A QUASI SUJA DISTRIBUTION Текст научной статьи по специальности «Математика»

A QUASI SUJA DISTRIBUTION

Rama Shanker1, Reshma Upadhyay1, Kamlesh Kumar Shukla2*

department of Statistics, Assam University, Silchar, Assam, India, Email: [email protected], [email protected]

2Department of Mathematics, Noida International University, Gautam Budh Nagar, India ^Corresponding Author email: [email protected]

Abstract

A two-parameter quasi Suja distribution which contains Suja distribution as particular case has been proposed for extreme right skewed data. Its statistical properties including moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, Renyi entropy measures, and stress-strength reliability have been derived and studied. The estimation of parameters using method of moments and maximum likelihood has been discussed. A simulation study has been presented to know the performance of maximum likelihood estimation. The goodness of fit of the proposed distribution has been presented.

Keywords: Suja distribution, Statistical Properties, parameters estimation, Goodness of fit.

I. Introduction

The search for a suitable distribution for modeling of lifetime data is very challenging because the lifetime data are stochastic in nature. The analysis and modeling of lifetime data are essential in almost every fields of knowledge including engineering, medical science, demography, social sciences, physical sciences finance, insurance, demography, social sciences, physical sciences, literature etc and during recent decades several researchers in statistics and mathematics tried to introduce lifetime distributions. Recently, Sharma et al [1] studied comparative study of several one parameter lifetime distributions and observed that there are some datasets which are extreme skewed to the right where these distributions were not giving good fit. In the search for a new lifetime distribution which can be used to model data from various fields of knowledge, Shanker [2] proposed a one parameter distribution Suja distribution which is defined by its probability density function (pdf) and cumulative distribution function (cdf) given by

f (x;d)=7q4 (1+x4) e~qx ;x >oq>o

(1.1)

F ( x;0) = 1 -

1 +

q x4 + ao' xJ + i2q x + 24qx o4+24

;x >0,0>0

(1.2)

Shanker [2] studied its statistical properties, estimation of parameter using method of moment and method of maximum likelihood and applications to some real lifetime data and observed that Suja distribution gives much closer fit than several one parameter lifetime distributions. Recently, Al-Omari and Alsmairan [3] obtained length-biased Suja distribution and studied its statistical properties and applications. Al-Omari et al [4] proposed power lenth-biased Suja distribution and discussed its properties and applications. Alsmairan and Al-Omari [5] derived weighted Suja distribution and discussed its statistical properties and applications to ball bearings data in safety engineering. Todoka et al [6] have studied on the cdf of various modifications of Suja distribution and discussed their applications in the field of analysis of computer- virus propagation and debugging theory.

The main objective of this paper is to propose a two-parameter quasi Suja distribution which contains Suja distribution as particular case. Its statistical properties including moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, Renyi entropy measures, and stress-strength reliability have been derived and studied. The estimation of parameters using method of moments and maximum likelihood methods has been discussed. A simulation study has been presented to know the performance of maximum likelihood estimation. Applications and goodness of fit of the proposed distribution have been discussed.

II. A Quasi Suja Distribution

The pdf and the cdf of quasi Suja distribution QSD are expressed as

f (x;0,a) = —^-(a + dx4)e-0x;x > 0,9 > 0,a > 0 (2.1)

V ' aO3 + 24 '

d4 x4 + 403 x3 + 12q2x2 + 240x

F ( x;d,a) = 1 -

1 + -

ad3 + 24

The survival function of QSD is given by

4x4 + 463x3 +1262x2 + 246x + (a63 + 24)

e-qx; x > 0,q> 0,a> 0 (2.2)

S ( x;d,a) =

a63 + 24

e-8x;x> 0,6 > 0,a > 0.

At a = q, the pdf and the cdf of QSD reduces to the corresponding pdf and cdf of Suja distribution. Like Suja distribution, QSD is also a convex combination of exponential distribution

with parameter #and gamma distribution with parameters (5,0) with mixing proportion

a03 P =-5--

a03 + 24

The nature of pdf and cdf of QSD for varying values of parameters are shown in the following figures 1 and 2 respectively. From the pdf plots of the QSD, it is clear that it is extreme skewed to the right.

Figurel: pdf plots of QSDfor varying values of parameters

Figure 2: cdf plots of QSD for varying values of parameters

III. Measures based on Moments

The r th moment about origin jr of QSD can be obtained as

, r lia — +(r + l)(r + 2)(r + 3)(r + 4)}

---(a-3 + 24) ;r = (3-1)

Taking r = 1,2,3 and 4, the first four raw moments of QSD can be expressed as

, a63 +120 , 2 (ad3 + 360) , 6 (a63 + 840) , 24 (a63 +1680)

M = 6(a63 + 24), ^ = 02 (a63 + 24) ' ^ = 63 (a 63 + 24) and ^ = 64 (a 63 + 24)

Now the relationship between central moments and raw moments gives the central moments as

a2 d6 + 528a O3 + 2880 A2 =-

A =

O2 (ad3 + 24)

2 (a369 +1512a2 66 +1728a 63 + 69120) 63 (a63 + 24)3

9 (a4012 + 2656a309 + 58752a2 06 + 1234944a 03 + 3870720)

= / \4

04 (a03 + 24)

The descriptive measures based on moments of QRD such as coefficient of variation (C.V), coefficient of skewness, ^^fbî), coefficient of kurtosis (b2) and index of dispersion (g) of QSD are obtained as

CV -Ja2 06 + 528a63 + 2880

' ' = " ad3 +120

b ¡u3 2 (a309 + 1512a2 06 + 1728a03 + 69120) (^2 ; (a2 q6 + 528a q3 + 2880)

p 9 (a4012 + 2656a309 + 58752a2 06 + 1234944a03 + 3870720)

b2 = 2 = / I 7 ^ \2

P2 (a2 06 + 528a 03 + 2880)

_ a2 06 + 528a q3 + 2880

g =

U 0(a03 + 24)(a03 +120)

The coefficient of variation, skweness, kurtosis and index of dispersion for varying values of parameters are shown in the following figures 3, 4, 5, and 6 respectively

--■ e-1.£ — e-2

9 = 2.E

0.5

—I— 1.0

—I—

1.5

—I—

2.0

2.5

3.0

3.5

4.0

> О

Figure3: Plots of Coefficient of variation (C.V) of QSDfor varying values of parameters

Fig. 4: Plots of Coefficient of skweness of QSD for varying values of parameters

- <x-0.

- - o-1

-" a = 1 •- a-2 a = 2.

I-1-1-1-1-1-1-Г

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Figure 5: Plots of Coefficient of kurtosis of QSD for varying values of parameters

— a= O.i

--a-1

--- a-1J • - a-2 a-2.!

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

— &=0.ï - - e-1

6=1.5 - 9-2 6=2.ï

0.5

1.0

—I— 1.5

2.0

2.5

3.0

3.5

4.0

Figure 6:Plots of Index of dispersion of QSD for varying values of parameters

R. Shanker, R. Upadhyay, K. K. Shukla RT&A, No 3 (69) A QUASI SUJA DISTRIBUTION_Volume 17, September 2022

IV. Reliability Measures

Let X be a random variable having pdf f (x) and cdf F (x). The 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

, x P(X < x + AxlX > x) f (x)

h (x)= lim^-1-= V , and

w Ax®0 Ax 1 - F (x)

i ^ 1

m(x) = E[X- x|X > x] = F(t)]dt = — JJtf (t)dt-x .

Now using the pdf and cdf of QSD, the hazard rate function, h (x) and the mean residual life function, m (x) of the QSD are thus obtained as

q4 (a+qx4)

h (x ) = and m (x) =

q4 x4+4q3 x3+i2q2 x2+24qx+(aq3+24)

Q4 x4 + 863x3 + 3662x2 + 966x + (a 63 +120)

6

64x4 + 463x3 +1262x2 + 246x + (a63 + 24)]

Obviously, we have h(0) =---= f (0)and m(0) = a-r = u'. The hazard rate

" V } a03 + 24 W 6(a63 + 24) 1

function and the mean residual life function of QSD for varying values of parameters are shown in figures 7 and 8 respectively.

Figure7: Plots of Hazard function of QSD for varying values of parameters

012345 012345

Figure8; Plots of Mean residual life function of QSD for varying values of parameters

V. Mean Deviations

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

mean deviation about the median and are defined as

¥ ¥

d (X) = J| x - /u\f (x) dx and d2 (X) = J|x - M\f (x) dx, respectively, where fl = E (X)

0 0

and M = Median (X). The measures d1 (X) and d2 (X) can be calculated using the relationships

J ¥

d1 ( X ) = | (/- x)f ( x ) dx +1 ( x -/)f ( x ) dx

0 J

J ¥

= /F (J) -J x f (x) dx - /[1 - F (/)] + J x f (x ) dx

0 /

¥

= 2 /F (// - 2 / + 2j x f (x) dx

/

M

= 2 mF (m) - 2f xf (x) dx (5.1)

and

d2 ( X ) = J (M - x) f ( x ) dx + J ( x - M ) f ( x ) dx

0 M

M ¥

-MF (M ) -J x f ( x) dx - M [1 - F (M )] + J x f ( x) dx

M

¥

0

= -ß + 2 J X f ( X ) dx

M

M

= ß-2 J x f ( x) dx

Using the pdf of QSD and the mean of QSD, we get

v {в р5 + 50 V + 200V + 60 в2р2 + (ав3 +120) в р + (а в3 +120)} e"

ер

J x f(x;0,a) dx = v

в(ав3 + 24 )

(5.2)

(5.3)

CO

M

-вы

\вМь +5вМ4 + 20вМ3 +60в2М2+(ав3 +120)вм + (ав3 + 120)} e~

f х/(х;в,а) dx = ß—-, >--i-^- (5.4)

0 1 j в(ав + 24)

Using expressions from (5.1), (5.2), (5.3), and (5.4), the mean deviation about mean, d1 (X) and the mean deviation about median, d2 (X) of QSD are obtained as

2 {qy + 86>y + 366V + 966 ^ + (a63 +120)} e'6

d ( x ) = ■

в (a в3 + 24)

(5.5)

l{e5M5 + 5в4Ы4 + 20в3М3 + 60в2М2 + (ав3 + 12й)вМ + (ав3 +1200} eeM

d ( X ) =-гт—Лл--------A (5.6)

в [а в3 + 24)

VI. Bonferroni and Lorenz Curves

The Bonferroni and Lorenz curves (Bonferroni [7] ) 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

1 q 1

B ( p ) =-J x f ( x ) dx = —

P/ 0 P

1q 1

and L ( p ) = —J x f ( x ) dx = —

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

0 q

¥ ¥ J x f ( x ) dx-J x f ( x ) d

M0 M

respectively or equivalently

1 p

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

1

p/

1 M

/-J x f ( x ) dx

M

J x f ( x ) dx

P»0

p

1

and L ( p ) = — | F ( x ) dx

№ 0

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

The Bonferroni and Gini indices are thus defined as

1

B = 1 -J B (p ) dp

(6.1) (6.2)

(6.3)

(6.4)

(6.5)

R. Shanker, R. Upadhyay, K. K. Shukla RT&A, No 3 (69) A QUASI SUJA DISTRIBUTION_Volume 17, September 2022

1

and G = 1 - 2 J L (p) dp (6.6)

0

respectively.

Using the pdf of QSD, we have

¥ {qy + 5q Y + 20qy + 60вУ + {ав3 +120) в q + (а в3 +120)} e-eq

f xf {x; в,а) dx = ±-—J-г--i-^- (6.7)

\ У } в (а в + 24)

Now using equation (6.7) in (6.1) and (6.2), we get

B (p) = -p

{q5q5 +5#y +20#y + 60#y + (ав3 +120)qq + (aq3 +120)} e

(ав3 +120)

(6.8)

\d5q5 + 594q4 + 20d3q3 + 6092q2 +(a93 +120) 9 q + (a 93 +120)} e~eq

L( p)=1 a+rn)

Now using equations (6.8) and (6.9) in (6.5) and (6.6), the Bonferroni and Gini indices are obtained as

(6.9)

{qy + 594q4 + 20qy + 60qy + (a 93 +120) q q + (a 93 +120)} e

B = 1--T—;-\-

(aq3 +120)

2 {q5q5 + 5qy + 2093q3 + 60qy +(a93 +120) q q + (a 93 +120)} e

-eq

qq

G =

(aq3+120)

-1

(6.10)

(6.11)

VII. Order Statistics

Let Xj,X2,...,Xn be a random sample of size n from QSD. Let X^ <X(2) < ... <X(n)denote the corresponding order statistics. The pdf and the cdf of the k th order statistic, say Y = X^k) are given by

n!

f (У ) =

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

Fk(y ){1 - F (y )Г7 (y)

n!

! n-к f n - к

and

(к -1)!( n - к)!

Z ~ IH'^1 Сy)f(y)

i=0

( n

j=k l=0 V J 0

n-jj

F (у) = Z\, Fj У ){1- F(y)r = ZZl " / ("U)

V - J

J=k V J 0

respectively, for k = 1,2,3,..., n. Thus, the pdf and the cdf of the k th order statistics of QSD are obtained as

and

( n\û4 (a + ûx4 ) & in - кЛ ( t

f (У (aû3 + 24)(к -1)\(n - к)\&è l ) (-1)

û4 x4 + 4û3 x3 + 12û2 x2 + 24ûx + (aû3 + 24)

1--3---

aû + 24

.-ex

к+l-1

^ ( * )=ZZ

j=k l=0

n n- j œ n v n - j ö 1 /j) ( -1)1

è j

в4 x4 + 4в3 x3 + 12в2 x2 + 24вх + (ав3 + 24 ) ав3 + 24 '

/вх

j+l

VIII. Stochastic Orderings 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) > hY (x) for all x

(iii) mean residual life order (X <mrl Y) if mX (x) £ mY (x) for all

x

(iv) likelihood ratio order (X £lr Y) if

fx ( * ) f ( * )

decreases in x. The following results due to Shaked

and Shanthikumar [8] are well known for establishing stochastic ordering of distributions

X £lrY ^ X <hrY ^ X <mrlY

u

(8.1)

or

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

Theorem: Let X ~ QSD (01,a1) and Y ~ QSD (02,al). Ifq > 02 and a1 = a2,

a > a2 and 01 = 02 then X £lr Y and hence X <hr Y, X <mrl Y and X <stY. Proof: We have

^_q4(a2q23 + 24)(a q 1 -q

fY ( *)

Now

в4 (aq3 + 24 ) èa +в2 x q4 (a2q23 + 24) " q {atf + 24)

ln Щ = ln

fY ( x)

+ ln

\x >0

œ a +q x4 ö èa2 + 02 X 0

- (01 -в2 У

This gives

d ln -

4 («201 -aA )

x

dx f ( x) « +01 x4 )(a2 + 02 x4 )

- (01 -02 )

d f ( x

Thus for 01 >02 and a1 = a2, or a1 >a2 and 01 = 02, —ln-^

dx f ( x)

< 0. This means that

X <,„ Y and hence X <hr Y, X <,r, Y and X <t Y.

mrl

IX. Renyi Entropy Measure

An entropy of a random variable X is a measure of variation of uncertainty. A popular

entropy measure is Renyi entropy [9]. If X is a continuous random variable having pdf f (.), then Renyi entropy is defined as

T

J- D

= log Ц fg (x) dx} , where g> 0 and gФ1.

Thus, the Renyi entropy of QSD can be obtained as

e4g

TR (g) = T^log

1 -g

!

(aq3 + 24);

-(a+qx4))

-вух

dx

1

1 -g 1

log

J

q4gag

(aq3 + 24)g

i q

1 + — X

a

Xdx

1 -g 1

1 -g 1

log

log

e4gag ^(g^fq

и

(aq3 + 24 )g j01 j 0

' л q4gagq

-x4 I e"

Xdx

a

* I g0

j=0 V j

(aq3 + 24)g

a1 0

Je~qgxx4]dx

1 -g

log

* (j 0

j=0 \J0

\ Q4g+ JagS

(aq3 + 24)

■i

e qgxx4j+1-1dx

1

1 -g 1

1 -g

log

log

jjfg^ q4g+]ag-] G (4j +1)

=0 j

(aq3 + 24)g (qgg

4 j+1

gg q4g-3y-iag-j G (4J +1)

=0 ^ J 0

(aq3 + 24)g (g)

4 j +1

X. 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 applied to it exceeds the strength, 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 in statistical literature it is known as

stress-strength parameter. 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 are independent strength and stress random variables having QSD with parameter (Q,a1) and (02,a2) , respectively. Then, the stress-strength reliability R of QSD can be obtained as

¥ ¥ R = P(Y < X) = JP(Y < X | X = x)fX (x)dx = J f (x;0a) F(x;02,a2)dx

= 1-

40320 qq24+20160 QQ23 (Q +Q2 )+8640Qq22 (Q +Q2 )2 + 2880Qq2 (Q +Q2 )3

+24 (aß24 + a2QQ23 + 24Q1 ) (Q1 + Q2 )4 + 24aQ23 (Q1 + Q2 )5 + 24aQ22 (Q1 + Q2 )6 +24aQ2 (Q1 + Q2 )7 + (a1a2Q23 + 24a1 ) (Q + Q2 )8

(aQQ + 24 )(a2Q23 + 24) (Q +Q2 )9

XI. Estimation of Parameters

In this section, the method of moments and the method of maximum likelihood for estimating parameters of QSD have been discussed.

I. Method of Moments

Since QSD has two parameters to be estimated, the first two moments about origin are required to estimate its parameters. We have

^ 2 (ad3 + 360)(ad2 + 24 )

= k (Say)

(m )2 K +120)2

Taking b = a03, above equation becomes 2 (b + 360)( b + 24) (b +120)2 2 (b2 + 384b + 8670)

b2 + 240b + 14400 " ( k - 2) b2 +(240k - 768) b + (14400k-17340) = 0 (11.1.1)

Now, for real root of b, the discriminant of the above equation should be greater than and equal to zero. That is

(240£ - 768)2 - 4(k - 2)(14400k -17340) > 0 ^ k £ 2.45.

r

m

This means that the method of moments estimate is applicable if k = —< 2.45, where

(1)

m2 is the second moment about origin of the dataset. Now taking b = a03 and equating the

population mean to the sample mean, we get the moment estimate 0 of 0 as

a03 +120 b +120 _ ~ b +120

= x ^ 00 =

0(a03 + 24) 0(b + 24) (b + 24) x'

Using the moment estimate of 0 in b = a03, we get the moment estimate cc of a as b b (b +124 )3 (x )3

a = ■

(6-)3 (b + 120)3

Thus the method of moment estimates a,) of parameters (0, a) of QSD are given by

(ô,àt ) =

b +120 b (b +124)3 (X)3 ^ (b + 24) x ' (b +120)3

where b is the value of the quadratic equation

(11.1.1).

II. Method of Maximum likelihood

Let x2, X3,..., Xn ) be a random sample of size n from QSD (Q,a). Then the likelihood function of QSD is given by

L =

a03 + 24

^^ (a + Q x 4 ) e~ nQx, where x is the sample mean.

' i=1

The log-likelihood function is thus obtained as

logL = n[4log0-log(a63 + 24)] + jjlog(a + 0xi4)-n0x .

i=1

The maximum likelihood estimates (0,a) of parameters (Q, a) are the solution of the following log-likelihood equations

S log L 4n 3n6a ^

5 =——+z-

x.

se

0 ae3 + 24 t! a + ex4

■-n x = 0

ô log L -n03

n

■I-

1

= 0

da ad3 + 24 tl a + 6xi 4 These two log-likelihood equations do not seem to be solved directly. We have to use Fisher's scoring method for solving these two log-likelihood equations. We have

S2 logL -4n 3na(aO4 -48O)

ÔO2

Z-

x

(aO3 + 24) «=i (a + Oxt4 )

52 log L

nd6

1

5a (a03 + 24) ¿=1 (a + 0xt 4 ) 52 log L -72n02 ^ xt4 d2 log L

-z-

50 da (a03 + 24)2 tl (a + 0xt4 )2 dadq ' The following equations can be solved for MLEs (3, a) of (0,a) of QSD

d2 ln L d2 ln L "d ln L "

d02 d0da 0-00 d0

d2 ln L d2 ln L a -a0 d ln L

d0da da2 0=00 _ da J0

where d0 and a0 are the initial values of 0 and a, respectively, as given by the method of

moments. . These equations are solved iteratively till sufficiently close values of Q and a are obtained.

R. Shanker, R. Upadhyay, K. K. Shukla RT&A, No 3 (69) A QUASI SUJA DISTRIBUTION_Volume 17, September 2022

XII. A Simulation Study

In this section, a simulation study has been carried out using R-software. Acceptance and rejection method is used to generate random number, where sample size, n = 40,60,80,100, value of q = 0.1,0.5,1.0,1.5 & (a = 1, a = 2) have been used for calculating Bias error (BE) and MSE (Mean square error) of parameter 6 and a which are presented in tables1 &2 respectively.

Table 1. BE and MSE for 6 and a at a =1

Sample в BE( в )(MSE( в )) BE(a) ( MSE (a))

40 0.1 0.026829 (0.02879335) 0.051492(0.11741)

0.5 0.0168297(0.0113295) 0.0291919(0.034087)

1.0 0.0004329(0.0007498) 0.0041920(0.0007029)

1.5 -0.0081702(0.0026701) -0.0208079(0.0173187)

60 0.1 0.01656832(0.0164705) 0.0363314(0.0791985)

0.5 0.0099016(0.00588256) 0.0196648(0.0232022)

1.0 0.0015683(0.00014757) 0.00299813(0.000539)

1.5 -0.0067650(0.0027459) -0.0136685(0.0112097)

80 0.1 0.01079101(0.00931567) 0.0619808(0.30733037)

0.5 0.00579101(0.00268286) 0.04948088(0.1958686)

1.0 -0.00045898(0.00001685) 0.03698088(0.1094068)

1.5 -0.006708987(0.0036008) 0.02448088(0.0479450)

100 0.1 0.008494785(0.007216137) 0.045858(0.21029559)

0.5 0.004494785(0.002020309) 0.035858(0.1285795)

1.0 -0.000505214(0.0002552) 0.025858(0.0668636)

1.5 -0.005505214(0.00303073) 0.015858(0.0251476)

Table 2. BE and MSE for в and a at a =2

Sample в BE (MSE) BE ( MSE) for alpha

40 0.1 0.01796973(0.0129164485) 0.17092052(1.1685530)

0.5 0.00796973 (0.0025406641) 0.14592052 ( 0.8517119)

1.0 -0.00453027(0.0008209337 ) 0.12092052 (0.5848709)

1.5 -0.01703027 (0.0116012033) 0.09592052(0.3680299)

60 0.1 0.016696288(0.0167259624) 0.031357314(0.058996870)

0.5 0.010029622( 0.0060355985) 0.014690648(0.012948908)

1.0 0.001696288 (0.0001726436) -0.001976019(0.000234279)

1.5 -0.006637045 (0.002643022) -0.018642686(0.020852983)

80 0.1 0.0107846385(0.0093046743) 0.05327416(0.22705089)

0.5 0.0057846385(0.0026769634) 0.04077416(0.13300257)

1.0 -0.000465361(0.0000173249) 0.02827416 (0.06395425)

1.5 -0.006715361(0.0036076864) 0.01577416(0.01990593)

100 0.1 0.0086072384(0.007408455) 0.037134273(0.137895421)

0.5 0.0046072384(0.002.122665) 0.027134273(0.073626875)

1.0 -0.000392761(0.0001.542617) 0.017134273(0.029358330)

1.5 -0.0053927616(0.002908188) 0.007134273(0.005089785)

It is obvious from tables 1 and 2 that as the sample size increases, both the BE and MSE decreases.

XIII. Applications

In this section, the goodness of fit of QSD has been discussed and compared with one parameter life time distributions including exponential distribution, Lindley distribution introduced by Lindley [10], Akash distribution proposed by Shanker [11], Suja distribution and two-parameter lifetime distributions including quasi Lindley distribution (QLD) of Shanker and Mishra [12] and Quasi Akash distribution of Shanker [13]. The pdf and the cdf of these distributions are presented in the table 3.

Table 3: pdf and the cdf of one parameter and two-parameter distributions

Distributions Pdf Cdf

QLD f ( x;0,a)= 9 (a + 9x ) e ~9x f ( -.M-1 -ç 1+a+i )

QAD f ( xq,a) = -q— (a + dx2 ) e"qx a0+2V ' , , œ dxidx+2)ö q F(x.6,a) = 1 1 + V ' e-0x V ' è aQ+2

LD f ( xq) = -^(1 + x)e-q F ( xq) = 1- , 0x ù _gx 1 +- e _ q+1j

AD f ( xq)=q (i+x2 ) F ( x; q) = 1 - è 1| qx (qx + 2) q2+2 )

Exponential f ( x;0) = 0e-qx F ( x; 6) = 1 - e-8x

The following dataset table 4 regarding the failure times of 50 electronic components which are extreme skewed to the right available in Murthy et al [14] has been considered to test the goodness of fit of the considered distributions.ML estimates of parameters of the considered distributions along with the values of — 2 log L, AIC, K-S and p values are presented in table 5. The fitted plots of the considered distributions for dataset in table 4 are shown in figure 9.

Table 4: Failure times data of 50 electronic components

0.036 0.058 0.061 0.074 0.078 0.086 0.102 0.103 0.114 0.116

0.148 0.183 0.192 0.254 0.262 0.379 0.381 0.538 0.570 0.574

0.590 0.618 0.645 0.961 1.228 1.600 2.006 2.054 2.804 3.058

3.076 3.147 3.625 3.704 3.931 4.073 4.393 4.534 4.893 6.274

6.816 7.896 7.904 8.022 9.337 10.940 11.020 13.880 14.730 15.080

Table 5: ML estimates of the parameters of the considered distributions along with values of — 2 log L, AIC, K-S and p-value

Distributions ML parameters - 2 logL AIC K-S p-value

q a

QSD 0.63822 179.0985 182.40 186.40 0.159 0.142

QLD 0.30731 35.0346 220.70 224.70 0.972 0.000

QAD 0.3680 44.9617 218.13 222.13 0.271 0.000

LD 0.59603 197.71 199.71 0.283 0.000

SD 1.12234 317.17 319.17 0.441 0.000

Exponential 0.29913 220.68 222.68 0.284 0.000

0 5 10 15

x

Figure 9 : Fitted plot of considered distributions on dataset

It is obvious from the goodness of fit in table 5 and the fitted plots in figure 9 that the QSD gives much closer fit as compared to other considered distributions for the failure time data of electronic components.

XIV. Conclusion

A two-parameter quasi Suja distribution (QSD) of which Suja distribution is a special case has been suggested. The proposed distribution is useful for extreme right skewed data. Its moments, skewness, kurtosis, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, Renyi entropy measures, and stress-strength reliability have been derived and studied. Method of moments and maximum likelihood estimation has been studied for estimating parameters. A simulation study has been presented. The goodness of fit of QSD has been presented with failure time data and the fit shows that QSD is the best distribution among the considered distributions.

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