GENERALIZATION OF LENGTH BIASED WEIGHTED GENERALIZED UNIFORM DISTRIBUTION AND ITS APPLICATIONS Текст научной статьи по специальности «Математика»

GENERALIZATION OF LENGTH BIASED WEIGHTED GENERALIZED UNIFORM DISTRIBUTION AND ITS

APPLICATIONS

Jismi Mathew

Department of Statistics, Vimala College (Autonomous), Thrissur, Kerala, India

jismijy@gmail.com

Abstract

In this article, a generalization of length biased weighted generalized uniform distribution called Marshall Olkin length biased weighted generalized uniform distribution is introduced and studied. Some of the statistical properties of the new distribution such as hazard rate function, compounding, quantile function, moments, Renyi and Shannon entropies are discussed. The maximum likelihood estimation of the model parameters is done and a simulation study is conducted for confirming the validity of the estimates and also introduced a minification process with respect to the model and explored its sample path behaviour for different combinations of parameters. Further, the stress strength analysis is carried out and the estimate of the reliability is obtained based on a simulation study.

Keywords: Entropy, Length biased weighted generalized uniform distribution, Maximum likelihood method, Order statistics, Quantiles, Stress strength analysis.

1. Introduction

The theory of weighted distributions provides a collective access for the problems of model specification and data interpretation. Weighted distributions take into account the method of ascertainment, by adjusting the probabilities of the actual occurrence of events to arrive at a specification of the probabilities of those events as observed and recorded [14].

The uniform distribution is considered as the simplest probability model and is connected to all the distributions. Many characterizations and modifications of the generalized uniform distribution have been introduced and explored by various researchers [see, 19, 10, 8, 18]. Rather and Subramanian [16] introduced and studied the properties of length biased weighted generalized uniform distribution.

The probability density function and cumulative distribution function of length biased weighted generalized uniform distribution (LBWGU) are respectively, given by

9LBWGu(x;8,Y) = {e+)eC1; 0<x<y,9>-1 (1)

GLBWGU(X-, 8,Y) = (-) ; 0<x<r, e>-l (2)

Marshall Olkin [11] introduced a new family of distributions by inserting a new shape parameter to the existing family of distributions. Let G(x) be the cumulative distribution function (cdf) of a random variable X, then the cdf of the Marshall and Olkin family of distributions is

F(x) =-^--(3)

Jismi Mathew RT&A, No 2 (73) GENERALIZATION OF WEIGHTED UNIFORM DISTRIBUTION_Volume 18, June 2023

The corresponding pdf of (3) is given by

f(x) =_Mfl__(4)

' V J [l-(l-ß)(l-G(x))]2' V '

where ß > 0 is a shape parameter. Clearly, for ß = 1, we obtain the baseline distribution, i.e., F(x) = G(x).

Many authors have introduced various univariate distributions belonging to the Marshall-Olkin family of distributions such as Marshall-Olkin Weibull [5], Marshall-Olkin semi Burr and Marshall-Olkin Burr [7], Marshall-Olkin Frechet distribution [9], Marshall-Olkin generalized exponential distribution [17] and Marshall-Olkin extended generalized Lindley distribution [2]. Recently, introduced Marshall-Olkin form of additive Weibull distribution [1], reliability test plan for the Marshall-Olkin length biased Lomax distribution [12] and Marshall-Olkin length biased Maxwell distribution and its applications [13].

The rest of this paper is planned as follows. In section 2, the Marshall-Olkin length biased weighted generalized uniform (MOLBWGU) distribution is given, with plots of the pdf and cdf. The statistical properties of the new distribution are studied in section 3, including hazard rate function, moments, quantile function, compounding properties, order statistics and Renyi and Shannon entropies. Estimation of the model parameters are discussed in section 4. In section 5, the application of the distribution in time series analysis is discussed. In section 6, the stress strength analysis is carried out using a simulation study. Concluding remarks are presented in section 7.

2. Marshall-Olkin Length Biased Weighted Generalized Uniform Distribution

Let X follows length biased weighted generalized uniform distribution. A new distribution can be defined by inserting (2) in (3). The cdf obtained is

(1)8+2

0<x<Y. (5)

i-(i-ß)(i-{f)S+2}

Based on (5), the survival function of the MOLBWGU distribution can be expressed as

B(i-£)e+2)

SMOLBWGU (x) 6, V, P) = 7 0+2\ , 0<X<Y. (6)

i-(i-p\H5) )

where 0 > -1 and p > 0.

By putting (1) and (2) in (4), we obtain the pdf of the MOLBWGU distribution as

fuoLBWGu (X) 8, Y, p) = , (e+2)fir~°-2x°"2, 0<x<Y. (7)

1-(1-P)[i-{^) +2)

We refer to this new distribution as the generalization of length biased weighted generalized uniform distribution with parameters 8, y and p.

The shape of the pdf f(x; 8,y,p) depends on parameter p. If p £ (0,1) then the pdf is a bell

( 0 + 2) B

shaped function on (0, y) with f(0;8,y,p) = 0 and f(y,8,y,P) =-. In the case of p > 1 then

( 0 + 2) B

the pdf is an increasing function on (0,y) with f(0,9,y,p) = 0 and f(y,Q,y,P) =—+—.

Remark 1. If p =1, we obtain length biased weighted generalized uniform distribution introduced by Rather and Subramanian [16].

Remark 2. When 8 = -1 and p = 1, MOLBWGU distribution reduces to uniform distribution over (0,7).

Remark 3. When y = 1 and p = 1, MOLBWGU distribution reduces to standard power function distribution.

Figure 1: Curves of the pdfofthe MOLBWGU distribution for different values of the parameters.

Figure 2 : Curves of the cdf of the MOLBWGU distribution for different values of the parameters.

3. Statistical Properties This section is devoted to some statistical properties of the MOLBWGU distribution.

3.1. Hazard Rate Function

(9 + 2)ßy-e-2xe+1

The hrf is given by, hM0LBWCU (x;9,y,ß)=-

where, 0 < x <y.

For p £ (0,1) and P > 1, the hrf is evidently increasing failure rate.

0.0 0.2 0.4 0.6 o.s 1.0

Figure 3: Curves of the hazard rate function of the MOLBWGU distribution for different values of the parameters. The reverse hazard function of MOLBWGU(8, y, P) is given by,

(9 + 2)Py-e-2xe+1

rMOLBWGU(x> = '

(0 + 2)3

The reverse hazard rate function decreases with r(0,8,y,p) = 0 and r(y,8,y,p) =—+—.

3.2. Compounding

The property that Marshall-Olkin family of distributions can be expressed as a compound distribution with exponential distribution as mixing density is useful in obtaining new parameter families of distribution in terms of existing ones, expressed Marshall-Olkin extended forms of Weibull, Lomax, linear exponential and exponential power family of distributions as a compound distribution [see 5, 4, 9].

Theorem 1. Let X be a continuous random variable with conditional survival function on A = 5

expressed as F(xlS) = (l - fye+2) e-(1-p)S§e+2, 0<x<y,

and let A follows a distribution function with probability density function

m(S) = ft e'V5, S>0. Then the random variable X has the MOLBWGU (8,y, P) distribution. Proof: The unconditional survival function of the random variable X is given by, F(x) = F(xlS)m(8)dS

= Ki-fy9+2)j~ e-w+(i-n$e+2vdS

_ P(.i-$e+2) 1-(1-P)[1-(§f+2) .

which is the survival function of the MOLBWGU (8, y,P) distribution.

Theorem 2. Let [Xi, i > 1} be a sequence of i.i.d. random variables with common survival function G'(x). Let T be a geometric random variable independently distributed of [Xi, i > 1} such that P(T = n) = P(1 - P)n-1,n = 1,2, ...,0<p <1 . Let YT = min1^TXi . Then [YT} is distributed as MOLBWGU(6, y, P) if and only if [Xt} follows LBWGU(8,y) . Proof: The survival function of the random variable YT is H'(x) =P(YT> x)

= H=i P(Yn > х)Р(Т = n) = In=i [G(x)]nß(1-ß)n-1

ßC(x)

1-(1-P)G'(x)

_ P(1-$S+2)

which is survival function of MOLBWGU (9, y,P) distribution.

Theorem 3. Let [Xi, i > 1} be a sequence of i.i.d. random variables with common survival function G'(x). Let T be a geometric random variable independently distributed of [Xi, i > 1} such that P(T =

n) = P(1-P)n-1,n = 1,2,...,0 < p <1 . Let ZT=max1^TXi . Then [ZT} is distributed as

1

MOLBWGU(9, y, p if and only if [Xi} follows LBWGU(9, y) distribution.

Proof: The distribution function of ZT is K(x) = P(ZT< x)

= YZ=1 P(Zn < x)P(T = n) = Zn=i [G(x)]np(1-p)n-1

PG(x)

1-(1-P)G(x)

1-(1-P){f)6+2 '

From this it follows that the survival function of the random variable ZT is

1(1-(pe + 2)

К(х) =

i-ii-w-ixn

which implies that ZT has MOLBWGU (6, y,1) distribution.

3.3. Order Statistics

Let (x1,x2,... ,xn) be a random sample of size n from MOLBWGU(9,y, P) distribution and let x1:n, x2:n,..., xn:n be the corresponding order statistics. Then the pdf of the jth order statistic for the MOLBWGU distribution is given by

fj-.n(X) =

(6 + 2)ßy-

U-iy.(n-j)\

1-(1-ß)(1-(§}e+2)

У

1-(1-ß){1-§e+2)

The MOLBWGU distribution has the following pdf for x1

f1:n(x) =

(e + 2)ßy~

1-(1-ß)(1-{^)e+2

ß(1-§e+2)

1-(1-ß)(1-{f}e+2)

and the pdf for xn.n is given by

fn:n(X) n

(e + 2)ßy~

{i-(iXi-®e+2))

У

i-(i-ß)(i-(;

-i

ß(1-$e+2)

1-(1-ß)(1-(f)

n-

, 0<х <y.

3.4. Quantile Function

The qth quantile of MOLBWGU(9, y,P) distribution isgiven by

= F-1

(«) = у(т^Т2' 0<4<i.

where F 1 (■) is the inverse distribution function.

n\

x

2

m-1

в-2.,в + 1

x

2

n-i

2

1

х

4

In particular the median of MOLBWGU (9, y, ft) distribution is givenby,

i

me dian(X) = y(-^Yr2. 3.5. Moments

If X has the MOLBWGU(8, y, ft) distribution, then the sth order moment is obtained as HXS) = r xs iB + Wr-*-^1 2dx

P(9 + 2) ry

{1-™{1-®e+2))

= fr_1_dx

rW + Qlo (PY0+2+(1-P)X0+2)2 '

where E denotes the expectation. Let xe+2 = u, above equation reduces to

yW + VJo (py0 + 2+il-p)u)2

From Prudnikov [15],

rb (x-a)a-1 (b-a)a B(a+k,n-k + 1)

ro (x-a) = -ar yn_ nC _

J a (cx+d)a+n+1 (ac+d^bc+d)^ k ibc+d)kiac+d)n-k'

where a, b, c and d are real numbers with (ac+d)(bc+d) > 0; Real part of a > 0 and B(a, b) = ^^(O^. Hence if s/9 + 2 is a positive integer, we have,

s

rre+2 uS+2 _ 1 vs/e+2 /n nr Rk

Jo ^6+2+1-^f au = r*+2+sp1+sKe+2) s/v + 2Ckp

Therefore

In particular,

B(1 + k+-^,1-k + ^~).

v 8+2 e+2J

E(*s) = ^¿^nT s/e + 2ckpkB(1 + k + 1+-2,1-k + Jt-j.

E(*) = Y4o+911/(o+2)n1kL0o+21/e + 2CkpkB(1 + k + 1+-2,1-k + J+-).

E(X2) = ^^e+vttlT 2/e + 2CkpkB(1 + k + —,1-k + —). 3.6. Renyi and Shannon Entropies

The Renyi entropy is defined as

= -^log f 1-r Jo

IR(y) = --log I fY(x)dx,y > 0,y & 1.

ry py(9 + 2)y xr(e+1)

J° yW+vr ^PYe+2+{1-P)xe+2)

Let u = xe+2. Therefore

Then T r(x)dx = r ^^__dx.

,y xr(e+1) , 1 rVe+2 uS+^Y-1)^1)

j dx — j /

J° {pye+2+il-p)xe+2fУ e+-Jo (pYe+2+il-p)UfY

Using the equation from Prudnikov [15] and if (y is a positive integer, the above integral

becomes

(y-1)(fl+3)

1 y (e+2) (/-1X6+3) r Rk

nk = 0 (0 + 2) ChP

1+(Y-1M±3lAjk=0 (0 + 2) ^k

ie + 2)yi0+2)+iY-l)iS+з)p1+ (9+2) ( ' k

B(1 + k + (HM!+1,1-k + O-^).

V (9 + 2) (9 + 2) J

Therefore the Renyi entropy is

1 =

' (1-y> , , (y-1)(e+3)

p(e+2)(9+2p-1) v (e+2) (Y-i)(6+3) k

Y(ie+9)y-1 Lk=0 (Q + 2) LjiP

v (9+2) (9+2) J\

Thus, the Shannon entropy is

E[-\ogf(x)] = -\og[p(e + 2)y~3(9+2)] -(e + 1)E[\og(X)] + 2E[\og(J3y9+2 + (1 - p)x9+2)].

4. Maximum Likelihood Estimation

The MLE method is used for the parameter estimation of MOLBWGU distribution. Let (xt,x2, ...,xn) be a random sample of size n from the MOLBWGU distribution. The likelihood function for the MOLBWGU distribution is,

L = (9 + 2)PY-0-2^!+1

from which the log-likelihood function is obtained as

\ogL = -2 Yn=! log (l -(1 - P)(i - y-9-2x9+2)) + (e + 1) Yn=! \og(xj) +n(-(e + 2)\og(Y) + \og(e + 2) + \og(J3)). The partial derivatives of this log-likelihood function is given by

d.ogL = (1-H)(Y-S-2log(Y)xf+2-Y-S-2x?+2log(xi))

d9 = 2^i = 1 1-(1-p)(1-Y-0-2xf+2)

+ 12=1 \og(xj) + n(^-\og(Y)).

dlogL _ „n (-9-2)(1-P)Y-0-3xf+2 (9 + 2)n

dY = 2 Li=1 1-(1-p)[1-Y-0-2x<l+2) Y . dlogL n n„n 1-Y-S-2xf+2

= -;-2Li=1

dp p Aji=1 1-{1-p)(1-Y-e-2xf+2)

The maximum likelihood estimator (6,Y,P) of the parameters (e,Y,P) can be obtained by solving the equations dJogL = 0, dJogL = 0 and dJogL = o.

1 d9 dY dp

4.1. Simulation Study

In this section, some simulation results are provided to study the behaviour of the MLEs in terms of the sample size. For this purpose, a Monte Carlo simulation study is conducted for MOLBWGU (e, y,P) distribution. The results are obtained from 1000 Monte Carlo replications and the simulations are carried out using the statistical software R. In each replication, a random sample of size 25, 50, 100, 150, 200 is generated for different combinations of e, y and P . The initial values of parameters are e = 1.2, y = 0.3, P = 1.5; e = 2, y = 1.5, P = 2.5; e = 0.5, y = 0.5, P = 1.5 and e = 2, y = 1, P = 2. Then computed mean of the MLEs of the parameters, biases and mean square errors (MSEs) of the parameter estimates. Tables 1, 2, 3 and 4 gives the values of the estimates, biases and MSEs of the corresponding parameters. From the tables, it can be seen that, as sample size increases the bias and MSE of the estimates decreases.

Table 1: Estimates, Biases and MSEs for e = 1.2, y = 0.3 and P = 1.5

Sample Size(n) Parameters Estimates Biases MSEs

e 1.2155 0.0155 0.0431

25 Y 0.3579 0.0579 0.1046

P 1.5034 0.0034 0.0009

e 1.2143 0.0143 0.0412

50 Y 0.3565 0.0565 0.1051

P 1.5022 0.0022 0.0003

e 1.2135 0.0135 0.0159

100 Y 0.3541 0.0541 0.1015

P 1.5015 0.0015 0.0003

e 1.2111 0.0111 0.0039

150 Y 0.3509 0.0509 0.1012

P 1.5009 0.0009 0.0002

e 1.2097 0.0097 0.0027

200 Y 0.3478 0.0478 0.1004

P 1.5001 0.0001 0.0002

Table 2: Estimates, Biases and MSEs for e = 2 , y = 1.5 and P = 2.5

Sample Size(n) Parameters Estimates Biases MSEs

e 2.6271 0.6271 0.4428

25 Y 1.5096 0.0096 0.2635

P 2.5242 0.0242 0.9715

e 2.4927 0.4927 0.3527

50 Y 1.5082 0.0082 0.0792

P 2.5211 0.0211 0.6226

e 2.3813 0.3813 0.3795

100 Y 1.5065 0.0065 0.0489

P 2.5175 0.0175 0.6535

e 2.3145 0.3145 0.1529

150 Y 1.5068 0.0068 0.0035

P 2.5148 0.0148 0.3614

e 2.2501 0.2501 0.1358

200 Y 1.5045 0.0045 0.0012

P 2.5122 0.0122 0.1428

Table 3: Estimates, Biases and MSEs for 9 = 0.5, y = 0.5 and ft = 1.5

Sample Size(n) Parameters Estimates Biases MSEs

e 0.5312 0.0312 0.0803

25 Y 0.5505 0.0505 1.5112

P 1.5091 0.0091 0.1583

e 0.5304 0.0304 0.0713

50 Y 0.5447 0.0447 0.8218

P 1.5082 0.0082 0.0685

e 0.5275 0.0275 0.0752

100 Y 0.5418 0.0418 0.7943

P 1.5079 0.0079 0.0082

e 0.5289 0.0289 0.0239

150 Y 0.5345 0.0345 0.5728

P 1.5074 0.0074 0.0047

e 0.5217 0.0217 0.0249

200 Y 0.5293 0.0293 0.5355

P 1.5066 0.0066 0.0032

Table 4: Estimates, Biases and MSEs for 0 =2, y =1 and ft = 2

Sample Size(n) Parameters Estimates Biases MSEs

e 2.1827 0.1827 0.3912

25 Y 1.1578 0.1578 0.2014

P 2.1666 0.1666 0.0009

e 2.0915 0.0915 0.3373

50 Y 1.1579 0.1579 0.2008

P 2.1643 0.1643 0.0004

e 2.0912 0.0912 0.1838

100 Y 1.1458 0.1458 0.2007

P 2.1712 0.1712 0.0003

e 2.0774 0.0774 0.1425

150 Y 1.1435 0.1435 0.2001

P 2.1626 0.1626 0.0003

e 2.0751 0.0751 0.0564

200 Y 1.1315 0.1315 0.0197

P 2.0125 0.0125 0.0002

5. Application in Autoregressive Time Series Modeling

In this section, some applications of MOLBWGU distribution in autoregressive time series modelling are provided. Now, we construct a first order autoregressive minification process with structure as follows,

X7

= ( £n, W.p.

lmin(xn-1,£n), w.p.

ß

1-ß, 0 < ß <1, n>1,

(8)

where { £n} is a sequence of i.i.d. random variables following LBWGU(e,Y) distribution independent of {xn -1,xn-2,...} . Then the process is stationary and is marginally distributed with MOLBWGU(e, y, P) distribution. This leads to the following theorem.

Theorem 4. In an AR (1) process with structure (8), {Xn, n > 0} defines a stationary AR (1) minification process with MOLBWGU(e,Y,P) marginal distribution iff {£„} is a sequence of independently and identically distributed random variable with LBWGU (e, y) distributon. Proof. Consider (8) in terms of survival function

FXn(x) = ßFe(x) + (1- ß)F„ (x)F£(x).

Under stationary equilibrium it reduces to

and hence

1-(1-ß)FEn(x)

ßFxW

1-(1-ß)Fx(x)

If £n follows LBWGU (S, y) from(9), we get

Fx(x) =---

Fx(x) =

Fsn(x) =

ßhn(x)

(9) (10)

1-(1-ß)(1-{-)

which is the survival function of M0LBWGU(8,y,ß) . Conversely, if we take

1-(1-p)(1-(f)e+2)

from (9) it can show thatF£n(x) is distributed LBWGU(e, y) with survival function (1 - (x)9+2).

5.1. Sample Path

To study the behavior of the process we simulate the sample path for various values of P, the properties of sample path shows that the MOLBWGU AR(1) minification process can be used for modelling a rich variety of real data. Sample path of MOLBWGU AR(1) process for y = 0.5, e = 0.9 and P= 0.4, 0.5, 0.6 and 0.8 is given in Figure 4.

Figure 4: Sample path of the MOLBWGU AR(1) process for y = 0.5, e = 0.9 and P= 0.4, 0.5, 0.6 and 0.8.

6. Stress Strength Analysis

The stress strength reliability analysis can be regarded as an assessment of reliability of a system in terms of random variables X and Y, where X represents strength and Y represents the stress. If the stress exceeds strength the system would fail and the system will function if strength exceeds stress. The stress strength reliability can be defined as R = P(X >Y ). Gupta [6] obtained various results on the MO family in the context of reliability modelling and survival analysis. Then, R = P(X>Y) = J+y(x > YIY = y)gY(y)dy

= te (-lnPi + Pi-1)

Let (x1,x2,...,xm) and (y1,y2,...,yn) be two independent random samples of sizes m and n from MOLBWGU distribution with tilt parameters /1 and /2 respectively, and common unknown parameters y and 8 . The log likelihood function is given by

Zm . . rim

^ ^ log (l - (1 - Pi)(l - y-e-2xf+2)) + (8 + 1)^ log(xj) + m(-(8 + 2)log(y) + 1og(8

+ 2) + log(fii)) - 2 ^ log (l -(1 - foil - y-e-2yf+2)) + (8 + 1) ^^log^)

+ n(-(8 + 2)log(y) + \og(8 + 2)+ log(fa)) 1

Then MLE of and /2 are the solutions of the nonlinear equations = 0 and dl°^L = 0. The elements of Information matrix are given by,

'11 = E(dpl) = sp?

122 = E (dp2) = sp?

I12 = I = -E (-^A = 0.

12 21 Kdp^p?)

By the property of MLE for m ^ and n ^ ,

(^m(p1 - fa), jn($2 - /2)) ^ N2(0, diag{a-1, a-1}).

where an = limm,n^ = ^2 and a22 = 1imm,n^~ '-f = ^2

Now from [6] the 95% confidence interval for R is given by R± 1.96/3^1(131, ¡32)J— +

where HWM =£ = [-2(Jh - /2) + (31 + and b2(XltX2) = £

Pi Uro o \ ro , O M„Pll _ Pi

(Pl-P2)3l P2

6.1. Simulation Study

For the simulation study, generate N=10,000 sets of X-samples and Y-samples from the MOLBWGU distribution with parameters (/1,y,8) and (/2,y,8) respectively. The combinations of samples of sizes m = 20,25,30 and n = 20,25,30 are studied. The validity of the estimate of R is considered by using the following measures, namely average bias of the estimate (b), average mean square error of the estimate (AMSE), average confidence interval of the estimate and coverage probability defined by,

1. Average bias (b) of the estimates of R:

R - R).

2. Average mean square error of the estimates of R:

±I?=1 (Ri-R)2.

3. Average length of the asymptotic 95% confidence interval of R:

lYl12(1.96)ßlibli(ßßli,ßß2i)J3 + 3.

4. The coverage probability of the confidence intervals given by the proportion of such interval that include the parameter R.

The numerical values obtained for the measures are presented in Table 5 and 6. The average bias decreases as the sample size increases. The coverage probability is close to 0.95 as the sample size increases. This simulation results shows that the average bias, average MSE, average confidence interval and coverage probability do not show much variability for various parameter combinations.

Table 5: Average bias and average MSE of the simulated estimates of R for y = 1 and 9 = 0.9

(ßl,ß2)

Average Bias (b)

Average Mean Square Error (AMSE)

(m n)

(0.2,0.4)

(0.4, 0.2) (0.5, 0.1) (0.4, 1.2) (0.2, 0.4)

(0.4, 0.2)

(0.5, 0.1)

(0.4, 1.2)

(20, 20) (20, 25) (20, 30) (25, 20) (25, 25) (25, 30) (30, 20) (30, 25) (30, 30)

0.0035 0.0066 0.0092 0.0024 0.0052 0.0077 0.0021 0.0048 0.0075

0.0825 0.0832 0.0839 0.0857 0.0877 0.0878 0.0887 0.0910 0.0908

0589

0605 0614 0588

0606 0611 0588 0602 0611

0.0286 0.0289 0.0279 0.0233 0.0241 0.0234 0.0199 0.0204 0.0205

0.0042 0.0041 0.0039 0.0041 0.0039 0.0038 0.0040 0.0038 0.0037

0.0115 0.0117 0.0117 0.0115 0.0118 0.0118 0.0116 0.0118 0.0118

0.0040 0.0041 0.0042 0.0040 0.0041 0.0041 0.0040 0.0040 0.0041

0.0052 0.0052 0.0050 0.0041 0.0043 0.0042 0.0036 0.0036 0.0037

Table 6: Average length of the confidence interval and coverage probability of the simulated 95% confidence intervals of R for y = 1 and 9 = 0.9

(ßl,ß2)

Average Confidence Length Coverage Probability

(m n) (0.2,0.4) (0.4, 0.2) (0.5, 0.1) (0.4, 1.2) (0.2, 0.4) (0.4, 0.2) (0.5, 0.1) (0.4, 1.2)

(20, 20) (20, 25) (20, 30) (25, 20) (25, 25) (25, 30) (30, 20) (30, 25) (30, 30) 0.3346 0.3167 0.3043 0.3179 0.2991 0.2860 0.3064 0.2867 0.2728 0.3505 0.3325 0.3201 0.3334 0.3145 0.3012 0.3215 0.3019 0.2879 0.3111 0.2960 0.2853 0.2951 0.2791 0.2675 0.2840 0.2671 0.2551 0.3239 0.3074 0.2956 0.3063 0.2889 0.2764 0.2939 0.2759 0.2631 0.9592 0.9585 0.9583 0.9582 0.9573 0.9568 0.9574 0.9557 0.9542 0.9839 0.9727 0.9611 0.9809 0.9613 0.9434 0.9720 0.9533 0.9264 0.9991 0.9898 0.9794 0.9799 0.9999 0.9696 0.9599 0.9599 0.9399 0.9805 0.9530 0.9299 0.9793 0.9533 0.9231 0.9783 0.9593 0.9333

7. Conclusion

In this paper, a generalization of LBWGU distribution namely MOLBWGU distribution is developed. Some of the statistical properties of the new distribution such as probability density function, hazard rate function, moments, quantile function, compounding, distribution of order statistics, Renyi and Shannon entropies are derived. We estimated the parameters of the distribution using maximum likelihood estimation method and a simulation study is conducted for proving the validity of the estimates.

Also we developed a minification process using the model and explored its sample path behavior for

different combinations of parameters. To check the impact of stress on strength of devices and systems,

the stress strength analysis is carried out and the estimate of the reliability is examined based on a

simulation study.

References

[1] Afify, A. Z., Cordeiro, G. M., Yousof, H. M., Saboor, A., and Ortega, E. M. (2018). The Marshall-Olkin additive Weibull distribution with variable shapes for the hazard rate. Hacettepe Journal of Mathematics and Statistics, 47 (2), 365-381.

[2] Benkhelifa, L. (2017). The Marshall Olkin extended generalized Lindley distribution: properties and applications. Communications in Statistics - Simulation and Computation, 46 (10), 8306-8330.

[3] Bhatt, M. B. (2014). Characterization of generalized uniform distribution through expectation. Open Journal of Statistics, 4 (8), 563-569.

[4] Ghitany, M. E., Al-Awadhi, F. A., and Alkhalfan, L. (2007). Marshall Olkin extended Lomax distribution and its application to censored data. Communications in Statistics - Theory and Methods, 36 (10), 1855-1866.

[5] Ghitany, M. E., Al-Hussaini, E. K., and Al-Jarallah, R. A. (2005). Marshall Olkin extended Weibull distribution and its application to censored data. Journal of Applied Statistics, 32 (10), 1025-1034.

[6] Gupta, R. C., Ghitany, M. E., and Al-Mutairi, D. K. (2010). Estimation of reliability from Marshall Olkin extended Lomax distributions. Journal of Statistical Computation and Simulation, 80 (8), 937-947.

[7] Jayakumar, K., and Mathew, T. (2008). On a generalization to Marshall Olkin scheme and its application to Burr type XII distribution. Statistical Papers, 49 (3), 421-439.

[8] Khan, M. I., and Khan, M. A. R. (2017). Characterization of generalized uniform distribution based on lower record values. ProbStat Forum, 10 (1), 23-26.

[9] Krishna, E., Jose, K. K., Alice, T., and Ristic, M. M. (2013). The Marshall-Olkin Frechet distribution. Communications in Statistics - Theory and Methods, 42 (22), 4091-4107.

[10] Lee, C. (2000). Estimations in a generalized uniform distribution. Journal of the Korean Data and Information Science Society, 11 (2), 319-325.

[11] Marshall, A. W., and Olkin, I. (1997). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrica, 84 (3), 641-652.

[12] Mathew, J. (2020 ). Reliability test plan for the Marshall-Olkin length biased Lomax distribution. Reliability: Theory and Applications, 15 (2), 36-49.

[13] Mathew, J., and Chesneau, C. (2020 a). Marshalla€"Olkm length biased Maxwell distribution and its applications. Mathematical and Computational Applications, 25 (4), 65.

[14] Patil, G. P. (2002). Weighted Distributions. Encyclopedia of Environmetrics, John Wiley and Sons, New York.

[15] Prudnikov, A. P., Brychkov, Y. A., and Marichev, O. I. (1986). Integrals and Series. Gordeon and Breach Sciences, Amsterdam, Netherland.

[16] Rather, A. A., and Subramanian, C. (2018). Characterization and estimation of length biased weighted generalized uniform distribution. International Journal of Scientific Research in Mathematical and Statistical Sciences, 5 (5), 72-76.

[17] Ristic, M. M., and Kundu, D. (2015). Marshall-Olkin generalized exponential distribution. Metron, 73 (3), 317-333.

[18] Subramanian, C., and Rather, A. A. (2018). Transmuted generalized uniform distribution. International Journal of Scientific Research in Mathematical and Statistical Sciences, 5 (5), 25-32.

[19] Tiwari, R. C., Yang, Y., and Zalkikar, J. N. (1996). Bayes estimation for the Pareto failure-model using Gibbs sampling. IEEE Transactions on Reliability, 45 (3), 471-476.