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Volume 20, Issue 2, June 2021, Pages 204 - 218

On Finite 3-Component Mixture of Rayleigh Distributions: A Classical Look

Authors
Muhammad Tahir1, *, Muhammad Ahsanullah2, Sidra Mohsin1, Muhammad Abid1
1Department of Statistics, Government College University, Faisalabad, Pakistan
2Professor Emeritus, Rider University, USA
*Corresponding author. Email: tahirqaustat@yahoo.com
Corresponding Author
Muhammad Tahir
Received 8 October 2018, Accepted 6 July 2020, Available Online 23 June 2021.
DOI
10.2991/jsta.d.210616.001How to use a DOI?
Keywords
Mixture distribution; Maximum likelihood estimation; Statistical properties; Entropies; Inequality measures; Order statistics
Abstract

In this study, we have discussed various statistical properties for 3-component mixture of Rayleigh distributions. Here initially, the main properties of mixture distributions are presented and analyzed. Second, some of the famous entropies, measures of inequality are also discussed. Also, the statistical properties of the density functions of rth-, 1st- and nth-order statistics are derived. Moreover, the parameters estimation of the considered mixture model under the maximum likelihood (ML) estimation is also performed using censored and complete data scheme. Finally, the results on ML estimation are also computed via Monte Carlo simulation study and as well as by using a real-life data set.

Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Mixture of distributions is arising naturally and discussed where a statistical population has more than two sub populations. In many applicable areas, the mixture representations have been compensated excessive care. However, the mixture of distribution that rises with a grouping of different distributions is said to be a mixture component, and the probabilities (weights) which are related to each of the component are known as the mixture weights.

In different practical situations, several authors have worked on the mixture modeling. Mostly, in the biology the direct applications of the mixture models are discussed by Bhattacharya [1] and by Gregor [2], in the medicine are presented by Chivers [3] and by Burekhardt [4], in the social sciences are observed by Harris [5], in an economics are mentioned by Jedidi et al. [6], in the reliability and survival are analyzed by Sultan et al. [7], in the life testing are suggested by Shawky and Bakoban [8], in the industrial engineering are observed by Ali et al. [9].

This study plan to develop a 3-component mixture of Rayleigh distributions for an effective modeling of time-to-failure data. Let consider a variable of interest y which follows a mixture distribution having q components and its density function is fy=∑m=1qwm fmy, where wmm=1,2,…,q is mth mixing proportions such that wq=1−∑m=1q−1wm and fmy is mth component density function. The pdf and cdf of 3-component mixture of Rayleigh distributions for the unknown mixing proportions w1 and w2 is defined as follows:

fy;Ω=w1f1y;θ1+w2f2y;θ2+w3f3y;θ3,w1,w2≥0,w1+w2≤1,(1)
1−R(y;Ω)=w11−R(y;θ1)+w21−R(y;θ2)+1−w1−w21−R(y;θ3),(2)
where Ω=θ1,θ2,θ3,w1,w2.
fmy;θm=yθm2exp−y22θm2,0<y<∞,θm>0,m=1,2,3,

The cdf Fmy;Ωm=1−Rmy;θm of the mth component density is given by

1−Rmy;θm=1−exp−y22θm2,0<y<∞,θm>0.(3)

2. STATISTICAL PROPERTIES

Here, we derive computable representations of some statistical properties associated with the 3-component mixture of Rayleigh distributions having pdf given in (1).

2.1. rth Moments about Origin

The rth moments about origin of a 3-component mixture of Rayleigh distributions for a random variable Y is given as follows:

EYr=w12r2θ1rr2+1+w22r2θ2rr2+1+1−w1−w22r2θ3rr2+1.(4)

2.2. Mean and Variance

The mean and variance of the considered mixture distributions are as follows:

EY=w1θ1π2+w2θ2π2+1−w1−w2θ3π2,(5)
Var(Y)=2w1θ12+2w2θ22+21−w1−w2θ32−w1θ1π2+w2θ2π2+1−w1−w2θ3π22.(6)

2.3. hth-Order Negative Moments

If we replaced −h instead of r in Equation (4), then it is converted into hth-order negative moments and it is defined as follows:

EY(−h)=2(−h)2w1θ1(−h)(−h)2+1+2(−h)2w2θ2(−h)(−h)2+1+2(−h)21−w1−w2θ3(−h)(−h)2+1.(7)

2.4. Factorial Moments

If we replaced (α−μ) instead of r in Equation (4), then it is converted into factorial moments and it is defined as follows:

EY(α−μ)=2(α−μ)2w1θ1(α−μ)(α−μ)2+1+w2θ2(α−μ)(α−μ)2+1+1−w1−w2θ3(α−μ)(α−μ)2+1.(8)

2.5. Quantile Function

The quantile function for a Y random variable is as follows:

yq=w1θ1−2ln(1−p)+w2θ2−2ln(1−p)+1−w1−w2θ3−2ln(1−p).(9)

2.6. Median

To solve the below mentioned Equation (10) for y, we obtain the median as follows:

p1exp−y22θ12+p2exp−y22θ22+1−p1−p2exp−y22θ32=12.(10)

2.7. Mode

To resolve the below mentioned Equation (11) for y, we obtain the mode as follows:

∑m=13p11θm2exp−y22θm2−∑m=13p11θm4y2exp−y22θm2=0.(11)

The numerical results of mean, median, mode, variance and coefficient of skewness for various choices of θ1,θ2,θ3,p1,p2 are offered in Table 1.

θ1, θ2, θ3, p1, p2 Mean Variance Median Mode Skewness
13, 14, 15, 0.1, 0.2 18.2984 91.6779 17.1902 14.5176 0.075906
13, 14, 15, 0.2, 0.3 17.9224 88.0297 16.837 14.1885 0.0759013
13, 14, 15, 0.3, 0.4 17.5464 84.3814 16.4837 13.8928 0.0759134
13, 14, 15, 0.4, 0.5 17.1704 80.7332 16.1305 13.6268 0.0759086
15, 14, 13, 0.2, 0.1 16.9197 78.5014 15.895 13.3907 0.0759137
16, 15, 14, 0.3, 0.2 18.549 94.339 17.4257 14.6759 0.0759014
17, 16, 15, 0.4, 0.3 20.1784 111.55 18.9563 15.9914 0.0759112
18, 17, 16, 0.5, 0.4 21.8077 130.135 20.4869 17.3345 0.0759137
12, 11, 10, 0.4, 0.2 13.7865 52.277 12.9515 10.8186 0.0759095
14, 13, 12, 0.4, 0.2 16.2931 72.8788 15.3063 12.8464 0.195787
16, 15, 14, 0.4, 0.2 18.7997 96.9142 17.6612 14.8668 0.179804
18, 17, 16, 0.4, 0.2 21.3063 124.383 20.016 16.8825 0.167591
Table 1

Mean, variance, median, mode and skewness.

It is revealed that the distribution of the 3-component mixture of Rayleigh distributions is positively skewed because the value of mean is greater than median and SK > 0 (cf. Table 1). As the values of θ1,θ2,θ3 increases the variance of the distribution is also increased for fixed value of p1 and p2 (cf. Table 1). Moreover, there is also an increase in the variance values as the value of p1 and p2 is also increased for fixed values of θ1,θ2,θ3(cf. Table 1).

3. RELIABILITY PROPERTIES OF THE PROPOSED MIXTURE DISTRIBUTIONS

3.1. Reliability Function

The reliability function of the proposed mixture distributions is given as follows:

Ry;Ω=w1R1y;θ1+w2R2y;θ2+1−w1−w2R3y;θ3,(12)
Ry;Ω=w1exp−y22θ12+w2exp−y22θ22+1−w1−w2exp−y22θ32.(13)

The graphical presentation of the reliability function for proposed mixture distributions is as follows:

3.2. Hazard Rate Function

The hazard rate function for proposed mixture distributions is as follows:

hy;Ω=fy;ΩRy;Ω=w1f1y;θ1+w2f2y;θ2+1−w1−w2f3y;θ3w1R1y;θ1+w2R2y;θ2+1−w1−w2R3y;θ3,(14)
h(y;Ω)=f(y;Ω)R(y;Ω)=w1yθ12exp−y22θ12+w2yθ22exp−y22θ22+1−w1−w2yθ32exp−y22θ32w1exp−y22θ12+w2exp−y22θ22+1−w1−w2exp−y22θ32.(15)

The graphical performance of the hazard rate function for proposed mixture distributions is given as follows:

3.3. Cumulative Hazard Rate Function

The cumulative hazard rate (CHR) function for proposed mixture distributions is defined as follows:

Hy;Ω=∫0yhy;Ωdy=−lnRy;Ω,(16)
Hy;Ω=−lnw1exp−y22θ12+w2exp−y22θ22+1−w1−w2exp−y22θ32.(17)

The graphical performance of the CHR function for proposed mixture distributions is shown as follows:

3.4. Mean Residual Life Function

The mean residual life (MRL) function for proposed mixture distributions is written as follows:

M(y;Ω)=1R(y;Ω)∫y∞(x−y)f(x;Ω)dx,(18)
M(y;Ω)=1R(y;Ω)E(Y)−∫0yuf(y;Ω)du−y,(19)
M(y;Ω)=w1θ1π2+w2θ2π2+1−w1−w2θ3π2−w1θ1π2erfy2θ1+w2θ2π2erfy2θ2+1−w1−w2θ3π2erfy2θ3w1exp−y22θ12+w2exp−y22θ22+1−w1−w2exp−y22θ32.(20)

The graphical performance of the MRL function for proposed mixture distributions is shown as follows:

3.5. Mean Waiting Time Function

The mean waiting time (MWT) function for proposed mixture distributions is defined as follows:

ω−(y;Ω)=y−11−R(y;Ω)∫0yuf(y;Ω)du,(21)
ω−(y;Ω)=y−w1θ1π2erfy2θ1−w2θ2π2erfy2θ2−1−w1−w2θ3π2erfy2θ31−w1exp−y22θ12−w2exp−y22θ22−1−w1−w2exp−y22θ32.(22)

The graphical presentation of the MWT function for proposed mixture distributions is shown as follows:

4. STATISTICAL FUNCTIONS

The different statistical functions such as moment generating function My(t), characteristic function ϕy(t), probability generating function G(α) and factorial moment generating function H0(δ) for proposed mixture distributions are given below, respectively:

My(t)=w1+tθ1π2expθ12t22erftθ12+1+w2+tθ2π2expθ22t22erftθ22+1  +1−w1−w2+tθ3π2expθ32t22erftθ32+1.(23)
ϕy(t)=w1+itθ1π2exp−θ12t22erfitθ12+1+w2+itθ2π2exp−θ22t22erfitθ22+1  +1−w1−w2+itθ3π2exp−θ32t22erfitθ32+1.(24)
G(α)=w1+lnαθ1π2expθ12(lnα)22erflnαθ12+1+w2+lnαθ2π2expθ22(lnα)22erflnαθ22+1  +1−w1−w2+lnαθ3π2expθ32(lnα)22erflnαθ32+1.(25)
H0(δ)=w1+ln(1+δ)θ1π2expθ12ln(1+δ)22erfln(1+δ)θ12+1  +w2+ln(1+δ)θ2π2expθ22ln(1+δ)22erfln(1+δ)θ22+1  +1−w1−w2+ln(1+δ)θ3π2expθ32ln(1+δ)22erfln(1+δ)θ32+1.(26)

5. MEASURES OF INEQUALITY

In this section, we have considered various measures of inequality such as Gini index (G), Lorenz curve Lp, Bonferroni curve BCp and Zenga index ξ for proposed mixture distribution. The mathematical expressions of these inequalities are as follows:

G=π2−1w1θ1+w2θ2+1−w1−w2θ3−1π2w1θ1+w2θ2+1−w1−w2θ3  −π2w12θ1+w22θ2+1−w1−w22θ3+π4w1w2θ1θ2+4w21−w1−w2θ2θ3+4w11−w1−w2θ1θ3(27)
L(p)=w1θ1π2+w2θ2π2+1−w1−w2θ3π2−1w1θ1π2erfy2θ1−w1yexp−y22θ12  +w2θ2π2erfy2θ2−w2yexp−y22θ22+1−w1−w2θ3π2erfy2θ3−1−w1−w2yexp−y22θ32.(28)
BC(p)=L(p)1−R(y;Ω),(29)
where
L(p)=w1θ1π2+w2θ2π2+1−w1−w2θ3π2−1w1θ1π2erfy2θ1−w1yexp−y22θ12  +w2θ2π2erfy2θ2−w2yexp−y22θ22+1−w1−w2θ3π2erfy2θ3−1−w1−w2yexp−y22θ32,(30)
and
1−Ry;Ω=1−w1exp−y22θ12−w2exp−y22θ22−1−w1−w2exp−y22θ32.(31)
ξ=1−η(y)−η(y)+,(32)
where
η(y)−=11−R(y;Ω)∫0yyf(y;Ω)dy and η(y)+=1R(y;Ω)y−∫0yyf(y;Ω)dy.

After substitution the values in (36), the simplified Zenga index is as follows:

ξ=1−w1θ1π2erfy2θ1−w1yexp−y22θ12+w2θ2π2erfy2θ2−w2yexp−y22θ22+1−w1−w2θ3π2erfy2θ3μ−w1θ1π2erfy2θ1−w1yexp−y22θ12+w2θ2π2erfy2θ2−w2yexp−y22θ22+1−w1−w2θ3π2erfy2θ3  −1−w1−w2yexp−y22θ32−1−w1−w2yexp−y22θ32w1exp−y22θ12+w2exp−y22θ22+1−w1−w2exp−y22θ321−w1exp−y22θ12+w2exp−y22θ22+1−w1−w2exp−y22θ32.(33)

6. ENTROPIES

The entropy is used as measure of uncertainty in various applied fields of science and engineering. The various type of entropy such as Shannon's entropy S(y;Ω), Rényi entropy LR(υ) and β-entropy Lβ(π) are considered in this study. The mathematical expression of these entropies is given below:

S(y;Ω)=−∫0∞w1yθ12exp−y22θ12+w2yθ22exp−y22θ22+1−w1−w2yθ32exp−y22θ32  logw1yθ12exp−y22θ12+w2yθ22exp−y22θ22+1−w1−w2yθ32exp−y22θ32.(34)
LR(υ)=1(1−υ)log∑j=0υ∑k=0jυjjkyυw1υ−jw2j−k1−w1−w2kθ1−2(υ−j)θ2−2(j−k)θ3−2k212θ12(υ−j)+12θ22(j−k)+12θ22k−1(35)
where υ>0 and υ≠1.
Lβ(π)=1(π−1)∑j=0π∑k=0jπjjkyπw1π−jw2j−k1−w1−w2kθ1−2(π−j)θ2−2(j−k)θ3−2k212θ12(π−j)+12θ22(j−k)+12θ22k−1(36)
where π>0 and π≠1.

7. DISTRIBUTIONS OF ORDER STATISTICS

In this section, we have provided the different function of order statistics such as kth, 1st and nth for proposed mixture distribution. The mean and variance of considered order statistics are also presented.

7.1. The kth-Order Statistic

The pdf of kth-order statistic for proposed mixture distribution is as follows:

hyk:n;Ω=n!(k−1)!(n−k)!1−R(y;Ω)k−1R(y;Ω)n−kf(y;Ω),(37)

After substitution in Equation (37), we get

hyk:n;Ω=n!(k−1)!(n−k)!∑j=0k−1∑l=0j∑r=0l−1jk−1   jjllrexp−ψ01−2yk22w1α01−1w2β01−11−w1−w2γ01−1  ∑e=0n−k∑f=0en−k   eefexp−ψ02−2yk22w1α02−1w2β02−11−w1−w2 γ02−1  w1ykθ12exp−yk22θ12+w2ykθ22exp−yk22θ22+1−w1−w2ykθ32exp−yk22θ32.(38)
where
ψ01−2=1θ12j−l+1θ22l−r+1θ32r,α01=j−l+1,β01=l−r+1,D01=l+1,ψ02−2=1θ12n−k−e+1θ22e−f+1θ32f,α02=n−k−e+1,β02=e−f+1,γ02=f+1.

7.2. The 1st-Order Statistic

Placing k=1 in Equation (38), we have obtained the pdf of 1st-order statistic for proposed mixture distribution as follows:

hy1:n;Ω=ny1∑g=131θg2∑e=0n−1∑f=0en−1    eefexp−ψ1g−2y122w1α1g−1w2β1g−11−w1−w2γ1g−1.(39)
where
ψ11−2=1θ12n−e+1θ22e−f+1θ32f,α11=n+1−e,β11=e−f+1,γ11=f+1,ψ12−2=1θ12n−1−e+1θ22e−f+1+1θ32f,α12=n−e,β12=e−f+2,γ12=f+1,ψ13−2=1θ12n−1−e+1θ22e−f+1θ32f+1,α13=n−e,β13=e−f+1,γ13=f+2.

7.3. The nth-Order Statistic

Replacing k=n in Equation (38), we have found the pdf of nth-order statistic for proposed mixture distribution as follows:

hyn:n;Ω=nyn∑g=131θg2∑j=0n−1∑l=0j∑r=0l−1jn−1    jjllrexp−ψ2g−2yn22w1α2g−1w2β2g−11−w1−w2γ2g−1,(40)
where
ψ21−2=1θ12j−l+1+1θ22l−r+1θ32r,α21=j−l+2,β21=l−r+1,γ21=r+1,ψ22−2=1θ12j−l+1θ22l−r+1+1θ32r,α22=j−l+1,β22=l−r+2,γ22=r+1,ψ23−2=1θ12j−l+1θ22l−r+1θ32r+1,α23=j−l+1,β23=l−r+1,γ23=r+2.

7.4. rth Moments about Origin of 1st- Order Statistic

The rth moment of 1st-order statistic about the origin is obtained as follows:

EY1r=n∑g=131θg2∑e=0n−1∑f=0en−1    eefw1α1g−1w2β1g−11−w1−w2γ1g−12r−12r−12+1ψ1w−2r−12+1.(41)

The mean and variance of 1st-order statistic are obtained as follows:

EY1=n∑g=131θg2∑e=0n−1∑f=0en−1    eefw1α1g−1w2β1g−11−w1−w2γ1g−1ψ1g−2.(42)
VarY1=nπ2∑g=131θg2∑e=0n−1∑f=0en−1    eefw1α1g−1w2β1g−11−w1−w2γ1g−1ψ1g−232  −n∑g=131θg2∑e=0n−1∑f=0en−1    eefw1α1g−1w2β1g−11−w1−w2γ1g−1ψ1g−22.(43)

7.5. rth Moments about Origin of nth- Order Statistic

The rth moment of nth-order statistic about the origin is obtained as follows:

EYnr=n∑g=131θg2∑j=0n−1∑l=0j∑r=0l−1jn−1    jjllrw1α2g−1w2β2g−11−w1−w2γ2g−12r−12r−12+1ψ1w−2r−12+1.(44)

The mean and variance of the nth order statistic are obtained as follows:

EYn=n∑g=131θg2∑j=0n−1∑l=0j∑r=0l−1jn−1    jjllrw1α2g−1w2β2g−11−w1−w2γ2g−1ψ2g−2.(45)
VarYn=nπ2∑g=131θg2∑j=0n−1∑l=0j∑r=0l−1jn−1    jjllrw1α2g−1w2β2g−11−w1−w2γ2g−1ψ2g−232  −n∑g=131θg2∑j=0n−1∑l=0j∑r=0l−1jn−1    jjllrw1α2g−1w2β2g−11−w1−w2γ2g−1ψ2g−22.(46)

8. ESTIMATION OF PARAMETERS

The ML method is used to estimate the unknown parameters of the proposed mixture distribution under type-I censored and complete sampling situations. The ML estimation under type-I censored and complete sampling situations are given below:

8.1. The Likelihood Function

The likelihood function for suggested mixture distribution under type-I censoring is written as follows:

LΩ|y∝∏h=1s1w1f1y1h∏h=1s2w2f2y2h∏h=1s31−w1−w2f3y3hR(t)n−s,(47)
where t represents the test termination time, s1,s2 and s3 denote the failures which are belonging to the subpopulations I, II and III, s=s1+s2+s3 indicates the uncensored observation and n−s denotes the censored observation.

After simplifying the Equation (47), we get

LΩ|y∝w1s1θ12s1w2s2θ22s21−w1−w2s3θ32s3exp−12θ12∑h=1s1y1h2exp−12θ22∑h=1s2y2h2exp−12θ32∑h=1s3y3h2  w1exp−t22θ12+w2exp−t22θ22+1−w1−w2exp−t22θ32n−s.(48)

8.2. ML Estimators and Their Variances for Censored Data

The ML estimators for Ω=θ1,θ2,θ3,w1,w2 of a 3-component mixture of Rayleigh distributions are obtained by solving the following nonlinear system of Equations (49–53).

∂lnLΩ|y∂θ1=−2s1θ1+∑h=1s1y1h2θ13−n−sw1t2exp−t22θ12θ13w1exp−t22θ12+w2exp−t22θ22+1−w1−w2exp−t22θ32=0.(49)
∂lnLΩ|y∂θ2=−2s2θ2+∑h=1s2y2h2θ23−n−sw2t2exp−t22θ22θ23w1exp−t22θ12+w2exp−t22θ22+1−w1−w2exp−t22θ32=0.(50)
∂lnLΩ|y∂θ3=−2s3θ3+∑h=1s3y3h2θ33−n−s1−w1−w2t2exp−t22θ32θ33w1exp−t22θ12+w2exp−t22θ22+1−w1−w2exp−t22θ32=0.(51)
∂lnLΩ|y∂w1=s1w1−s31−w1−w2+n−sexp−t22θ12−exp−t22θ32w1exp−t22θ12+w2exp−t22θ22+1−w1−w2exp−t22θ32=0.(52)
∂lnLΩ|y∂w2=s2w2−s31−w1−w2+n−sexp−t22θ22−exp−t22θ32w1exp−t22θ12+w2exp−t22θ22+1−w1−w2exp−t22θ32=0.(53)

It is tough to find out the closed form for the ML estimators. The normal equations do not have explicit solutions and they have to be obtained numerically. The Mathematica (Wolfram [10]) software is used to find the ML estimates (MLEs) of θ1,θ2,θ3,w1 and w2.

The simplest large sample approach is to assume that the MLE θ^1,θ^2,θ^3,w^1,w^2 are approximately multivariate normal distribution with mean Ω=θ1,θ2,θ3,w1,w2 and covariance matrix I−1Ω, where I−1Ω is the inverse of the observed information matrix. Therefore, ML variances (MLVs) are on the main diagonal of an inverted information matrix as given below:

I−1=−E∂2l∂θ12θ^1,θ^2,θ^3,w^1,w^2∂2l∂θ1∂θ2θ^1,θ^2,θ^3,w^1,w^2∂2l∂θ1∂θ3θ^1,θ^2,θ^3,w^1,w^2∂2l∂θ1∂w1θ^1,θ^2,θ^3,w^1,w^2∂2l∂θ1∂w2θ^1,θ^2,θ^3,w^1,w^2∂2l∂θ2∂θ1θ^1,θ^2,θ^3,w^1,w^2∂2l∂θ22θ^1,θ^2,θ^3,w^1,w^2∂2l∂θ2∂θ3θ^1,θ^2,θ^3,w^1,w^2∂2l∂θ2∂w1θ^1,θ^2,θ^3,w^1,w^2∂2l∂θ2∂w2θ^1,θ^2,θ^3,w^1,w^2∂2l∂θ3∂θ1θ^1,θ^2,θ^3,w^1,w^2∂2l∂θ3∂θ2θ^1,θ^2,θ^3,w^1,w^2∂2l∂θ32θ^1,θ^2,θ^3,w^1,w^2∂2l∂θ3∂w1θ^1,θ^2,θ^3,w^1,w^2∂2l∂θ3∂w2θ^1,θ^2,θ^3,w^1,w^2∂2l∂w1∂θ1θ^1,θ^2,θ^3,w^1,w^2∂2l∂w1∂θ2θ^1,θ^2,θ^3,w^1,w^2∂2l∂w1∂θ3θ^1,θ^2,θ^3,w^1,w^2∂2l∂w12θ^1,θ^2,θ^3,w^1,w^2∂2l∂w1∂w2θ^1,θ^2,θ^3,w^1,w^2∂2l∂w2∂θ1θ^1,θ^2,θ^3,w^1,w^2∂2l∂w2∂θ2θ^1,θ^2,θ^3,w^1,w^2∂2l∂w2∂θ3θ^1,θ^2,θ^3,w^1,w^2∂2l∂w2∂w1θ^1,θ^2,θ^3,w^1,w^2∂2l∂w22θ^1,θ^2,θ^3,w^1,w^2−1
I−1=Varθ1∧Covθ1,∧θ2∧Covθ1,∧θ3∧Covθ1,∧w1∧Covθ1,∧w2∧Covθ2,∧θ1∧Varθ2∧Covθ2,∧θ3∧Covθ2,∧w1∧Covθ2,∧w2∧Covθ3,∧θ1∧Covθ3,∧θ2∧Varθ3∧Covθ3,∧w1∧Covθ3,∧w2∧Covw1∧,θ1∧Covw1∧,θ2∧Covw1∧,θ3∧Varw1∧Covw1∧,w2∧Covw2∧,θ1∧Covw2∧,θ2∧Covw2∧,θ3∧Covw2∧,w1∧Varw2∧
where
∂2lnLΩ|y∂θ12=2s1θ12−3∑h=1s1y1h2θ14+3n−sw1t4exp−t22θ12w2exp−t22θ22+1−w1−w2exp−t22θ32θ17w1exp−t22θ12+w2exp−t22θ22+1−w1−w2exp−t22θ322.
∂2lnLΩ|y∂θ22=2s2θ22−3∑h=1s2y2h2θ24+3n−sw2t4exp−t22θ22w1exp−t22θ12+1−w1−w2exp−t22θ32θ27w1exp−t22θ12+w2exp−t22θ22+1−w1−w2exp−t22θ322.
∂2lnLΩ|y∂θ32=2s3θ32−3∑h=1s3y3h2θ34+3n−s1−w1−w2t4exp−t22θ32w1exp−t22θ12+w2exp−t22θ22θ37w1exp−t22θ12+w2exp−t22θ22+1−w1−w2exp−t22θ322.
∂2lnLΩ|y∂w12=−s1w12−s31−w1−w2 2+n−sexp−t22θ12−exp−t22θ32w1exp−t22θ12+w2exp−t22θ22+1−w1−w2exp−t22θ322.
∂2lnLΩ|y∂w22=−s2w22−s31−w1−w2 2+n−sexp−t22θ22−exp−t22θ32w1exp−t22θ12+w2exp−t22θ22+1−w1−w2exp−t22θ322.

8.3. ML Estimators and their Variances for Complete Data

All the censored observations become uncensored if t approaches to ∞ and sl approaches to nll=1,2,3. The proficiency of the ML estimators is increased due to the fact that in complete data all the observations are merged in our data. The mathematical expressions of the ML estimators and as well as their variances expression based on complete data are shown below:

θ∧1=∑h=1n1y1h22n1,θ∧2=∑h=1n2y2h22n2,θ∧3=∑h=1n3y3h22n3,w∧1=n1n,w∧2=n2n,Varθ1∧=∑h=1n1y1h28n12,Varθ2∧=∑h=1n2y2h28n22,Varθ3∧=∑h=1n3y3h28n32,Varw1∧=n1n3n2n1+n3,Varw2∧=n2n3n2n2+n3.

9. MONTE CARLO SIMULATION

In this section, we have used Monte Carlo simulation to find the MLEs and MLVs for the proposed mixture distribution. Through the following steps, we obtained the MLEs and MLVs, as follows:

  1. First we generate w1n, w2n and w3n observation from f1y;θ1, f2y;θ2 and f3y;θ3, respectively.

  2. A censored sample is selected at a fixed t and observations which are greater than t will be considered as censored observations. When generated an uncensored (complete) sample then this step is neglected.

  3. The Steps 1 and 2 are repeated 1000 times for all selected choices of parameters.

  4. To find the MLEs and MLVs of θ1,θ2,θ3,w1 and w2 based on the samples obtained in Step 3.

The abovementioned Steps 1–4 used for different sample size n = 50, 100, 200, 500, parameters values θ1,θ2,θ3,w1,w2=13,14,15,0.3,0.5,16,15,14,0.5,0.3 and test termination time t=24,32. In such condition, the choice of t=24,32 was done to have 10% to 20% censored rate in the resulting sample.

From Tables 2 and 3, if θ1>θ2>θ3 and w1>w2, it is observed that parameters θ2,θ3 and w1 are over-estimated but θ1 and w2 are under-estimated at different values of n and t. On the other hand, if θ1<θ2<θ3 and w1<w2, it is pointed that parameters θ2,θ3 and w2 are over-estimated whereas θ1 and w1 are under-estimated at various values of n and t. Also, the amount of over-estimation of θ1,θ2,θ3,w1 and w2 is smaller for a large n at different values of t, and an opposite behavior was noticed for a small t at a given value of n. Moreover, the parameters θ1,θ2,θ3,w1 and w2 were pointed under-estimated to a smaller degree when the true values of θ1,θ2,θ3 were larger at various choices of t for a fixed n. The difference of the MLEs of parameters θ1,θ2,θ3,w1 and w2 from the nominal values becomes the least with the increase of n at a fixed t.

t n θ^1 θ^2 θ^3 w^1 w^2
24 50 11.80089 14.61541 15.01973 0.311001 0.450384
24 100 12.14090 14.47142 15.79121 0.309475 0.459040
24 200 12.32481 14.41900 15.71900 0.306542 0.471284
24 500 12.64047 14.30591 15.49235 0.305908 0.483244
32 50 12.30870 14.50639 15.28540 0.307408 0.473077
32 100 12.572412 14.32247 15.64205 0.305157 0.482970
32 200 12.70826 14.24910 15.39890 0.303980 0.489530
32 500 12.81997 14.11887 15.19401 0.302209 0.496001
Table 2

The MLEs values of the proposed mixture distribution for θ1=13,θ2=14,θ3=15,w1=0.3,w2=0.5 and t=24,32 under censored data.

t n θ^1 θ^2 θ^3 w^1 w^2
24 50 14.95881 15.60520 14.95971 0.442387 0.314019
24 100 15.20851 15.47148 14.84121 0.459401 0.311701
24 200 15.32170 15.40986 14.55412 0.470451 0.310951
24 500 15.71638 15.29501 14.42015 0.482201 0.308541
32 50 15.41810 15.55680 14.21510 0.461050 0.309451
32 100 15.54207 15.39238 14.74209 0.485094 0.306105
32 200 15.70844 15.25910 14.43770 0.489451 0.304404
32 500 15.85159 15.13786 14.21241 0.495001 0.302199
Table 3

The MLEs values of the proposed mixture distribution for θ1=16,θ2=15,θ3=14,w1=0.5,w2=0.3 and t=24,32 under censored data.

From Tables 4 and 5, it is revealed that the θ1,θ2 and θ3 are under-estimated as the value of t→∞. The degree of under-estimation of θ1,θ2 and θ3 is lesser for larger values of n. The degree of under-estimation or over-estimation of θ1,θ2 and θ3 is greater for censored data than the complete data at various choices of n. It is also revealed that for t→∞ the difference in the values of MLEs and assumed values of parameters will be reduced with the increase in n.

n θ^1 θ^2 θ^3 w^1 w^2
50 12.88630 13.89810 14.68280 0.30000 0.50000
100 12.90350 13.93310 14.90500 0.30000 0.50000
200 12.96220 13.97070 14.94290 0.30000 0.50000
500 12.96660 13.99730 14.99860 0.30000 0.50000
Table 4

The MLEs values of the proposed mixture distribution for θ1=13,θ2=14,θ3=15,w1=0.3,w2=0.5 under complete data.

n θ^1 θ^2 θ^3 w^1 w^2
50 15.91150 14.93810 13.87220 0.50000 0.30000
100 15.94880 14.95030 13.92910 0.50000 0.30000
200 15.95560 14.97220 13.98150 0.50000 0.30000
500 15.98630 14.98730 13.99820 0.50000 0.30000
Table 5

The MLEs values of the proposed mixture distribution for θ1=16,θ2=15,θ3=14,w1=0.5,w2=0.3 under complete data.

10. APPLICATION BASED ON REAL DATA SET

In this section, we present the real-life application of the proposed mixture distribution based on three components, i.e., Transmitter Tube (V805), Transmitter Tube and Indicator Tube (V600) related to the aircraft (cf. Davis [11]). Davis [11] showed that the distribution of these components is exponential. For an exponential random data x, the suitable transformation y=2x gives the Rayleigh random data y. Thus, the suggested mixture distribution can be a fair choice to model the abovementioned mixture lifetime data. In addition, it is not known that which component of an aircraft radar set fails until the condition of disappointment of a set of radar arises before or at t=600 hours. The summary of the data for t=600 is as follows:

n=1340,s1=866,s2=337,s3=83,s=s1+s2+s3=1286,n−s=54,∑k=1s1y1k2=2∑k=1s1x1k=268160,∑k=1s2y2k2=2∑k=1s2x2k=100750,∑k=1s3y3k2=2∑k=1s3x3k=32500.
when t→∞, then the summary of a complete data set is as follows:
n=1340,n1=903,n2=337,n3=100,n=n1+n2+n3=1340,∑k=1n1y1k2=2∑k=1n1x1k=322660,∑k=1n2y2k2=2∑k=1n2x2k=100750,∑k=1n3y3k2=2∑k=1n3x3k=59500.

The MLEs and MLVs are shown in Table 6.

t θ^1 θ^2 θ^3 w^1 w^2
MLEs 600 12.87401 11.73240 16.62847 0.642813 0.240017
MLVs 600 0.054882 0.116176 0.809115 0.000055 0.000048
ML estimates ∞ 13.36638 12.22623 17.24819 0.673880 0.251492
Variances ∞ 0.049463 0.110891 0.743750 0.000050 0.000043
Table 6

MLEs and MLVs using lifetime mixture under censored and complete data.

From Table 6, it is observed that the ML estimators based on complete data are more efficient than the ML estimator using censored data due to the lesser values of MLVs.

11. CONCLUSION

In this study, we have discussed some basic statistical properties, various statistical functions, some important entropies and different order statistics for the proposed 3-component mixture of Rayleigh distributions. A Monte Carlo simulation is used to evaluate the performance of the unknown parameter based on the ML estimator under censored and uncensored sampling schemes. To explain a practical application of the proposed mixture model, a real-life example has also been analyzed.

From simulated results, it has been observed that an increase in t under a fixed n yield very efficient ML estimators and vice versa. It is also noticed that the parameters are over-estimated to a small degree with relatively larger n. However, the degree of over-estimation (under-estimation) of parameters is smaller for a relatively large parameter value and vice versa. Finally, it is concluded that the results are more efficient under complete data as compared to censored data due to associated least MLVs.

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

AUTHORS' CONTRIBUTIONS

Conceptualization: Muhammad Tahir, Muhammad Abid; Formal analysis: Muhammad Abid, Sidra Mohsin; Methodology: Muhammad Tahir; Original draft: Muhammad Tahir; Review & Editing: Muhammad Abid, Mohammad Ahsanullah.

ACKNOWLEDGMENTS

The authors are grateful to the anonymous reviewers for their valuable suggestions that helped in improving the initial version of the manuscript.

REFERENCES

8.A.I. Shawky and R.A. Bakoban, J. Appl. Sci. Res., Vol. 5, 2009, pp. 1351-1369.
10.M. Wolfram, Version 9.0, Champaign, IL, USA, 2012.
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Journal
Journal of Statistical Theory and Applications
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Pages
204 - 218
Publication Date
2021/06/23
ISSN (Online)
2214-1766
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risenwbib
TY  - JOUR
AU  - Muhammad Tahir
AU  - Muhammad Ahsanullah
AU  - Sidra Mohsin
AU  - Muhammad Abid
PY  - 2021
DA  - 2021/06/23
TI  - On Finite 3-Component Mixture of Rayleigh Distributions: A Classical Look
JO  - Journal of Statistical Theory and Applications
SP  - 204
EP  - 218
VL  - 20
IS  - 2
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.210616.001
DO  - 10.2991/jsta.d.210616.001
ID  - Tahir2021
ER  -