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ENHANCING ENGINEERING SCIENCES WITH UMA DISTRIBUTION: A PERFECT FIT AND VALUABLE
CONTRIBUTIONS
R. A. Ganaie1, C. Subramanian2, V. P. Soumya3, R. Shenbagaraja4, Mahfooz Alam5, D. Vedavathi Saraja6, Rushika Kinjawadekar7, Aafaq A. Rather8*, Showkat A. Dar9
12, 3 6Department of Statistics, Annamalai University, Tamil nadu, India 4 Department of Statistics, Dr. M.G.R. Govt. Arts and Science College for Women, Tamil Nadu, India ^Department of Statistics, Faculty of Science and Technology, Vishwakarma University, Pune, India
7Marathwada Mitra Mandal, Pune, India 8*Symbiosis Statistical Institute, Symbiosis International (Deemed University), Pune-411004, India 9Department of Computer Science and Engineering, GITAM University Bangalore Campus, India [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], 8*[email protected], [email protected]
Abstract
In this study, we introduce a novel class of distributions called the length biased Uma distribution. This distribution is a specific instance of the broader weighted distribution family, known for its versatility in various applications. We explore the structural properties of the length biased Uma distribution and propose a robust parameter estimation technique based on maximum likelihood estimation. To assess its efficacy, we apply the newly introduced distribution to two real-world datasets, evaluating its flexibility and performance in comparison to existing models. The results obtained demonstrate the potential of the length biased Uma distribution as a valuable addition to the repertoire of statistical distributions, offering valuable insights for a wide range of practical applications.
Keywords: Length biased distribution, Uma distribution, Reliability analysis, Order statistics, Maximum likelihood estimation.
1. Introduction
In probability distribution theory, the concept of weighted probability models have attained a great importance for modeling different lifetime data sets occurring from various practical and applied fields like engineering, medical sciences, insurance, finance etc. There are several situations were classical distributions may not provide best fit to lifetime data. In such situation attempt has been made to generalize standard distribution by introducing an extra parameter to it. This extra parameter can be introduced through various techniques. One of such technique is of weighted technique. Fisher [7] introduced the concept of weighted distribution to model the ascertainment bias which was later formalized by Rao [16] in a unifying theory for problems were the observations fall in non-experimental, non-replicated and non-random categories. The weighted distributions are remarkable
for efficient modeling of statistical data and prediction obviously when classical distributions are not suitable. The weighted distributions were formulated in such a situation to record the observation according to some weight function. The weighted distribution reduces to length biased distribution when the weight function considers only the length of units of interest. The probability of selecting an individual in a population is proportional to its magnitude is called length biased sampling. However, when observations are selected with probability proportional to their length, resulting distribution is called length biased distribution. The concept of length biased distribution was introduced by Cox [5] in renewal theory. Length biased sampling situation occurs were a proper sampling frame is absent. In such cases items are sampled at a rate proportional to their lengths so that the larger values could be sampled with higher probability.
A significant and remarkable contribution has been done by several authors to develop some important length biased probability models along with their applications in various fields. Oluwafemi and Olalekan [15] discussed on length and area biased exponentiated weibull distribution based on forest inventories. Ekhosuehil et al. [6] proposed the weibull length biased exponential distribution with statistical properties and applications. Atikankul et al. [2] discussed on the length biased weighted Lindley distribution with applications. Ratnaparkhi and Nimbalkar [18] presented the length biased lognormal distribution and its application in the analysis of data from oil field exploration studies. Mathew [14] proposed the reliability test plan for the Marshall Olkin length biased Lomax distribution. Mustafa and Khan [13] obtained the length biased power hazard rate distribution with its properties and applications. Chaito et al. [3] discussed on the length biased Gamma-Rayleigh distribution with applications. Al-omari and Alanzi [1] introduced the inverse length biased Maxwell distribution with statistical inference and application. Ghorbal [11] discussed on properties of length biased exponential model with applications. Ganaie and Rajagopalan [9] presented the weighted power Garima distribution with applications in blood cancer and relief times. Recently, Chaito and Khamkong [4] discussed on length-biased weibull-Rayleigh distribution for application to hydrological data.
Uma distribution is a recently introduced one parametric continuous lifetime distribution proposed by Shanker [19]. Its various statistical properties like moments, mean residual life function, hazard rate function, reverse hazard function, stochastic ordering, coefficient of variation, skewness, kurtosis and index of dispersion have been discussed. For estimating its parameter the method of maximum likelihood estimation has been discussed.
2. Length Biased Uma (LBU) Distribution
The probability density function of Uma distribution is given by
q4 f \
f{x;&) =-11 + x + x3 I e" ^; x > 0, 9> 0 (1)
93 +92 + 6 ^ '
and the cumulative distribution function of Uma distribution is given by
6x (c
1 + —y
F ( x;6>) = 1 -
^ Ox \02x2 + 39x + 92 + 6 ^ 3 . r,2
\
e Ox ; x > 0, 6> 0 (2)
eD + e2 + 6 j
Consider X be the non-negative random variable with probability density function f (x) .Let its nonnegative weight function be w( x), then the probability density function of weighted random variable is given by
, , , w(x)f(x) fw(x) = m ( * , X > 0'
E(w( x))
Where w(x) be the non - negativeweight functionand E(w(x)) = Jw(x) f (x)dx < For various forms of weight function w(x) obviously when w(x) = xc, resulting distribution is called weighted distribution. In this paper, we have considered the weight function at w(x) = x to obtain the length biased version of Uma distribution and its probability density function is given by
* (x) = f) (3)
E(x)
Where E(x) = J xf (x, 0)dx 0
<93 + 262 + 24
E( x) = --3-2---(4)
<9(<93 +<92 + 6)
By using the equations (1) and (4) in equation (3), we will obtain the probability density function of length biased Uma distribution as
x95 ( - 6x
fl (x) = —5-2-11 + x + x3 I e (5)
' (63 + 2 62 + 24) V '
and the cumulative distribution function of length biased Uma distribution can be obtained as
x
Fl (x) = J ft (x)dx 0
x x05 I 31 - 6x F (x) = J-11 + x + x3 I e dx
0 (63 + 262 + 24)
F ( x) =-
(03 + 202 + 24)
1 x 51 3 1 - 0x
F ( x) =-J x6 1l + x + x I e dx
(03 + 202 + 24) 0
( \
95 XJxe-0xdx + 95 xx2 e-&dx + 95 xx4 e- 0xdx V 0 0 0 ;
(6)
dt t Put 0x = t ^ 9dx = dt ^ dx = —, Also x = —
9 9
When x ^ x, t ^ 0x and When x ^ 0, t ^ 0
After the simplification of equation (6), we will obtain the cumulative distribution function of length biased Uma distribution as
F (x) = —5-1-(93r(2, 0x) + 92r(3, 0x) + /(5, 0x) I (7)
(0 + 202 + 24) V '
1
3. Reliability Analysis
In this section, we will discuss about the reliability function, hazard rate function, reverse hazard rate function and Mills ratio of the proposed length biased Uma distribution.
3.1 Reliability function
The reliability function is termed as survival function and the reliability function of executed length biased Uma distribution can be determined as
R( x) = 1 - Fl (x)
R(x) = 1--1-i93 y(2, 6x) + 92 y(3, 6x) + y(5, 6x) 1 (8)
(63 + 262 + 24) ^ J
3.2 Hazard function
The hazard function is also known as failure rate or force of mortality and is given by
h( x)=fxL 1 - F (x)
--(9)
(63 + 262 + 24) - (93y(2, 6x) + 92y(3, 6x) + y(5, 6x))
3.3 Reverse hazard function The reverse hazard rate function is given by
h ( ^ fl (x)
K(x)=-
F, (x)
», /- ^ x05(1 +x +x3) e-6X nm
hr (x) = —----(10)
(<93y(2, 6x) + <92 y(3, 6x) + /(5, 6x))
3.4 Mills Ratio
The Mills Ratio of proposed model is given by
MR _ 1 _ (^3/(2, 6x) + 02y(3, 6x) + /(5, 6x)) . hr (x) x05(1 + x + x3) e-6x
0.00 0.02 0.04 0.06 OOS
0 2 4 6 6
Fig. 3: Reliability plot of LBU distribution Fig. 4: Hazard rate plot of LBU distribution
4. Order Statistics
Suppose X(1), X(2),..., X(n) be the order statistics of a random sample X1, X2,..., Xn from a continuous population with cumulative distribution function FX(x) and probability density function fx (x) then the probability density function of rth order statistics X(r) is given by
fx(r)(x ) -
n!
(r — 1)! (n - r)!
fx (x){Fx (x)) r — 1 (1 — Fx (x)>
(12)
Now substituting the equations (5) and (7) in equation (12), we will obtain the probability density function of rth order statistics X(r) of length biased Uma distribution as
V . . Ar-1
fx(r)(x) --
(r - 1)!(n - r)!
x0
y (6 3 + 26 2 + 24)
L 3) — 6x
(1 + x + x3) e
1
y (63 + 26 2 + 24)
diy(2, 6x) + 02r(3, 6x) + y(5,6x)
1 --
(6 3 + 26 2 + 24)
e3y(2, 6x) + e2y(3, 6x) + y(5, 6x)
Therefore, the probability density function of higher order statistic X(n) of length biased Uma distribution can be determined as
n
n—r
1
X
fx(n)(x) -
nxd
3 2 (d3 + 262 + 24)
L 3] -6x
Ii + x + x3 I e
sn-1
3 2 ^ (63 + 26 2 + 24)
03y(2, 6x) + e2y{3, 6x) + y(5, 6x)
and probability density function of first order statistic X(i) of length biased Uma distribution can be
determined as
fx(1) (x ) = ■
nxd'
'x(i)— 3 2
(63 + 26 + 24)
L 3 \ - 6x
Ii + x + x3 I e
,n - i
i--
3 2 ^ (63 + 26 + 24)
d3y(2, 6x) + d2y(3, 6x) + y(5, 6x)
5. Test for Length biasedness of Length biased Uma distribution
Consider X1, X2,...., Xn be the random sample of size n from length biased Uma distribution. To analyse its flexibility consider the hypothesis.
H0 : f (x) = f (x ; 9) against H1 : f (x) = f (x ; 9)
In order to determine, whether the random sample of size n comes from Uma distribution or length biased Uma distribution, the following test statistic rule is employed.
a = Li= "fi (x ;d) L0 ^ f (x ;d)
L n
A- -1 = n Lo i-i
(
xt d(d3 +d2 + 6) (6 3 + 26 2 + 24)
A--
L
j3 , ¿>2
d(d3 +d2 + 6) (63 + 262 + 24)
n
n xt
i-i
We should refuse to accept the null hypothesis, if
A-
d(d3 +d2 + 6) (63 + 26 2 + 24)
n xi > k i-i
(13)
(14)
(15)
(i6)
Equivalently, we should also refrain to retain the null hypothesis where
* n
A - n xi > k i-i
(
(63 + 262 + 24) d(d3 +d2 + 6)
n
* n * *
A - n xi > k , Where k - k i-i
n
3 2 > (63 + 20 + 24)
3 2 ^ d(d + d2 + 6) J
(i7)
Whether, the 2!og ^ is distributed as chi-square distribution with one degree of freedom if the sample is large of size n and also p-value is determined by using chi-square distribution. Thus, we should refuse to accept the null hypothesis, if the probability value is given by
* \ * n n
A >y J Where y =nxi is less than a specifiedlevel of significance and nxi is the experimental
i=1 i=1
value of the statisticA .
i
j
n
L
o
n
n
\
J
\
6. Statistical properties
In this section, we will derive about the different structural properties of length biased Uma distribution those include moments, harmonic mean, moment generating function and characteristic function.
6.1 Moments
Let X be the random variable following length biased Uma distribution with parameter 0, then the rth order moment E(X r) of proposed distribution can be obtained as
E(Xr) = ur' = J xrfi (x)dx
E(Xr) = ur' = J x (
E( Xr ) = Mr' = J E(Xr) = u ' =
x05
0 (63 + 262 + 24)
» xr +1 05
1 + x + x3 | e 6x dx
0fn3 , i/)2
(63 + 26 2 + 24)
1 + x + x3 | e 6x dx
0-
(63 + 262 + 24)0
! r +1 (, 3 | - 6x , J x | 1 + x + x | e dx
E(Xr) = u ' =
05
(03 + 26 2 + 24)
( r + 2) -1 - 6x , 7 ( r + 3) -1 - 6x J xv ' e dx + J xv ' e dx
0 0
, 7 ( r + 5) -1 - 6x, + J xv ' e dx
- 0
(18)
(19)
(20) (21)
(22)
After simplification, we obtain from equation (22)
E(Xr ) =ur' =
6 3 r(r + 2) + 6 2 r(r + 3) + r(r + 5) 0r (63 + 262 + 24)
(23)
Now substituting r = 1, 2, 3 and 4 in equation (23), we will obtain the first four moments of length biased Uma distribution as
2 63 + 662 +120
E(X) = U1' =
0(63 + 262 + 24)
E(X 2) = ^2' =
E(X 3) = u,' =
E(X 4) = ^4' =
663 + 246 2 + 720 02(63 + 262 + 24) 2463 +12062 +5040 03(63 + 26 2 + 24) 12063 +72062 +40320 04(63 + 262 + 24)
Variance = -
3 2 663 + 2462 + 720
2 3 2 0 (6 + 262 + 24)
32 263 + 662 +120
3 2
-0(63 + 262 + 24) -
(24)
(25)
(26) (27)
0
r
S.D(a) =
3 2 663 + 2462 + 720
92 (63 + 262 + 24)
3
2
26 + 662 +120 v9(63 + 262 + 24) j
6.2 Harmonic mean
The harmonic mean for proposed length biased Uma distribution can be obtained as
(1 I 7 1 H M = E\- |=J - f (x)dx V x J 0 x
71 x9 HM = J1
5
0 x (63 + 262 + 24)
1 + x + x3 | e 6 dx
HM = J
9-
0(63 + 262 + 24)
1 + x + x3 | e 6dx
H M = -
7/ _qx
J( 1 + x + x | e xdx
H M = -
(93 + 262 + 24) V 0
(63 + 262 + 24)0
7 (2) - 2 - 6x, 7 (2) -1 - 6Xj 7 (4) -1 - 6x , J x e dx + J ^ e dx + J x e dx
After simplification, we obtain
HM =
9(92 + 62 + 6) (63 + 26 2 + 24)
(28)
(29)
(30)
(31)
(32)
(33)
6.3 Moment generating function and characteristic function
Let X be the random variable following length biased Uma distribution with parameter 0, then the moment generating function of executed distribution can be determined as
Mx (t) = E(etx)= ]etxfl(x)dx (34)
Using Taylor's series, we obtain
Mx (t) =J
(tx)2 ^ 1 + tx + +... 2!
f, (x)dx
7 7 tj
My (t) = J s — xJ f (x)dx xw 0jJ=0 J! Jl
7 t
7 tJ
Mx (t) = Z —
J=0 J!
Mx (t) = Z-Mj'
J=0 j! J
( 3 ? 6 3rc^-+2)+62 rq-+3)+r(j+5)
9J (63 + 26 2 + 24)
5
9
5
9
0
0
0
Mx (t) =-
E (o3r(J + 2) + 02r(J + 3) + r(J + 5)
(63 + 262 + 24) j=0 j!0J
Similarly, the characteristic function of length biased Uma distribution can be determined as
(Px (t) = Mx (it)
(36)
Mx (it ) = -
1
(03 + 202 + 24) J=0 J! 9J
E Ji—T f03^(J + 2) + 02W + 3) + r(J + 5)
(37)
7. Bonferroni and Lorenz Curves
The bonferroni and Lorenz curves are also termed as income distribution or classical curves are mostly being used in economics to measure the distribution of inequality in income or poverty. The bonferroni and Lorenz curves can be written as
1 q
B(p) =--Jxf (x)dx
PMi 0
1 q -1 L(p) = pB(p) = — J x f (x)dx and q = F (p) Mi'0 I
Where /u^ =
20 3 + 60 2 +120
0(03 + 20 2 + 24)
<9(03 + 20 2 + 24) q B(p) =-^-:-J x
x05
p (203 + 60 2 +120) 0 f 03 + 20 2 + 24
1 + x + x3 I e 0xdx
, 0(03 + 202 + 24) q x2 05
B( p) =-^-:-J "
p (203 + 60 2 +120) 0 f03 + 20 2 + 24j
i , , 31 - 0x i 1 + x + x I e dx
B( p) = -
06
p (203 + 602 +120) 0
J x2 f1 + x + x3 e 0x dx
B( p) =
p(203 + 60 2 +120)
q x<3) - 1 e - ^ dx + q x(4) -1 e - ^dx + q x(6) -1 e" °x dx
(38)
(39)
(40)
(41)
(42)
After simplification, we obtain
B(p) =
L( p) =
0
p (203 + 602 +120)
(r(3, 0q) + r(4, 0q) + y(ß, 0q))
(203 + 602 +120)
(K3, 0q) + r(4, 0q) + ï(6, 0q))
(43)
(44)
8.
Maximum Likelihood Estimation and Fisher's Information Matrix
In this section, we will discuss the method of maximum likelihood estimation to estimate the parameters of length biased Uma distribution. Consider Xi, X2,...., Xn be a random sample of size n from the length biased Uma distribution, then the likelihood function can be defined as
1
6
0
V
/
6
0
L(x) = n fl (x)
i=1
L(x) = n i=1
xi 9-
63 + 26 2 + 24
1 + x - + x- I e
31„- 6xi
(45)
t5m
L(x) =
(63 + 262 + 24)n i=1
n n \xi I1 + xi + xi31 e " 6xi
(46)
The log likelihood function is given by
3 ? n
log L = 5n log 9 - n log( 63 + 262 + 24) + Z log x
i=1
3'
+ Z log 11 + x. + x.3 I-9Z xi
(47)
By differentiating log likelihood equation (47) with respect to parameter 9 and must satisfy the following normal equation
3 log L 5n
-=--n
39 9
\
(39 2 + 49) 63 + 26 2 + 24)
- Z x = 0
i
i = 1
(48)
The above likelihood equation is too complicated to solve it algebraically. Therefore, we use numerical technique like Newton-Raphson method for estimating required parameter of proposed distribution. To use the asymptotic normality results for determining the confidence interval. We have that if (A = 9)) denotes the MLE of (1 = 9). We can determine the results as
Vn(A-A) ^ N(0, I_1(A))
-1
Where I (A) is Fisher's informatio n matrix. ie .,
1
I (A) = -1
n
( (-i2
\\
Here, we can define that
E
fd2 log L ^
392
5n
92
(
--n
]3 . -">,32
3 log L
E -g-
V V 39 JJ
2
(63 + 26 2 + 24) (69 + 4) - (39 2 + 49) (63 + 26 2 + 24)2
(49)
Since a being unknown, we estimate I 1(a) by I !(a) and this can be used to obtain asymptotic confidenc interval for 9.
r-1/
9. Application
In this section, we have applied the two real data sets in length biased Uma distribution to determine its goodness of fit and then comparison has been developed in order to realize that the length biased Uma distribution provides quite satisfactory results over Uma, Shanker, Garima and Lindley distributions. The two real data sets are given below as
The following first real data set reported from Lawless [12] represents the failure times in hours of an accelerated life test of 59 conductors without any censored observation. and the data set is given below in table 1.
n
n
2
V
J
Table 1: Data regarding the failure times in hours of an accelerated life test of 59 conductors by Lawless (2003)
2.997 4.137 4.288 4.531 4.700 4.706 5.009 5.381 5.434 5.459 5.589 5.640
5.807 5.923 6.033 6.071 6.087 6.129 6.352 6.369 6.476 6.492 6.515 6.522
6.538 6.545 6.573 6.725 6.869 6.923 6.948 6.956 6.958 7.024 7.224 7.365
7.398 7.459 7.489 7.495 7.496 7.543 7.683 7.937 7.945 7.974 8.120 8.336
8.532 8.591 8.687 8.799 9.218 9.254 9.289 9.663 10.092 10.491 11.038
The following data set obtained from Folks and Chhikara [8] and studied by Gadde et al. [10] represents runoff amounts at Jug Bridge, Maryland and the data set is given as under in table 2
Table 2: Data regarding the runoff amounts at Jug Bridge, Maryland reported by Gadde et al. (2019)
0.17 1.19 0.23 0.33 0.39 0.39 0.40 0.45 0.52 0.56
0.59 0.64 0.66 0.70 0.76 0.77 0.78 0.95 0.97 1.02
1.12 1.24 1.59 1.74 2.92
To compute the model comparison criterions along with estimation of unknown parameters, the technique of R software is applied. In order to compare the performance of length biased Uma distribution over Uma, Shanker, Garima and Lindley distributions, we use criterion values like AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion), AICC (Akaike Information Criterion Corrected), CAIC (Consistent Akaike Information Criterion), Shannon's entropy H(X) and -2logL. The distribution is better which shows corresponding criterions AIC, BIC, AICC, CAIC, H(X) and -2logL values smaller as compared with other distributions. For determining the criterion values AIC, BIC, AICC, CAIC, H(X) and -2logL given below following formulas are used.
AIC = 2k - 2log L, BIC = k log n - 2log L, 2kn
AICC = AIC+
CAIC = -2 log L + -
and
H ( X ) = -
2k (k + 1)
n-k-1 2log L
n - k-1 n
Where n is the sample size, k is the number of parameters in statistical model and -2logL is the maximized value of log-likelihood function under the considered model.
Table 3: Shows MLE and S. E Estimate for Data set 1 and Data set 2
Data set 1
Distributions MLE S.E
Length Biased Uma 0 = 0.69944543 0 = 0.04014727
Uma 0 = 0.54873604 0 = 0.03478584
Shanker 0 = 0.27241572 0 = 0.02435319
Garima 0 = 0.21941841 0 = 0.02422882
Lindley 0 = 0.25722036 0 = 0.02393044
Data set 2
Length biased Uma 0 = 3.5827918 0 = 0.3376072
Uma 0 = 2.3356047 0 = 0.2501545
Shanker d - i.5i6488i 0 - 0.2i88903
Garima d - i.5342883 d - 0.2647i62
Lindley d - i.6358862 d - 0.2574658
Table 4: Shows Comparison and Performance of fitted distributions for Data 1
Distributions -2logL AIC BIC AICC CAIC H(X)
LBU 261.0041 263.0041 265.0817 263.0742 263.0742 4.4237
Uma 274.1516 276.1516 278.2291 276.2217 276.2217 4.6466
Shanker 308.9524 310.9524 313.03 311.0225 311.0225 5.2364
Garima 335.3371 337.3371 .339.4147 337.4072 337.4072 5.6836
Lindley 316.7054 318.7054 320.783 318.7755 318.7755 5.3678
Table 5: Shows Comparison and Performance of fitted distributions for Data 2
Distributions -2logL AIC BIC AICC CAIC H(X)
LBU 32.62635 34.62635 35.84523 34.8002 34.8002 1.3050
Uma 42.32838 44.32838 45.54726 44.5022 44.5022 1.6931
Shanker 40.29075 42.29075 43.50963 42.4646 42.4646 1.6116
Garima 40.39727 42.39727 43.61614 42.5711 42.5711 1.6158
Lindley 39.60251 41.60251 42.82138 41.7764 41.7764 1.5841
From results given above in table 4 and table 5, it has been clearly revealed and observed that the length biased Uma distribution has lesser AIC, BIC, AICC, CAIC, H(X) and -2logL values as compared to the Uma, Shanker, Garima and Lindley distributions. Hence, it can be concluded that the length biased Uma distribution leads to a better fit over Uma, Shanker, Garima and Lindley distributions.
9. Conclusion
This article introduces a novel distribution known as the length-biased Uma distribution, which has been carefully developed and compared to its baseline distribution. Throughout the study, various essential characteristics of the length-biased Uma distribution, including moments, reliability function, hazard rate function, reverse hazard function, shapes of PDF, CDF, hazard, and reliability function, order statistics, Bonferroni, and Lorenz curves, have been thoroughly explored and presented.
Furthermore, the article employs the maximum likelihood estimation method to estimate the parameters of the length-biased Uma distribution, enhancing the robustness and applicability of the proposed model.
To assess its practical effectiveness, the newly introduced distribution has been tested with two real-world datasets. The results demonstrate the superiority of the length-biased Uma distribution over existing distributions such as Uma, Shanker, Garima, and Lindley distributions. This superiority is evidenced by the significantly improved performance and satisfactory outcomes obtained from the proposed length-biased Uma distribution.
In conclusion, the study provides valuable insights into the characteristics and applications of the length-biased Uma distribution, making it a promising addition to the field of statistical modeling and probability theory. The findings open up new avenues for researchers and practitioners to explore the potential of this distribution in various real-world scenarios and data analyses.
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