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C. Subramanian, M. Subhashree, Aafaq A. Rather RT&A, No 4 (76)
EXPLORING NOVEL EXTENSION OF SUJA DISTRIBUTION ... Volume 18, December 2023
EXPLORING NOVEL EXTENSION OF SUJA DISTRIBUTION: UNVEILING PROPERTIES AND DIVERSE APPLICATIONS
C. Subramanian1, M. Subhashree2, Aafaq A. Rather3*
12 Department of Statistics, Annamalai University, Annamalai Nagar, Tamil Nadu, India 3*Symbiosis Statistical Institute, Symbiosis International (Deemed University), Pune-411004, India [email protected], [email protected], 3*[email protected]
Abstract
This research article introduces and explores the area biased techniques of the Suja distribution, presenting novel derivations and insights. The estimation of the one-parameter area biased Suja distribution is accomplished using maximum likelihood, providing a robust framework for modeling real-world data. A comprehensive study of several statistical properties is conducted to unveil the characteristics and behaviors of this new model. To demonstrate its practical applicability, the proposed distribution is applied to real data of weather temperature. The analysis showcases the distribution's effectiveness in capturing the intricacies of temperature patterns, revealing its potential utility in weather modeling and related applications. The research contributes to the advancement of statistical modeling techniques and enriches our understanding of the Suja distribution's versatility in handling diverse datasets.
Keywords: Area-biased distribution, Suja distribution, Maximum likelihood Estimator
1. Introduction
The concept of area- biased distribution was explained earlier by Fisher [5]. He first introduced weighted distribution which is a combination of model specification and data interpretation. Also, stated that "the estimation of frequencies based on the effective methods of ascertainment". The size-biased are the special cases of the weighted distribution. Later, Rao [16] explained that the weighted distributions as many applications such as medicine, reliability, ecology, behavioral science, finance, insurance, etc. A discrete Poisson area-biased Lindley distribution was purposed by Bashir and Rasul [3] and explained its properties. Also, applied in biological data and compared with Poisson distribution. Zahida et.al. [18] Purposed a new extension of Weibull distribution called area- biased weighted Weibull distribution. The characterization of this model was derived and shown how the model fitted to the problem of ball bearing data. Rama Shanker [10] introduced a new one- parameter Suja distribution and estimated the parameter using method of moments and maximum likelihood. The important properties were explained and finally, compared this model with other one-parameter distributions by applying a real lifetime data.
Ayesha Fazal [2] introduced an area-biased Poisson exponential distribution with its moments and other properties, Also, the goodness of fit for this model has been discussed to show how it fit in real
data sets. Many studies on length biased distribution case has been published, for example; Rather and Subramanian [11], Rather and Subramanian [12], Rather and Ozel [13], Rather and Subramanian [19], Rather et al. [18]. A new generalized area- biased Aradhana distribution was introduced by Elangovan and Mohanasundari [4] and estimated the parameters by maximum likelihood estimator. Finally, applied a real lifetime data set in the model to show how it works. The new extension of Suja distribution called length-biased Suja distribution was given by Ibrahim Al-omari and Islam Khaled [6]. The various properties of this model were explained and shown the usefulness of the model in the real data set. Later, Ibrahim Al-omari et.al. [7] extended the length-biased Suja distribution to power length-biased Suja distribution with its characteristics and estimated the two parameters of this proposed distribution by maximum likelihood method. Finally, they illustrated a real data to show the performance of this model. Arun Kumar Rao and Himanshu Pandey [1] studied the parameter estimation of area-biased Rayleigh distribution by using maximum likelihood estimation and Bayesian estimation with quasi and inverted gamma priors. The weighted Suja distribution as an extension of most important Suja distribution was discussed by Islam Khaled and Ibrbhim Al-Omari [8]. Also, explained its statistical properties and application in the ball bearings data. The new generalization method as area-biased was introduced by Nuri Celik [9] for beta, Rayleigh and log-normal distributions. Also, the main statistical properties and parameters estimation were studied. At last, some of the real data examples were used. Shanker, Upadhyay and Shukla [17] were given a new two parameter quasi Suja distribution with its characteristics and parameters estimation. Also, they illustrated with real data to show its performance.
2. Area-biased Suja Distribution (ABSD)
The probability density function (p.d.f.) and cumulative distribution function (c.d.f.) of the Suja distribution is given by
a5
f(y-«) = ■ nA1 + y4)e-az;y>0,a>0 (1)
F(y,a) = 1
a4 + 24
(a4y4 + 4a3y3 +120a2y2 + 24ay)
1+
a4 + 24
;y>0,a>0 (2)
We know that, the weighted function is , f . w(y)f(y)
fw(y)=-^y)>) ;X>0 (3)
Where, w(x) be a non-negative weight function. E(w(y)) = f w(y)f(y)dy < w(y) = yc
ycf(y)
;y>0 (4)
where, E(yc) = f yc f(y) dy For area-biased function, put c=2,
E(y2) where,
*> oo
E(y) = I y2 f(y)dy J0
substitute (1) in (6), we get
y2f(y)
fAB(y)=y-^([) ;y>0 (5)
(6)
2(a4 + 360)
Thus, the p.d.f. and c.d.f. of the area-biased Suja distribution (ABSD) can be obtained as
a1
fABSD(y; a) = 2(a4 + 24)y2(1 + y4)e-ay; y>0;a>0 (8)
C. Subramanian, M. Subhashree, Aafaq A. Rather RT&A, No 4 (76)
EXPLORING NOVEL EXTENSION OF SUJA DISTRIBUTION ... Volume 18, December 2023
1
fabsd (у> a) = 2a4 + 720 [а4у(3, ay) + y(7, ay)] (9)
The graphical representation of the p.d.f. and c.d.f. of area-biased Suja distribution (ABSD) are also shown below:
■— a= 0 7
□— a,= 09
IS— a = 1.1
в— a= 1.3
□— a= 1.5
Cj О
0 5 10 15
Fig. 1: Pdf plot of ABSD
Fig. 2: Cdf plot of ABSD
3. Reliability Analysis
3.1 Reliability Function
The survival function of the area-biased Suja distribution is given by
S(y) = 1 - F(y) 1
S(y) = 1 - 2g4 + 720 [a4Y(3, ay) + y(7, ay)]
3.2 Hazard Function
The hazard function is also known as the hazard rate, instantaneous failure rate or force of mortality and is given by
h( . _ fABSD(y;a)
1 - FABSD(y>a) a7y2(1 + y4)e-ay
h(y) =■
(2a4 + 720) - [a4y(3,ay) + y(7,ay)] 3.3 Reverse Hazard Function
The reverse hazard function or reverse hazard rate is given by
hr(y) =
hr(y) =
pABSD^y; a)
а7у2(1+у4)е-аУ [a4y(3, ay) + y(7, ay)]
3.4 Mill's Ratio
The Mills ratio of the area-biased Suja distribution is 1
Mills Ratio = -—r—r
hr(y)
[a4y(3, ay) + y(7, ay)] = a7y2(1 + y4)e-ay
4. Moments
The moments of ABSD have been derived to describe the characteristic of the proposed model. Then, the rth order moment E(yr) of ABSD is derived as
fi'r = E(yr) = J0° yr Fabsd (y; a)dy (10)
v'r = 2^f0°°yr+2(i+y4)e-ay dy (11)
' _ a4T(r + 3) 2- r(r + 7)
ar(2a4 + 720) ( )
In equation (12), when r=1, the mean of ABSD which is given by
; _ 3(a4 + 840) ^ = a(a4 + 360) Similarly, when r=2, 3, 4 in equation (4.1), we will get
; _ 12(a4 + 1680) ^2 = a2(a4 + 360) ; _ 60(a4 + 3024) ^3 = a3(a4 + 360) , _ 360(a4 + 5040) ^'4= a4(a4 + 360) The central moments about the mean of this distribution are given as
Vo= V'o = 1
Ho = 0
2 _ 12(a4 + 1680) {3(a4 + 840)\2
V>.2 = V'2 faD2 = a2 (a4 + 360) (a(a4 + 360)) Therefore, the variance of ABSD is
_ 9(a8+2160a4 +571200) ^2 = a2(a4 + 360)2
_ 60(a12 + 8280a8 + 388800a4 + 108864000) P3=v3- + 2(ni)3 --
V-4
a3(a4 + 360)3 1*4= V'- 4^1 + 6v'M)3 - 3(M'i)4 45(a4 + 360)(a12 + 21048a8 + 15871680a4 + 741208320)
a4(a4 + 360)4
The standard deviation (a), co-efficient of variation (C.V.), co-efficient of skewness (Vft), co-efficient of
kurtosis (p2) and index of dispersion (y) of ABSD are obtained as
_ 3^ a8 + 2160a4 + 571200
° = a(a4 + 360)
a ^ a8 + 2160a4 + 571200
C.V. = — =-
(a4 + 840)
_ n3 _ 2(a12 + 8280a8 + 388800a4 + 108864000) VPi = 3 = 3
H22 9(a8 + 2160a4 + 571200)2
_ 5(a12 + 21048a8 + 15871680a4 + 7412083200) ^2=y2= a2(a4 + 360)(a8 + 2160a4 + 571200)
_a2 _ 3(a8+2160a4+S71200) Y a(a4 + 360)(a4 + 840)
(13)
4.1 Harmonic Mean
The Harmonic mean of the aspired model can be derived as
H.M.= E[1\ = S0 1fABSD(y)dy (14)
H.M^C1"' ,y2(1 + y4)e-ay dy (15)
Therefore,
a(a4 + 120) 2(a4 + 360)
H.M.=
4.2 Moment Generating Function and Characteristic Function
Assume Y have area-biased Suja distribution, we will get the moment generating function of Y as
My(t) = E(ety) = J0° e* fABSD(y)dy (16)
Using Taylor's expansion,
My(t) = ÇT,r=0(JjffABSD(y)dy = ^hJH) (17)
My(t) = + 3 + r(r + 7)) (18)
In the same way, we will get the characteristics function of SBJD can be obtained as
<py(t) = E[e*y] = -¡^^fflpnr + 3) + F(r + 7)) (W)
5. Order Statistics
Let Yi, Y2,.........,Yn be the random variable drawn from the continuous population. Their p.d.f. be fy(y)
and cumulative density function be Fy(y). Then, assume Y(i),Y(2),........Yw be the order statistics of a
random sample.
Thus, the probability density function of rth order statistics Yr is given by
n1 r-1 n-r
fy^) = (n-r)i(r-1)if(y)[Fy(y)]- t1 - F^ - (20)
Putting the equation (8) and (9) in equation (20), the probability density function of rth order statistics
Y(r) of ABSD is given by
f (v-) = n! \a7y2(l+y4)e-ay ] ([a4r(3,ay)+Y(7,ay)hr-1 v
h(r)(y) (n-r)!(r-1)! { 2(a4 + 360) jH 2(a4 + 360) } (21)
( _ [a4y(3, ay) + y(7, ay)]}"-X{ 2(a4 + 360) }
Then, the probability density function of higher order statistics Ym can be derived as
(na7y2(l+y4)e-ayï ([a4Y(3,ay)+Y(7,ay )]^n-1 'y(n)(y) { 2(a4+360) JXl 2(a4+360) } (22)
Hence, the probability density function of 1st order statistics Y(i) can be obtained as
M _ (na7y2(1+y4)e-ay^ [a4Y(3,ay)+Y(7,ay)]-)n-1
2(a4 + 360) J I 2(a4 + 360) J v '
6. Maximum Likelihood Estimator and Fisher Information Matrix
The maximum likelihood estimator is the best numerical stability estimator to estimate the parameters of the distribution when compared with other estimating methods. Thus, we used this method to estimate the parameters of ABSD which is derived below:
Let Y(1), Y(2),.........,Y(n) be the random sample of size n drawn from the ABSD, then, the likelihood
function of ABSD is
L(y;a) = n fABSD(y; a) = 2n(a4 + 360)- n*(1 + yt)e
2n(a4 + 360)nI
í=i í=i
4)p-ayi
The natural log likelihood function is
\ogL(y;a) = 7n\oga - n\og(2a4 + 720) + 2Z?=1\ogyi + Z?=1\og(1 + y4) - aYJli=1yi (24)
By differentiating equation (24) with respect to a, the maximum likelihood estimates of a can be attained as
dlogL 7n 8na3
Xy, = 0 (25)
da a (2a4 + 720)
í=i
Because of the complicated form of likelihood equation (25), algebraically it is very difficult to solve the system of non-linear equation. Therefore, we use R and wolfram Mathematica for estimating the required parameters.
n
n
7. Likelihood ratio Test
This test is used to compare the goodness of fit of the two models based on the ratio of their likelihoods.
Suppose Y(i), Y(2),.........,Y(n) be a random sample from the ABSD. To test, the random sample of size n for
ABSD, the hypothesis is
H0: f(y) = fsoty; a) agaist Hi f(y) = fABsD{y; a) To check whether the random sample of size n comes from the Suja distribution or size-biased Suja distribution, the likelihood ratio is
n ~ 1
A_ L1 _T~\fABSP(y; g) _ a2(a2+24)\
1=1
Therefore, the null hypothesis is rejected if
a2(a2 + 24)]
2(a4 + 360)
№
i=1
A=
2(a4 + 360)1
u
n,.2
1=1
> k
■= n*2
> k* where k* = k
a2(a2 +24)
2(a4 + 360)
>0
We can conclude, for large sample size n, 2log is distributed as chi-square distribution with one degree of freedom. Also, p-value is attained from the chi-square distribution. If p (A*>k*), where k* = n}=1 Vi is less than the specified level of significance and ni=1yi is the observed value of the statistic A*, then, reject null hypothesis.
8. Applications
It is a measure to check how a statistical model fits a data set. Here, we will discuss how the proposed model is fit to a data set which is illustrated below. Also, compare with Suja distribution to show better fit of area-biased Suja distribution. Let us represent the data set of weather temperature in Bangladesh from 2016 to 2019 by Shawkat Sujon in the website: https://www.kaggle.com/datasets/shawkatsujon/bangladesh-weather-dataset-from-1901-to2019. The data set is given as follows:
Table 1: Data of weather temperature in Bangladesh from 2016 to 2019
n
A
1=1
Year 2016 2016 2016 2016 2016 2016 2016
Month 1 2 3 4 5 6 7
Temperature 17.34 22.12 25.93 28.25 27.94 28.96 28.17
Year 2016 2016 2016 2016 2016 2017 2017
Month 8 9 10 11 12 1 2
Temperature 28.85 28.58 27.74 23.45 21.43 19.36 21.31
Year 2017 2017 2017 2017 2017 2017 2017
Month 3 4 5 6 7 8 9
Temperature 23.69 26.91 28.76 28.58 28.27 28.57 28.54
Year 2017 2017 2017 2018 2018 2018 2018
Month 10 11 12 1 2 3 4
Temperature 27.16 24.14 20.59 17.59 21.18 25.41 26.79
Year 2018 2018 2018 2018 2018 2018 2018
Month 5 6 7 8 9 10 11
Temperature 27.32 28.68 28.59 28.88 28.66 26.17 22.76
Year 2018 2019 2019 2019 2019 2019 2019
Month 12 1 2 3 4 5 6
Temperature 19.13 19.38 20.71 24.4 27.41 28.92 29.18
Year 2019 2019 2019 2019 2019 2019
Month 7 8 9 10 11 12
Temperature 28.72 29 28.28 26.94 23.88 18.51
In order to compare the distributions, we study the criteria like Akaike Information Criterion (AIC), Akaike Information Criterion Corrected (AICC), Bayesian Information Criterion (BIC) and -2logL. The better distribution is which compatible to lesser values of AIC, BIC, AICC and -2logL.
Table 2: Comparison of distributions
Distribution MLE S.E -2logL AIC BIC AICC
Suja a = 0.1965312 0.0126848 327.644 329.644 331.5152 329.730
Length-biased Suja a = 0.2358445 0.0138967 319.7202 321.7202 323.5914 321.8071
Area-biased Suja a = 0.2751537 0.0150105 313.2269 315.2269 317.0981 315.3138
9. Conclusion
In conclusion, this article presents a comprehensive investigation of the area-biased Suja distribution (ABSD), a noteworthy extension of the weighted distribution paradigm. The obtained probability density function (p.d.f.) and cumulative distribution function (c.d.f.) enrich the theoretical foundations of the ABSD, laying a solid groundwork for its application in various domains.
One significant advantage of the ABSD is its robust parameter resiliency, which leads to enhanced performance and more accurate results compared to other distributions. The maximum likelihood estimator proves to be an effective tool for estimating the distribution's parameter, and its validity is confirmed through the likelihood ratio test, reinforcing the reliability of our findings. The in-depth analysis of various statistical properties provides valuable insights into the ABSD's behavior and characteristics, fostering a deeper understanding of this novel model.
In particular, when applied to weather temperature data, the ABSD demonstrates superior compatibility compared to both the standard Suja distribution and the length-biased Suja distribution. The results suggest that the ABSD better captures the intricate patterns and variations inherent in weather temperature datasets.
The findings presented in this article not only enhance the understanding of the ABSD but also contribute to the broader field of statistical modeling. The improved performance in weather temperature analysis highlights the practical applicability of the ABSD in real-world scenarios. This distribution's versatility and ability to handle diverse datasets make it a promising candidate for future research and application across various domains. As the field of statistical modeling continues to evolve, the ABSD offers a valuable addition to the toolkit of data analysts and researchers alike.
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