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Ashin K Shaji, Rani Sebastian RT&A, No 4 (76)
SOME APPLICATIONS OF TRANSMUTED LOG-UNIFORM DISTRIBUTION Volume 18, December 2023
SOME APPLICATIONS OF TRANSMUTED
LOG-UNIFORM DISTRIBUTION
Ashin K Shaji, Rani Sebastian
•
Department of Statistics
St.Thomas' college (Autonomous)
Thrissur, Kerala, 680001, India
ashinkidangan@gmail.com
ranikanjiram@gmail.com
Abstract
As a generalization of the Log Uniform distribution, Transmuted Log - Uniform distribution is
introduced and its properties are studied.We obtained graphical representations of its pdf, cdf, hazard
rate function and survival function. We have derived statistical properties such as moments, mean
deviations, and the quantile function for the Transmuted Log-Uniform distribution. We also obtained
the order statistics of the new distribution. Method of maximum likelihood is used for estimating
the parameters. Estimation of stress strength parameters is also done. We applied the Transmuted
Log-Uniform distribution to a real data set and compared it with Transmuted Weibull distribution and
Transmuted Quasi-Akash distribution. It was found that the Transmuted Log-Uniform distribution
was a better fit than the Transmuted Weibull distribution and Transmuted Quasi-Akash distribution
distributions based on the values of the AIC, CAIC, BIC, HQIC, the Kolmogorov-Smirnov (K-S) goodness-
of-fit statistic and the p-values.
Keywords: Transmuted distribution, Transmuted Log- Uniform distribution,Stress- strength
parameters
1. Introduction
The Transmuted family of distributions was first introduced by Shaw and Buckley (2007) and
has since been widely used in various fields including finance, engineering, and environmental
sciences. According to quadratic rank transmutation map (QRTM) technique approach, a random
variable X is said to have a Transmuted distribution, if its cdf is given by,
D(x) = (1 + A)G(x) - A(G(x))2; -1 < A < 1 (1)
where G(x) is the c.d.f of the base distribution.
The corresponding probability density function (p.d.f) with parameter A is given by:
d(x)= g(x)(1 + A - 2AG(x)); -1 < A < 1 (2)
where A is a scale parameter.
There are different families of distributions which are useful for developing flexible compound
distributions for solving real life problems. Transmuted distributions have emerged as the
superior option, surpassing their standard counterparts in terms of flexibility and performance.
Some of the models studied were Transmuted Exponential Lomax distribution by Abdullahi and
Ieren [1], Transmuted complementary Weibull Geometric distribution by Afify [2], the Transmuted
Weibull Lomax distribution by Afify [3], the Transmuted Weibull distribution by Aryaland Tsokos
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[5], Transmuted Additive Weibull distribution by Elbatal and Aryal [6], Transmuted Quasi Akash
distribution by Hassan [8], Transmuted Exponentiated Gamma distribution by Hussian [9],
Transmuted modified Weibull distribution by Khanand King [11], Transmuted Inverse Weibull
distribution by Khan [10], Transmuted Gompertz distribution by Khan [12], Transmuted Lindley
distribution by Mansour [13], Transmuted Rayleigh distribution by Merovci [15], Transmuted
Pareto distribution by Merovci and Puka [14], Transmuted Inverse Exponential distribution by
Oguntunde and Adejumo [16] and Transmuted generalized Uniform distribution by subramanian
[18] etc.
In this article we present a new generalization of Log-Uniform distribution called the Trans-
muted Log-Uniform distribution.
2. Transmuted Log-Uniform Distribution
A Log-Uniform distribution is a probability distribution where the logarithm of the random
variable is uniformly distributed.
A random variable X is said to have the Log-Uniform distribution with parameter Я if its
probability density function is defined as,
g(x)
________1
x ln(b)-ln(a)
0;
if,a < x < b,0 < a < b,a,b <E R
otherwise
(3)
where a and b are the parameters of the distribution and they are location parameters that define
the minimum and maximum values of the distribution on the original scale and In is the natural
Log function (the logarithm to base e).
The corresponding Cumulative distribution function (c.d.f.) is,
G(x)
ln(b)-ln(a); if , a < X < b,° < a < b (1, b G R
0, otherwise
(4)
The cdf of a Transmuted Log-Uniform distribution,
F(x)
(1 + Я)
0;
Mi)
ln( b)
(Я)
Mi)
ln( b)
2
if |Я| < 1,a < x < b,
0 < a < b, a, b <E R
otherwise.
The pdf of Transmuted Log-Uniform distribution is
f(x)
(1+Я) (2X)ln(x) .
(x)ln(b) (x)(ln(b))2;
0;
if |Я| < 1,a < x < b,
0 < a < b, a, b <E R
otherwise
(5)
(6)
The survival function of Transmuted Log-Uniform distribution is given by:
S(x)
(ln(b))2 - (1 + X)ln(f )ln(b) - (X)(ln(f ))2
(ln( b ))2
(7)
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SOME APPLICATIONS OF TRANSMUTED LOG-UNIFORM DISTRIBUTION Volume 18, December 2023
Figure 1: Plot of cdf of the Transmuted Log-Uniform distribution
Figure 2: Plot of pdf of the Transmuted Log-Uniform distribution
Figure 3: Plot of survival function of the Transmuted Log-Uniform distribution
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SOME APPLICATIONS OF TRANSMUTED LOG-UNIFORM DISTRIBUTION Volume 18, December 2023
The failure rate function or hazard function of Transmuted Log-Uniform distribution is:
h(x)
(1 + A)ln(b) - (2A)ln(x)
xKMb))2 - (1 + A)ln(x)ln(b) - (X)(ln(x))2]
(8)
Figure 4: Plot of hazard rate function of the Transmuted Log-Uniform distribution
• Special Case:
If we putA = 0, then Transmuted Log-Uniform distribution reduces to Log-Uniform
distribution.
3. Statistical properties
3.1. Moments
Let X is a random variable following Transmuted Log-Uniform distribution with parameters a,b,A
and then the rth moment for a given probability distribution is given by:
Fr
b
r
x
(1 + A)
%ln a
2A ln x
x(ln b )2
dx
E(Xr) = Fr
(1 + A)(br - ar)
r ln( a)
2Aar b b b 1
(Mi))? [(a>ln(a - (ar> + <)
Mean of the Transmuted Log-Uniform distribution is obtained as:
F1
(1+A)(b-a)
ln( Ь)
2Aa
(ln(ba ))2
(b)ln(b) - (b) + 1
(9)
3.2. Quantile function
The Quantile function of Transmuted Log-Uniform distribution is obtained by inverting distribu-
tion function.
P
(1 + A)
ln( x)
ln( a)
A
ln(x)l2
ln( b)
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x = a exp
(1 + A) + +A2 + 2Я + 1 - 4Яp
ln Ь
2A
The median, 2nd quartile is obtained by putting p = 1 in (10)
"(1 + A) + 7X2+11 ln b
x = a exp
2A
(10)
(11)
3.3. Mean deviation
Let X follows Transmuted Log-Uniform distribution with mean ц and median M.
• Mean Deviation from the Mean is given by:
Гb
$i(x)= |x - ц|f (x)dx = 2ц(¥(ц) - 1) + 2Т(ц)
Ja
where ц is the mean of the distribution and
Г b
Т(ц) = xf (x)dx
Ju
T(u)
(1 + A) (2A)ln( x)
(x)ln( I) (x)(ln( I))2-
dx
T(u)
(1 + A)(b - ц) (2A)
ln( Ь)
(ln( Ь ))2
b( ln b - Л - J ln U - 1
Similarly, the Mean Deviation about Median is:
/* b
h(x)= lx - M|f (x)dx = 2T(M) - ц
a
(12)
(13)
(14)
where M is the median of the distribution and ц is the mean of the distribution and
Г b
T( M) = xf (x)dx
JM
T( M)=(1 + A)(b - M) -^ \b( ln b - Л - m( ln M - 1
( ) ln(b) (ln(b))2[ V a ) \ a
(15)
The mean deviations about mean is obtained by substituting the mean, cdf and T(ц) in (12). The
mean deviations about median is obtained by substituting the mean, cdf and T(M) in (14).
3.4. Order Statistics
Let X(!),X(2),X(3),...,X(n) denote the order statistics of a random sample Xx,X2,X3,...,Xn drawn
from the continuous distribution with pdf fx(x) and cdf FX(x), then the pdf of rth order statistics
X(r) is given by:
fX(r)(x)
n!
(r - 1)!(n - r)!
f (x)[F(x)](r-1)[1 - F(x)](
n-r)
(16)
b
x
ц
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Using the equations (5) and (6) the probability density function of rth order statistics X(r) of
Transmuted Log-Uniform distribution is given by:
fx(r)(x, a, b, A)
n!
(r - l)!(n - r)!
(1 + A)
x ln( b)
ln(x)
1 - ((1 + A)- (A)
■ ln(b)
(2A) ln( X)
x(ln( b ))2
(1 + A)
t-( X)
ln( b)
-A
t-( X)
t-( b)
- 2 - (n-r)
)
t-( X)
ln( b)
- 2- (r-1)
(17)
4. Parameter estimation
In this section, we discuss the method of maximum likelihood(ML) for the estimation of the
unknown parameters a, b, A of Transmuted Log-Uniform distribution. Let Xi,X2,X3,...,Xn be the
random sample of size n drawn from Transmuted Log-Uniform distribution, then the likelihood
function is given by:
n 1 n
L(xi; a, b, A) = П — П
i=i —i i=i
1 + A
ln( ~a )
The log-likelihood function is given by:
2Aln( —■)
(M b ))2
lnL(xi; a, b, A)
ln
n
Щ
i=1 xi
n
+ ln П
i=1
1 + A
ln(~a )
2Aln( xt)
(ln( b ))2
(18)
(19)
Therefore, the maximum likelihood estimator of a, b and A which maximize equation (19), must
satisfy the following normal equations given by
dlnL » ln(b)(1 + A) - 4Aln(xi) + 2Aln(b) q da it! [(1 + A) ln(b) - 2A ln(^)] ln(b) (20)
dlnL * 4Aln(xt) - ln(b)(1 + A) 0 db U [(1 + A) ln(b) - 2A ln(^)]b ln(b) (21)
dlnL * ln(b) - 2 ln(xl) o dA (1 + A) ln(b) - 2A(ln *) (22)
Solving this system of equations, in a, b, в gives the MLEs of a, b, A as a,b,A.
5. Estimation of Stress-Strength parameter
In this section, the procedure of estimating reliability of R = P(X2 < X1) model is considered. The
expression R = P(X2 < X1) measures the reliability of a component in terms of probability and
the random variables X1 representing the stress experienced by the component does not exceed
X2 which represents the strength of the component. If stress exceeds strength, the component
would fail and vice-versa.
In order to estimate the stress-strength parameter, considering two random variables X and Y
with Transmuted Log-Uniform (A1, a, b) and Transmuted Log-Uniform (A2, a, b) distributions
respectively. We assume that X and Y are independent random variables and the stress-strength
parameter is obtained in the form:
R = P(Y < X)
X<Y
f (x, y)dxdy
f (x; A1, a, b)F(x; A2, a, b)dx
0
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where f (x, y) is the joint probability density function of random variables X and Y, having
Transmuted Log-Uniform distribution so that:
R
'l [(j + Д,)(fU/!)) - A/l"(x/»)i
\\n(b/a)
2 n
On simplification we get:
R
\ln(b/a)J
(Aj — A2 + 3)
6
(1 + A2) 2A2ln(x/a)"
x ln(b/a) x(ln(b/a))2_
dx
(23)
To compute the maximum likelihood estimate of R, we need to compute the maximum likelihood
estimate of Aj and A2.Suppose Xj, X2,..., Xn is random sample of size n from the Transmuted Log-
Uniform (Aj, a, b) and Yj, Y2,..., Ym is an independent random sample of size m from Transmuted
Log-Uniform (A2, a, b). Then the likelihood function of the combined random sample can be
obtained as follows:
L = Пf (xi; Aj, a, b) Пf (yd A2, a, b)
i=! i=j
nn
L = П^ П
i=j xi i=j
The log-likelihood function is given by:
j + Aj 2Adn( )
j + Aj 2Ajln( xi) nx п i=jxi i=j 4+A2 2A2ln( xi)'
|_ln( b) (ln(b))2 J [ln( b) (ln( b ))2J
(24)
InL = In
n
П^
i=j xi
+ In П
i=j
in(ь) (in(b))2
+ In
m
ПХ
i=j xi
+ In П
i=j
j + A2 2AM xi)
ln( b) (ln( b ))2
(25)
The maximum likelihood estimate (MLE) of Aj and A2 can be obtained by solving the following
equations:
dlnL ^ ln( b) — 2 ln( £)
E
г (j + Aj) ln(b) — 2Aj (ln £)
dAj
dlnL
~dX2 i= a+a2) ln(a)—2A2(ln xi)
E
ln( b) — 2 ln( £)
0
0
(26)
(27)
From the equations (26) and (27), we can obtain the ML estimates Aj and AX- The corresponding
ML estimate of R is computed from (23) by replacing Aj and A2 by their ML estimates
R
(Aj — A2 + 3)
6
(28)
This can be used in estimation of stress-strength for the given data.
n
m
m
6. Simulation study and Data analysis
6.L Simulation study
Simulation studies are an important tool for statistical research. These help researchers assess the
performance of a model, understand the different properties of statistical methods and compare
them. Here we take distinct combinations of parameters a, b, A with sample size as bias and the
mean square error(MSE) of the parameter estimates.
The simulation is done by using different true parameter values. The chosen true parameter
values are as follows: •
• a = 7.5,b = UA = —0.25
As the n increases, MSE decreases for the selected parameter values given in table j.
Moreover, the bias is close to zero as the sample size increases. Thus, as the sample size
increases the estimates tend to be closer to the true parameter values.
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Table 1: Simulation study at a = 7.5, b = 16,A = -0.25
n Parameter Estimate Bias MSE
a 8.5628 1.0628 1.1296
250 b 14.8466 1.1533 1.3302
A 0.3504 0.6004 0.3604
a 8.5470 1.0470 1.0962
350 b 15.8661 0.9338 0.1790
A 0.1411 0.3912 0.1530
a 8.3061 0.8061 0.6499
500 b 16.2075 0.2075 0.0430
A 0.0465 0.2965 0.0879
a 7.4086 0.0913 0.0083
600 b 16.2010 0.2010 0.0404
A -0.2993 0.0493 0.0024
a 7.4955 0.0044 0.0000193
750 b 16.0647 0.0647 0.004182
A -0.2533 0.00331 0.0000109
6.2. Data analysis
The data set given in Table 2 represents the relief times (in minutes) of twenty patients receiving
an analgesic Gross and Clark (1975). We fit the Transmuted Log-Uniform distribution to a real life
data set and compare the results with the Transmuted Quasi Akash distribution and Transmuted
Weibull distribution.
Table 2: Relief times of 20 patients receiving an analgesic
0
1.1 1.4 1.3 1.7 1.9
1.8 1.6 2.2 1.7 2.7
4.1 1.8 1.5 1.2 1.4
3.0 1.7 2.3 1.6 2.0
Table 3: AIC, CAIC, BIC,and HQIC statistics of the fitted model in data set
Distribution AIC CAIC BIC HQIC
Transmuted Log-Uniform Distribution 6.0016 8.1258 5.9790 8.1834
Transmuted Quasi Akash Distribution 49.79 51.78 50.18 50.50
Transmuted Weibull Distribution 63.3218 65.446 63.299 65.503
From the table 3, it has been observed that the Transmuted Log-Uniform Distribution possesses
the lesser AIC, CAIC, BIC,and HQIC values as compared to Transmuted Quasi Akash distribution
and Transmuted Weibull distribution. To check the model goodness of fit we had considered the
Kolmogorov-Smirnov (K-S) test (goodness-of-fit) statistics for the relief times of patients receiving
an anelgesic data.
To determine the Goodness of fit of the models, the magnitude of K-S Statistic is obtained. Since
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SOME APPLICATIONS OF TRANSMUTED LOG-UNIFORM DISTRIBUTION Volume 18, December 2023
the p-value of fitted model is highest than the other distributions we have considered.Therefore
the results indicate, that the proposed model performed better than other models.
7. Conclusion
In this study, we introduced a new distribution called the Transmuted Log-Uniform distribution.
The distribution was generated using the Transmuted technique and the Log-Uniform distribution
as the base distribution with two parameters. We obtained graphical representations of its pdf,
cdf, hazard rate function and survival function. We have derived statistical properties such
as moments, mean deviations, and the quantile function for the Transmuted Log-Uniform
distribution. We also obtained the order statistics of the new distribution.
We used the maximum likelihood method to estimate the parameters of the distribution and
the stress strength parameters. We performed a simulation study to validate the estimates of
the model parameters, and it was observed that the distribution showed the least bias, with the
values of mean square error decreasing as the sample size increased. Finally, we applied the
Transmuted Log-Uniform distribution to a real data set and compared it with Transmuted Weibull
distribution and Transmuted Quasi-Akash distribution. It was found that the new distribution
was a better fit than these distributions based on the values obtained for the AIC, CAIC, BIC,
HQIC, the Kolmogorov-Smirnov (K-S) goodness-of-fit statistic, and the p-values obtained for the
models.
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