ) = — [ x f (x ) dx = —

/ o /

respectively or equivalently

1 p

B (p ) =-f F- (x ) dx

P/ 0

and

1 p

L (p ) = - f F 1 ( x) dx u 0

respectively, where / = E (X) and q = F 1 (p) .

1 /

/-J x f ( x ) dx

q

(4.3.1)

(4.3.2)

(4.3.3)

(4.3.4)

The Bonferroni and Gini indices are thus define

1

B = 1 -J B (p ) dp

0

1

G = 1 - 2 J L (p ) dp

(4.3.5)

(4.3.6)

respectively

Using pdf (2.1), we get

J x f ( x;6) dx

| (q5 + 4q4 + 6q3 + 4q2 + q) 65 + (Sq4 + 16q3 + 18q2 + 8q +1) 64 - (20q3 + 48q2 + 36q + 8) 63 + (60q2 + 96q + 36) 62 + (120q + 96) 6 +120]

,-6q

6 (64 + 463 +1262 + 246 + 24)

(4.3.7)

Now using equation (4.3.7) in (4.3.1) and (4.3.2), we get

( q5 + 4q4 + 6q3 + 4q2 + q) 65 + ( Sq4 + 16q3 + 18q2 + 8q +1) 64

+ ( 20q3 + 48q2 + 36q + 8 )63 +( 60q2 + 96q + 36)62 +(120q + 96 )6 +120

B ( p >=p

1 -

>e

6q

64 + 86 + 3662 + 966 +120

0

q

L ( p ) = 1 -

(q5 + 4q4 + 6q3 + 4q2 + q) 65 + (Sq4 + 16q3 + 18q2 + 8q +1) 64 + ( 20q3 + 48q2 + 36q + 8)63 +( 60q2 + 96q + 36) 62 +(120q + 96)6 +120

(4.3.8)

re

64 + 863 + 3662 + 966 +120

(4.3.9)

Now using equations (4.3.8) and (4.3.9) in (4.3.5) and (4.3.6), the Bonferroni and Gini indices of Om distribution (2.1) are obtained as

( q5 + 4q4 + 6q3 + 4q2 + q) 65 + (Sq4 + 16q3 + 18q2 + 8q +1) 64 1

-(20q3 + 48q2 + 36q + 8)63 +(60q2 + 96q + 36)62 +(120q + 96 )6 +120

B = 1 -

re

64 + 863 + 3662 + 966 +120

G =

(q5 + 4q4 + 6q3 + 4q2 + q) 65 + (Sq4 + 16q3 + 18q2 + 8q +1) 64 + ( 20q3 + 48q2 + 36q + 8)63 +( 60q2 + 96q + 36)62 +(120q + 96 )6 +120

(4.3.9)

ye

64 + 86 + 3662 + 966 +120

■-1

(4.3.10)

IV. Stochastic ordering

Stochastic ordering of positive continuous random variables is an important tool for judging their comparative behavior. A random variable X is said to be smaller than a random variable Y in the

(i) stochastic order (x <st y ) if Fx (x) > FY (x) for all x

(ii) hazard rate order (X <hr Y) if hx (x) > h (x) for all x

(iii) mean residual life order (X <mrl Y) if mx (x) < m7 (x) for all x

fx ( x)

(iv) likelihood ratio order (x <lr y)if ^ decreases inx .

fY ( x )

The following results due to Shaked and Shanthikumar (1994) are well known for establishing stochastic ordering of distributions

X <,Y ^ X <hr Y ^ X <mrJ

U

X <st Y

The Om distribution is ordered with respect to the strongest 'likelihood ratio' ordering as shown in the following theorem:

Theorem-. Let X ~ Om distribution (0) and Y ~ Om distribution (0). If 0 > 0, then X <lr Y and hence X <hr Y, X <mrl Y and X Y . Proof. We have

*(x;0) _ (V + ^ + + + 24) -(0-

-e-Q2; x; x >0 jy ( xA> q2 5 q + Q +! 2Q2 + 24Q + 24)

Now

- Q5 A4 + 403 + 1202 + 240 + 24)

1 v 2 2 2 2 ' -(0-0)

ln fx() = ln ln fy (x-A) ln

e25 (04 + 403 +1202 + 240! + 24) This gives — ln f^ = - (0 - 0 ). Thus for 0 > 0, — ln f^ < 0. This means that

dX fT ( 2) V 1 ! 1 —x fY (x02)

X <ir Y and hence X <hr Y, X <mrf Y and X Y.

V.Distribution of Order Statistics

Let X,X2X be a random sample of size n from Om distribution (2.1). Let X^ < X^ < ... < X^ denote the corresponding order statistics. The pdf and the cdf of the k th order statistic, say Y = X^ are given by

n! , w w -k

fy(>)=(k-T)nn-k)7F-'(>)i1 -F(>f(>)

n! n-k(n-k\ u , .

! - )(- 1)Fk+l-1 (y) f (y)

(k -1)!( n - k )! 1=

and

F (y ) = Zin IF1 (y ){i - F (y)}

j=k v J

\

n- J

n

n"zifn V n - j N

III j,

j=k i=0 Vj

l

(- 1)F+l (y),

respectively, for k = 1,2,3,..., n.

Thus, the pdf and the cdf of k th order statistics of Om distribution (2.1) are obtained as n!9s (1 + x)4 _

/ (y ) =

(e4 + 49s + 1ie2 + 249 + 24) (k -1)! (n - k)!

k (n - k\ ,i (-1)

XI[ i

i=0 V l

J(1+x)4 e4+4 (1+x)3e3 +12 (1+x)2 e2 + 24 (1+x)e+241 _ex I e4 + 4e3 + 12e2+24e + 24 \

k+l-1

and

Fr (y) = II

j=k l=0

— n-j (

n - j l

\

(-1)1

1 -

(1+x )4 e4 + 4 (1+x )3 e3 +12 (1+x)2 e2 + 24 (1+x )e+24 J

j+i

e4 + 4e3 + 12e2 + 24e+24

VI. Stress-Strength Reliability

The stress- strength reliability describes the life of a component which has random strength X that is subjected to a random stress Y. When the stress Y applied to it exceeds the strength X, the component fails instantly and the component will function satisfactorily till X > Y. Therefore, r = p (Y < x) is a measure of component reliability and is known as stress-strength reliability in

statistical literature. It has wide applications in almost all areas of knowledge especially in engineering such as structures, deterioration of rocket motors, static fatigue of ceramic components, aging of concrete pressure vessels etc.

Let X and Y be independent strength and stress random variables having Om distribution (2.1) with parameter 9x and 9z respectively. Then the stress-strength reliability R of Om distribution can be obtained as

m

R = P (Y < X) = J P (Y < X | X = x)/x (x) dx

0

m

= J/(x;9) F(x;92)dx

0

(e4 + 49z 3 + 129z2 + 249z + 24) (e + 92)s +(89z 4 + 289z3 + 729z 2 +1209z + 96 )(e + 9z)? (56e4 + 1689z3 + 360e2 + 4809z + 288)(e +e)6 (336e24 + 8409z3 + 14409z2 + 14409z + 576) (e +e)5 (1680e4 + 3360e23 + 43209z2 + 28809z + 576)(e + 9z)4 (67209z4 + 100809z3 + 86409z2 + 2880e2)(ei + 9z)3

(201609z4 + 20160e3 + 86409z 2 )(e +9z)2 +(403209z4 + 201609z 3 )(e + 9z) + 403209z

e5

(e4 + 4e3 + 129z + 24e + 24) (e4 + 4e23 + 129z 2 + 249z + 24) (e + e2)

9

Rama Shanker & Kamlesh Kumar Shukla RT&A, No 4 (51) OM DISTRIBUTION WITH PROPERTIES AND APPLICATIONS_Volume 13, December 2018

V. Maximum likelihood estimation

Let (x,x2,...,Xn)be a random sample of size n from Om distribution (2.1). The likelihood function L of Om distribution can be expressed as

L =

f tf Y » „

_"_ nil + x )4 e~"ex

v04 + 403 + 1202 + 240 + 24J ifV

--5--I TT (1 + 4x + 6x2 + 4x3 + x4) e"0x

04 + 40 +120 + 240 + 24J 11 '

The log likelihood function is thus given by

f 05 I "

logL = nlog —----- \y log(1 + 4x + 6x2 + 4x3 + x4)-n0x .

,04 + 40 +120 + 240 + 24 J V 1 1 1 l}

The maximum likelihood estimates (MLE) 0 of parameter 0 is the solution of the log-likelihood

equation-g— = 0 and is given by

d0

d log L 5 n n (403 +1202 + 240 + 24) _

- =---a-;-;- n x = 0.

d0 0 04 + 40 +120 + 240 + 24

This gives a fifth degree polynomial equation in 0 as

x05 +(4x-1)04 + 4(3x -2)03 +12(2x -3)02 + 24(x -4)0-120 = 0.

This equation can be easily solved using any numerical iterationmethod namely, Newton-Raphson method, Regula Falsi method or Bisection method. In this paper Newton-Raphson method has been used to estimate the parameter 0 from above equation. It should be noted that equating the population mean to the corresponding sample mean, the method of moment estimate is the same as method of maximum likelihood.

VI. Data analysis

In this section the goodness of fit of Om distribution has been discussed with following two real lifetime datasets from engineering.

Data Set 1: The data is given by Birnbaum and Saunders (1969) on the fatigue life of 6061 - T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second. The data set consists of 101 observations with maximum stress per cycle 31,000 psi. The data ( x are presented below (after subtracting 65).

5 25 31 32 34 35 38 39 39 40 42 43

43 43 44 44 47 47 48 49 49 49 51 54

55 55 55 56 56 56 58 59 59 59 59 59

63 63 64 64 65 65 65 66 66 66 66 66

67 67 67 68 69 69 69 69 71 71 72 73

73 73 74 74 76 76 77 77 77 77 77 77

79 79 80 81 83 83 84 86 86 87 90 91

92 92 92 92 93 94 97 98 98 99 101 103

105 109 136 147

Data Set 2: This data set is the strength data of glass of the aircraft window reported by Fuller et al

(1994)

18.83 20.8 21.657 23.03 23.23 24.05 24.321 25.5 25.52 25.8 26.69 26.77

26.78 27.05 27.67 29.9 31.11 33.2 33.73 33.76 33.89 34.76 35.75 35.91

36.98 37.08 37.09 39.58 44.045 45.29 45.381

For these two datasets, Om distribution has been fitted along with other one parameter lifetime distributions. The ML estimate, value of —2ln L, Akaike Information criteria (AIC), K-S statistics and p-value of the fitted distributions are presented in tables 3 and 4.. The AIC and K-S Statistics are computed using the following formulae: AIC = —2ln L+2k and

K-S = Sup |-Fn (x) — F (x)|, where k = the number of parameters, n = the sample size , F (x) is

x

the empirical (sample) cumulative distribution function, and F (x) is the theoretical cumulative

distribution function. The best distribution is the distribution corresponding to lower values of —2ln L, AIC, and K-S statistics and higher p-value

Table 3: MLE's, - 2ln L, AIC, K-S and p-values of the fitted distributions for dataset 1

Distributions MLE (d ) S.E ( 0 ) —2logL AIC K-S P-Value

Om 0.07211 0.00322 924.64 926.64 0.138 0.043

Shambhu 0.08755 0.00357 918.61 920.61 0.117 0.131

Devya 0.07289 0.00326 924.26 926.26 0.333 0.000

Amarendra 0.05824 0.00213 934.38 936.38 0.163 0.010

Suja 0.07317 0.00327 924.21 926.21 0.136 0.049

Akshaya 0.05769 0.00288 935.11 937.11 0.164 0.008

Rama 0.05854 0.00293 934.05 934.05 0.162 0.012

Aradhana 0.04327 0.00249 952.58 954.58 0.196 0.001

Sujatha 0.04356 0.00251 951.78 953.78 0.195 0.001

Akash 0.04387 0.00253 950.97 952.97 0.194 0.001

Shanker 0.02925 0.00206 980.97 982.97 0.248 0.000

Lindley 0.02887 0.00204 983.11 985.11 0.252 0.000

Exponential 0.01463 0.00145 1044.87 1046.87 0.366 0.000

Table 4: MLE's, - 2ln L, AIC, K-S and p-values of the fitted distributions for dataset 2

Distributions MLE ( 0 ) S.E ( 0 ) —2logL AIC K-S P-Value

Om 0.15718 0.01262 228.81 230.81 0.230 0.061

Shambhu 0.19339 0.01417 223.40 225.40 0.199 0.148

Devya 0.16087 0.01292 227.68 229.68 0.422 0.000

Amarendra 0.12829 0.01210 233.41 235.41 0.257 0.027

Suja 0.16227 0.01303 227.25 229.25 0.223 0.077

Akshaya 0.12574 0.01129 234.44 236.44 0.263 0.022

Rama 0.12978 0.01165 232.79 234.79 0.253 0.030

Aradhana 0.09432 0.00978 242.22 244.22 0.306 0.004

Sujatha 0.09561 0.00990 241.50 243.50 0.303 0.005

Akash 0.09706 0.01005 240.68 242.68 0.298 0.006

Shanker 0.64716 0.00820 252.35 254.35 0.358 0.000

Lindley 0.06299 0.00800 253.98 255.98 0.365 0.000

Exponential 0.03245 0.00582 274.53 276.53 0.458 0.000