POWER WEIGHTED SUJATHA DISTRIBUTION WITH PROPERTIES AND APPLICATION TO SURVIVAL TIMES OF PATIENTS OF HEAD AND NECK CANCER Текст научной статьи по специальности «Электротехника, электронная техника, информационные технологии»

POWER WEIGHTED SUJATHA DISTRIBUTION WITH

PROPERTIES AND APPLICATION TO SURVIVAL TIMES OF PATIENTS OF HEAD AND NECK CANCER

Rama Shanker

Department of Statistics, Assam University, Silchar, Assam, India Shankerrama2009@gmail.com

Kamlesh Kumar Shukla

Department of Community Medicine, Noida International Institute of Medical Sciences, Noida International University, Gautam Budh Nagar, India

Kkshukla22@gmail. com

Abstract

In this paper a power weighted Sujatha distribution, which includes power Sujatha distribution, weighted Sujatha distribution and Sujatha distribution as particular cases, has been proposed. Its statistical properties including behavior of density function, moments, hazard rate function, and mean residual life function have been discussed. Estimation of parameters has been discussed using the method of maximum likelihood. A simulation study has been presented to know the performance of maximum likelihood estimates of parameters. Application of the proposed distribution have been explained with a real lifetime data relating to patients suffering from head and neck cancer and goodness of fit shows quite satisfactory fit.

Keywords: Sujatha distribution, Weighted Sujatha distribution, Power Sujatha distribution, Hazard rate function, Mean residual life function, Maximum Likelihood estimation

1. Introduction

It has been observed that the survival times of patients suffering from head and neck cancer needs special consideration to find a suitable distribution which can be used to model the data. During recent decades several one parameter, two-parameter and three-parameter lifetime distributions have been proposed in statistics literature to model survival times of patients suffering from head and neck cancer and observed that all proposed distributions are not very much suitable due to theoretical or applied point of view. It has been observed that, in general, the survival times of patients suffering from head and neck cancer are stochastic in nature and while discussing the goodness of fit of several one parameter , two-parameter and three-parameter well-known distributions which were earlier proposed by different researchers that these distribution does not give good fit.

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PWSD WITH PROPERTIES & APPLICATION TO SURVIVAL TIMES... Volume 18, September 2023

Shanker [1] proposed a one parameter Sujatha distribution having its probability density function

(pdf) and cumulative distribution function (cdf) as

93

fi(y;9) = 82 + 8 + 2il+y+y2)e-9yy>°-d>° (11

f3(x;9,a)=-^^xa-1(1 + xa+x2a)e-0x ;x > 0,9 > 0,a > 0 (1.5)

F3 (y; 9, a) = l2- [l + 9yy272)] e-eya; y > 0,9 > 0,a > 0 (1.6)

Fi(y;9) = 1-[l + 9ylZ+e9+T] e-9y, y > 0,9 > 0(1.2) (12)

Shanker and Shukla [2], taking a weight function xa-1in (1.1), proposed a two-parameter weighted Sujatha distribution (WSD) defined by its pdf and cdf

na+2 -,,a-i

f2 (y; 9, a) = " -(1 + y + y2)e-0y;y >0,9>0,a>0 (1.3)

J2v • J 02 + a0 + a(a+1) r(a) v J J J J v '

( . {02 + a0 + a(a+1)}r(a,0y) + (0y)a(0y+0 + a+1)e-ey F2(y;9,a) = 1--{02+ae+a(a+1)]r(a)-, (1.4)

where r(a,z) = f e-yya-1dy;y > 0, a > 0 is the upper incomplete gamma function.

1

Shanker and Shukla [3], taking a power transformation x = yP in (1.1), proposed a two-parameter power Sujatha distribution(PSD) defined by its pdf and cdf

a03 02+0+2'

0ya(0ya+0 + 2)~\

3(y;U,a) = 1-[1+ 02 + 0+2

Ghitany et al. [4] proposed a two-parameter weighted Lindley distribution (WLD) having parameters 9 and a defined by its probability density function (pdf) and cumulative distribution function (cdf)

na+i ,,a-i

f4(y; 9, a) = 0—1y;-T) (1 + y)e-0y; y>0,9>0,a>0 (1.7)

F4(y;9,a) = 1 ^(0l^)г(^ey\l(0y)ae-ey .y > 0,9 > 0,a > 0, (1.8)

' (0 + a)r(a) J v '

where r(a) and r(a,z) are the complete gamma function and the upper incomplete gamma function. Its structural properties including moments, hazard rate function, mean residual life function, estimation of parameters and applications for modeling survival time data has been discussed by Ghitany et al. [4]. Shanker et al. [5] discussed various moments based properties including coefficient of variation, coefficient of skewness, coefficient of kurtosis and index of dispersion of weighted Lindley distribution and its applications to model lifetime data from biomedical sciences and engineering.

Ghitany et al. [6] proposed a power Lindley distribution (PLD) having parameters 9 and a defined by its pdf and cdf

f5(y; 9, a) = -^ya-1(1 + ya)e-0ya;y > 0,9 > 0,a > 0 (1.9)

F5(x; 9, a) = 1- [1+e^e-0xa;x > 0,9>0,a>0 (1.10)

Note that the PLD is a convex combination of Weibull (a, 9) and a generalized gamma

0

(2, a, 9)distribution with mixing proportion—. Ghitany et al. [6] has discussed the properties of PLD including the shapes of the density, hazard rate functions, moments, skewness and kurtosis measures, estimation of parameters using maximum likelihood estimation and application to model a real lifetime data from engineering. Recall that at a = 1 both WLD in (1.7) and PLD in (1.9) reduce to Lindley distribution introduced by Lindley [7] having pdf and cdf

0

f6(y;9)=-^-(1+y)e-0y;y>0,9>0 (1.10)

0+1

F6(y;9) = 1- [1+-0L]e-0y;y > 0,9 > 0 (1.11)

Ghitany et al. [8] have discussed its various statistical and mathematical properties and application. Shanker et al. [9] have detailed study on modeling of lifetime data using both exponential and

Rama Shanker and Kamlesh Kumar Shukla RT&A, No 3 (74)

PWSD WITH PROPERTIES & APPLICATION TO SURVIVAL TIMES... Volume 18, September 2023

Lindley distributions and observed that there are many lifetime data where exponential

distribution gives better fit than Lindley distribution.

Zakerzadeh and Dolati [10] have introduced a three parameter generalized Lindley distribution (GLD) having pdf and cdf given by

na+1 Va-1

na,1 .— i

f7(x-,9,a,p) = y—^-)(a + px)e-ex;x> 0,8 > 0,a> 0,fi > 0 (1.12) F7(x;9,a,p) = i+.W+WW+Wf'-» ;x >0,8>0,a> 0,fi > 0 (1.13)

Lindley distribution, gamma distribution and weighted Lindley distribution (WLD) are particular cases of (1.9) at (a = fi = 1), (fi = 0) and (fi = a), respectively. Shanker [11] obtained various raw moments and central moments of GLD and discussed properties based on moments including coefficient of variation, skewness, kurtosis and index of dispersion of GLD and its comparative study with generalized gamma distribution (GGD) introduced by Stacy [12] to model various lifetime data from engineering and biomedical sciences and concluded that in many cases GGD gives much better fit than GLD. Shanker and Shukla [13] have detailed comparative study on modeling of real lifetime data from engineering and biomedical sciences using GLD and generalized gamma distribution (GGD) introduced by Stacy (1962) [12] and concluded that there are several lifetime data where GGD gives much better fit than GLD.

In the present paper, a three - parameter power weighted Sujatha distribution (PWSD) which includes PSD, WSD and Sujatha distribution as particular cases, has been proposed and discussed. Its raw moments been obtained. The survival function and the hazard rate function of the distribution have been derived and their shapes have been discussed for varying values of the parameters. The estimation of its parameters has been discussed using maximum likelihood method. Finally, the goodness of fit and the application of the distribution have been explained through a real lifetime data relating to patients suffering from head and cancer and the fit has been compared with other one parameter, two-parameter and three-parameter lifetime distributions.

2. Power weighted Sujatha distribution

i

Assuming the power transformation X = YP in the pdf of WSD (1.3), the pdf of the random variable Xcan be obtained as

f8(x-,e,a,B) =—^—^-^(l+xP +x2P)e-0x^;x > 0,9 > 0,a >0,fi >0 (2.1)

We would call the distribution in (2.1) as the power weighted Sujatha distribution (PWSD) and we denote it as PWSD(0,a,fi). It can be easily verified that the PWSD(0,a,fi) in (2.1) reduces to Sujatha distribution, WSD and PSD for (a = fi = 1),(fi = 1) and (a = 1), respectively. It can be easily verified that PWSD is a convex combination of generalized gamma distributions with different parameters, namely GGD (6, a, fi), GGD (6, a + 1,fi) and GGD (6, a + 2,fi). That is

f5(x; 8,a,p) = pigi(6,a,fi) + P2g2(8,a + 1,fi) + (1-Pi~ P2)g3(8,a + 2,fi),

Where

62 ad

Pi = P2

62 + a6 + a(a+1)' ^2 02 + a0 + a(a+1)

g1(9,a,p) =^-^xiSa-1e-Bx-,x> 0,9 > 0,a > 0,0 > 0

BQa+i

g2(6,a + 1,0) = rfia + -q xP(a+1)-1e-9xP; x > 0,8 > 0,a >0,fi>0

ona+2

g3(8,a + 2,fi) = ^ + ^ xP(«+2)-1e-exil; x > 0,8 > 0,a >0,fi>0

Graphs of density function of PWSD for varying values of parameters 8, aandfi have been drawn and presented in figure 1. From the figure 1, It was observed that pdf of PWSD is increasing with

increased value of theta at fixed value of alpha and beta, whereas its value is decreasing with increased value of alpha at fixed valued of theta and beta.

Figure 1: Graphs of the pdf of PWSD for varying values of parameters^, aandfi

3. Moments

The th moment about origin of PWSD (2.1) can be obtained as

Vr' = ,a X xpa+r-1 {1 + XV+ x2^)e-ex"dx

ÛÛ&+2 I" rœ pœ pœ

P I e-exPx^a+r-1 dx + I e-6xP xPa+P+r-1 dx + I e-9xPx^a+2^+r-1 dx

Jo Jo Jo

{ e + ad + a(a + 1)}r(a)

Assuming и = , we have du = в/Зх^ 1dx, x = (^)^and dx =

du

p-1

Thus, we have

H-r

eu

{в + ав + а(а+1)}Г(а)

ва+т

+-

:-1 J0

Г „

a+T1du +

ea+P

rœ

fo e

,, a+t+l-l ,

-Uu P du

в^Р

rm a+^r+2-1 ,

I e Uu p du

o

ва

{в + ав + а(а+1)}Г(а) ва+1

{в + ав + а(а+1)}Г(а)

г

ва+т1

, г(а^+1) , r(a+J+2)

+ Г +

ва+р

г

ва+Р+1

в2 r(a+jj) + er(a+jj+1)+r(a+^+2)

г

ва+т+1

e2+e(a^)+(a^)(a+r1)r(a+Tj)

вР{в + ав + а(а+1)}

в2р2 + вр(ар+г) + (ар+г)(ар+р + г) '

Г (а.)

1,2,3,.

_Ц

Г(а)

(3.1)

9 + a9 + a(a+ï)}

Substituting r = 1,2,3 and 4 in (3.1), the first four moments about the origin of PWSD can be obtained. Again using the relationship between moments about origin and central moments, central moments can be obtained. Since the expressions for central moments are complicated, central moments are not being given.

i

1

i

r

r

r

4. Hazard Rate Function The survival function 5(x; 8, a, fi)of PWSD can be obtained as S(x;9,a,fi) = P(X>x) = J" f5(t; 9, a, fi) dt

=--j" tf3a_i (i + tp + t2H)e-0tP dt

fiea+2

{ в + ав + а(а + 1)}Г(а)

Jrœ рЖ рЖ

t?a-1e-etPdt+ I tPa+P-1 e-etP dt + I tPa+2P-1 e-eti! dt

X J X J r

Taking и = and t = (u) Ogives dt = and thus we have

du

J-

Pu P

S(x\6,a,p) =

{ в +ав + а(а + 1)}Г(а)

ва+2

l

r

œ а Ё2-1 du fœ n P«+P-1 du е-вии P I е-вии P

r

P-1 и P

Pa+2P-1 du

+ I е-вии P

P

P

I

j y

p-1 и P

œ

p-1 и p

{в + ав + а (а + 1)}Г(а)

ва+2

I

j y

œ pœ pœ

e-8uua-1du + I e-8uuadu + I e-8uua+1du

J vfi J yI

{в + ав + а (а + 1)}Г(а)

\_JXP J XP J XP

Г(а, вх?) е-вхР(вх^)а + аГ(а.вх^)

+

ва ва+1 е-вх^(вхР)а(вхР + а + 1) + а(а + 1)Г(а,вхИ)

в2г(а,вхР)+в{е-вхР (вхР)а+аГ(а,вхР)}+е-вхР (вхР)а (вхР + а+1)+а(а+1)г(а,вхР)

[в + ав + а(а+1)]Г(а)

_ {в2 + ав + а (а + 1)}Г(а,вхР) + ве-вхР (вхР)а + е-вхР (вхР)а(вхР + а + l) = {в + ав + а (а + 1)}Г(а)

\в2 + ав + а(а+1)}г(а,вхР)+е-вхР(вхР + в + а+1)(вхР)а . ^г - .

= 1-—т----———, where Г(а, вхР ) is the

{в + ав + а(а+1)}Г(а) v J

upper incomplete gamma function defined as Г(а,вхР) = j™xРyа-1е-Уdy;а> 0,вхР > 0. It can be easily verified that at (p = 1),(а = 1)and (а = p = 1), the survival function of PWSD reduce to the survival function of WSD, PSD and Sujatha distribution, respectively.

Rama Shanker and Kamlesh Kumar Shukla RT&A, No 3 (74)

PWSD WITH PROPERTIES & APPLICATION TO SURVIVAL TIMES... Volume 18, September 2023 Thus the cdf of PWSD can be obtained usingF5 (x; e,a,ß) = 1 — S(x; в, a, ß). The hazard rate function, h(x; в, a,ß), of PWSD can be given by

h(x; e,a,ß) =

f5(x:e,a,ß) S(x-fi,a,ß)

ßea+2xßa-i(1+xß + x2ß)e-exß

{02 + a0 + a(a+1)}r(

Graphs of h(x; 9, a,fi)for varying values of parameters 9, aand/3 are shown in figure 2.

Figure 2: Graphs of hazard rate function of PWSD for varying values of parameters9, aandp.

5. Mean Residual life function

The mean residual life function of PWSD can be obtained as

1 rm

[G2 + aG + a (a + 1)}r(a,Gxp) + e-9xJ(Gxp + G + a + 1)(Gxp)a e-9tJdt — x

Jr OO

tpa (i + tp + t2P)e-etí¡

X

pec

[G2 + ad + a (a + 1)}r(a,Gxp) + e-9xJ(Gxp + G + a + 1)(Gxp)a

e-etP tP*dt + I e-9tJ tpa+pdt + I e-9tJ tpa'

X X X

2 P d

- x

Taking u = tPand t = (u)pgives dt = dp_t and thus we have

du

J-

Pu J

m(x; G, a,p) =

f

Jxl

[G2 + aG + a (a + 1)}r(a,Gxp) + e-9xJ(Gxp + G + a + 1)(Gxp)a

JxJ

e*uU~p-1dU + fe-»uU"'P"-1dU+l e->uu"f"-ldu

J

J

Gc

[G2 + aG + a (a + 1)}r(a,Gxp) + e-9xJ(Gxp + G + a + 1)(Gxp)a ( a + 1,Gxp) r(a + 1+1,Gxp) r(a+1 + 2,Gx

1

Ga+p

+

1

Ga+p+1

+

1

Ga+p+2

92r(a+1,9xJ)+9r(a+1+1,9xJ)+r(a+1+2,i 9J[92 + a9 + a(a+1)}r(a,9xJ)+e-exJ(9xJ + 9 + a+l)(9xJ)a

■ — x.

X

X

G

X

X

Graphs of mean residual function of PWSD for varying values of parameters 9, a andfl have been drawn and presented in figure 3. From the figure 3, It was observed that mean residual of PWSD is decreasing with increased value of theta at fixed value of alpha and beta.

Figure 3: Graphs of mean residual life function of PWSD for varying values of parametersd, aandfi.

6. Maximum Likelihood Estimation of parameters

Suppose (x1,x2,x3,...,xn) be a random sample of size n from PWSD (2.1). The natural log likelihood function is thus obtained as

n

InL = ^ lnf8 (xi; 8, a,fi)

i=i

= n[lnfi + (a+ 2) Ind - ln(e2 + ad + a2 + a) - InT (a)] + (fia - 1) Yn=i ln(*i)

+ in=iin(i + xiP + x2)-ezn=iXip.

The maximum likelihood estimates (MLEs) (6,a,j) of parameters (6,a,j) of PWSD are the solution of the following nonlinear log likelihood equations

dlnL n(a + 2)

n(2S + a)

dS

s

S2 +aS + a2 +a

-^> = 0

dlnL da

= n n e

n(S + 2a + 1) S2 + aS + a2 + a

nip (a) + ß^ ln(Xi) = 0

i=i

d n L

=n+a^lnxi+^

=1 =1

xi>(l + 2xi>)ln(xi)

1 + xi> + xi2>

e^x/ ln(xi) = 0

=1

where ip(a) = —lnT (a) is the digamma function.

These three natural log likelihood equations do not seem to be solved directly, because they cannot be expressed in closed forms. The (MLE's) (9,a,ß) of parameters (S,a,ß) can be computed directly by solving the natural log likelihood equation using Newton-Raphson iteration method available in R-software till sufficiently close values of S, a and ß are obtained. The initial values of parameters S, a and ß are taken as S = 0.5,a = 0.5 and ß = 1.5.

n

1=1

n

n

7. A Simulation Study In this section, a simulation study has been carried to check the performance of maximum likelihood estimates by taking sample sizes (n = 40,80,120,160) for values of 6 = 0.5,1.0,1.5,2.0 and a = 0.5 jj = 1.0. Similarly a = 0.5,2.0,4.0,5.0&6 = 1.5, j = 1.0, and j = 0.1,0.5,1.5,2.0 & 6 = 1.5, a = 2.5. Acceptance and rejection method is used to generate random number for data simulation using R-software. The process were repeated 1,000 times for the calculation of Average Bias error (ABE) and MSE (Mean square error) of parameters 6, aand jare presented in tablesl, 2 &3 respectively.

Tablel. ABE and MSE ofMLE parameters 6,aand j for fixed valueof a = 0.5, j = 1.0

n S ABE (Si) MSE(§) ABE (a) MSE (a) ABE(ß) MSE(ß)

40 0.5 0.0183 0.0134 -0.0909 0.0005 0.0249 0.0495

1.0 0.0058 0.0014 -0.0409 0.0669 0.0198 0.0316

1.5 -0.0066 0.0017 -0.0909 0.3305 0.0074 0.0044

2.0 -0.0192 0.0147 -0.1159 0.5374 0.0248 0.0495

80 0.5 0.0131 0.0138 -0.0465 0.0006 0.0248 0.0495

1.0 0.0068 0.0038 -0.0215 0.0369 0.0198 0.0316

1.5 0.0006 0.0003 -0.0464 0.1729 0.0073 0.0043

2.0 -0.0056 0.0025 -0.0589 0.2784 0.0011 0.0001

120 0.5 0.0132 0.0210 -0.0295 0.0002 0.0112 0.0151

1.0 0.0091 0.0098 -0.0129 0.0199 0.0078 0.0074

1.5 0.0049 0.0028 -0.0296 0.10491 -0.0004 0.0002

2.0 0.0007 0.0000 -0.0379 0.1723 -0.0046 0.0025

160 0.5 0.0101 0.0164 -0.0219 0.0000 0.0080 0.0105

1.0 0.0070 0.0078 -0.0095 0.0144 0.0056 0.0049

1.5 0.0038 0.0024 -0.0219 0.0773 -0.0006 0.0007

2.0 0.0007 0.0093 -0.0282 0.0127 -0.0037 0.0023

Table2.ABE and MSE of MLE parameters 9, aand fi for fixed value ofd = 1.5,fi = 1.0

N a ABE(9) MSE (8) ABE (a) MSE (a) ABE(fi) MSE(fi)

40 0.5 0.0157 0.0099 -0.0903 0.0003 0.0594 0.1412

2.0 0.0033 0.0004 -0.04033 0.0650 0.0494 0.0977

4.0 -0.0092 0.0034 -0.0903 0.3263 0.0244 0.0238

5.0 -0.0217 0.0188 -0.1153 0.5320 0.0119 0.0056

80 0.5 0.0078 0.0049 -0.0451 0.0002 0.0297 0.0706

2.0 0.0016 0.0002 -0.0201 0.0325 0.0247 0.0488

4.0 -0.0046 0.0016 -0.0451 0.1632 0 .0122 0.0119

5.0 -0.0108 0.0094 -0.0576 0.2660 0.0059 0.0028

120 0.5 0.0071 0.0060 0.0293 0.0449 0.0185 0.0413

2.5 0.0029 0.0010 0.0168 0.0341 0.0152 0.0277

4.0 0.0293 0.1034 0.0012 0.0002 0.0068 0.0056

5.0 0.0377 0.1704 0.0054 0.0035 0.0027 0.0008

160 0.5 0.0056 0.0052 -0.0217 0.0000 0.0138 0.0305

2.5 0.0025 0.0010 -0.0124 0.0245 0.0113 0.0204

4.0 -0.0005 0.0005 -0.0217 0.0757 0.0051 0.0041

5.0 -0.0037 0.0022 -0.0280 0.1256 0.0019 0.0005

Table 3. ABE and MSE of MLE parameters 6, aand fi for fixed value ofd = 1.5, a = 2.5

N fi ABE(Q) MSE(G) ABE(a) MSE (a) ABE(fi) MSE(J3)

40 0.1 0.79609 25.3479 0.7246 26.3839 0.0121 0.0059

40 0.5 0.7835519 24.5581 0.7621 23.2352 0.0021 0.0001

40 1.5 0.7710519 23.7808 0.7246 21.0050 -0.0228 0.0208

40 2.0 0.7585519 23.0160 0.6996 19.5807 -0.0353 0.0499

80 0.1 1.24654 93.2320 1.2157 97.3935 0.0051 0.0015

80 0.5 1.2382 91.9897 1.2407 92.3639 -0.0015 0.0001

80 1.5 1.2298 90.7557 1.2157 88.6793 -0.0182 0.0198

80 2.0 1.2215 89.5300 1.1990 86.2645 -0.0265 0.0422

120 0.1 0.3259 12.7464 0.3011 13.0932 0.0025 7.7334

120 0.5 0.3217 12.4226 0.3136 11.8053 -0.0007 7.5789

120 1.5 0.3175 12.1029 0.3011 10.8831 -0.0091 9.9985

120 2.0 0.3134 11.7874 0.2928 10.2891 -0.0132 2.1209

160 0.1 0.1996 6.3794 0.1856 6.8906 0.0018 5.7328

160 0.5 0.1965 6.1813 0.1950 6.0855 -0.0006 5.8974

160 1.5 0.1934 5.9863 0.1856 5.5145 -0.0068 7.5232

160 2.0 0.1903 5.7944 0.1794 05.1495 -0.0099 1.5942

8. Applications

The applications and goodness of fit of the power weighted Sujatha distribution (PWSD) has been discussed for one real dataset reported by Efron [14] relating to the survival times of a group of patients suffering from Head and Neck cancer disease and treated using a combination of radiotherapy and chemotherapy (RT+CT).The goodness of fit is compared with exponential distribution, Lindley distribution, Weibull distribution introduced by Weibull [15], Gamma distribution, Generalized exponential distribution (GED) introduced by Gupta and Kundu [16] , Power Lindley distribution (PLD), Weighted Lindley distribution (WLD), Power Sujatha distribution (PSD), weighted Sujatha distribution (WSD), and Generalized Lindley distribution

(GLD ). The dataset is as follow:

12.20 23.56 23.74 25.87 31.98 37 41.35 47.38 55.46 58.36

63.47 68.46 78.26 74.47 81.43 84 92 94 110 112

119 127 130 133 140 146 155 159 173 179

194 195 209 249 281 319 339 432 469 519

633 725 817 1776

In order to compare the goodness of fit of the considered distributions for the dataset, values of —2 In L, AIC (Akaike information criterion), K-S Statistic ( Kolmogorov-Smirnov Statistic) and p-value for two datasets have been computed The formulae for AIC and K-S Statistics are as follows: AIC =—2 InL + 2k, and ^ — 5 = Sup|i^(x) — F0(x)|, wherekbeing the number of parameters

X

involved in the respective distributions, n is the sample size and /Vj(x)is the empirical distribution function. The best distribution corresponds to the lower values of—2 In L, AIC and K-S statistic. Note that the estimates of parameters of the considered distributions are based on maximum likelihood estimates. The initial values of the parameters for ML estimates of PWSD have been selected as 8 = 1.5, a = 0.5 and/3 = 1.5, as the log-likelihood function is non-linear. The ML estimate of parameters of the considered distributions for dataset is given in table 4. The goodness of fit by K-S statistics for dataset with considered distributions are presented in tables 5. From the goodness of fit given in table 5, it is crystal clear that PWSD gives much closure fit to dataset relating to survival times of patients suffering from head and neck cancer and hence it can be considered as an important distribution for modeling data relating to survival times of patients suffering from head and neck cancer. The variance-covariance matrix of the parameters (8, a,/) of PWSD for dataset is given in table 6. The survival plots for the dataset, fitted plot of cdf for given dataset, and the fitted plot of given dataset are presented in figures 4, 5 and 6 respectively.

Table 4: Summary of the ML estimates of parameters for dataset

Model ML Estimates

8 a 3

PWSD 26.9742 50.3682 0.1325

GLD 0.0047 0.0524 5.0750

PSD 0.1539 0.5690

WSD 0.0089 0.01637

WLD 0.00531 0.21191

PLD 0.05301 0.68893

GED 0.00482 1.09367

Gamma 0.00489 1.08501

Weibull 0.00710 0.92327

Lindley 0.00892

Exponential 0.00447

Table 5: Summary of Goodness of fit by K-S Statistic for dataset

Model -2 InL AIC K-S p-value

PWSD 555.41 561.41 0.104 0.572

GLD 564.09 570.09 0.150 0.248

PSD 559.45 563.45 0.135 0.512

WSD 579.96 583.96 0.350 0.000

WLD 565.91 569.91 0.161 0.181

PLD 560.78 564.78 0.118 0.529

GED 563.93 567.93 0.145 0.280

Gamma 564.10 568.10 0.149 0.249

Weibull 563.71 567.71 0.298 0.005

Lindley 579.16 581.16 0.219 0.025

Exponential 564.01 566.01 0.145 0.282

Table 6: Variance-covariance matrix of the parameters 9, aand fi of PWSD for dataset

Gafi

6 983.70956 1475.4914 -2.33334' a 1475.4914 2219.87726 -3.47289 2.33334 -3.47289 0.005661

Figure 4. Survival plots for the given dataset.

1.0 -

0.8 -

0.6 -

0.4 "

0.2 -

0.0 -

ecdf(x)

_ _ —*.....

| - Observed --PWSD

j

500

1000

Figure 5. Fitted plot of CDF on given dataset

CO

o

0 500 1000 1500

x

Figure 6. Fitted plot of pdf of PWSD on dataset

9. Concluding Remarks

In this paper a three-parameter power weighted Sujatha distribution (PWSD) which includes weighted Sujatha distribution (WSD), power Sujatha distribution (PSD) and Sujatha distribution as particular cases, has been proposed. Its statistical properties including behavior it density function, moments, hazard rate function, mean residual life function have been discussed. Estimation of parameters has been discussed using the method of maximum likelihood. A simulation study has been presented to know the performance of maximum likelihood estimates of parameters. Application of the proposed distribution have been explained with a real lifetime data which is related to patients suffering from head and neck cancer and its goodness of fit has been compared with other lifetime distributions. The proposed distribution shows better fit over the other one parameter, two-parameter and three-parameter distributions.

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