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Aijaz Ahmad, M. A. Lone and Afaq A. Rather
EJAZ DISTRIBUTION A NEW TWO PARAMETRIC
DISTRIBUTION FOR MODELLING DATA
RT&A, No 1 (77)
Volume 19, March 2024
EJAZ DISTRIBUTION A NEW TWO PARAMETRIC
DISTRIBUTION FOR MODELLING DATA
Aijaz Ahmad
•
Department of Mathematics, Bhagwant University, Ajmer, India
aijazahmad4488@gmail.com
M. A. Lone*
•
Department of Statistics, University of Kashmir, Srinagar, India
murtazastat@gmail.com
Aafaq. A. Rather
•
Symbiosis Statistical Institute Symbiosis International (Deemed University), Pune, India
aafaq7741@gmail.com
Abstract
This paper introduces a novel probability distribution known as the Ejaz distribution (ED), which
is characterized by two parameters. The study offers a comprehensive analysis of this distribution,
including an examination of key properties such as moments, moment-generating functions, order
statistics, and reliability functions. Additionally, the paper explores the graphical representation of
essential functions like the probability density function, cumulative distribution function, and hazard
rate function, enhancing our visual understanding of their behavior. The distribution's parameters are
estimated using the widely accepted method of maximum likelihood estimation. Through real-world
examples, the paper highlights the practical applicability of the Ejaz distribution, demonstrating its
performance and relevance in diverse scenarios.
Keywords: Moments, Reliability analysis, oder statistics, maximum likelihood estimation, Data
analysis.
1. Introduction
In numerous fields such as economics, engineering, finance, insurance, demography, biology, and
environmental and medical sciences, various statistical distributions have been widely utilized to
describe and predict observed phenomena. However, the data encountered in these disciplines
often exhibit complex behaviors and diverse shapes, characterized by varying degrees of skewness
and kurtosis. Consequently, many of the conventional standard distributions have limitations
when it comes to accurately representing these data. As a result, the application of these classical
distributions may not yield satisfactory fits. Hence, numerous researchers have endeavored to
enhance these established classical distributions to achieve greater adaptability in modeling data
from a wide array of academic domains. In recent times, researchers have been actively engaged
in the development of new families of continuous probability distributions known for their
remarkable flexibility. This innovation involves the incorporation of extra parameters into the
191
Aijaz Ahmad, M. A. Lone and Afaq A. Rather
EJAZ DISTRIBUTION A NEW TWO PARAMETRIC
DISTRIBUTION FOR MODELLING DATA
RT&A, No 1 (77)
Volume 19, March 2024
foundational distributions. These novel families of lifetime distributions have gained prominence,
particularly in fields like economics, engineering, finance, insurance, demography, biology, and
environmental and medical sciences, where data frequently exhibit intricate behaviors, diverse
shapes, skewness, and kurtosis variations. The integration of additional parameters empowers
these distributions to offer a more adaptable and versatile framework for modeling complex
data. By doing so, they overcome the limitations of traditional standard distributions, enabling
researchers to better capture and predict real-world phenomena with precision. Thus, these
newly proposed lifetime distribution families have become invaluable tools for data analysis and
modeling in a wide range of disciplines. In recent years, researchers have introduced modifica-
tions to enhance the adaptability of conventional distributions when interpreting diverse datasets.
These changes aim to improve the accuracy of data analysis across different fields by tailoring
distribution characteristics to specific dataset requirements. For reference Aijaz et al. [1-3], Terna
Godfrey Ieren [18], Albert Luguterah [4], Topp-Leone Rayleigh distribution by Fatoki olayode
[9],Amal S. Hassan et al. [5], Frank Gomes-silva et al. [10], Brito et al.[7], Morad Alizadeh et al.
[15], Shanker et al. [17],Lindley [14], Flaih, A et al. [11], Akhter, Z et al. [6], G.M. Corderio et
al. [13]. The formulated distribution is versatile and suitable for modeling various data types,
including left-skewed, right-skewed, and symmetric datasets. This versatility is evident when
examining probability density function (PDF) plots, as they demonstrate that this distribution can
offer the most optimal fit for complex datasets. Whether the data exhibits a pronounced tail on
the left, a tail on the right, or a balanced symmetry, this distribution's flexibility allows it to adapt
and provide a robust representation. Its ability to accommodate a wide range of data patterns
makes it a valuable tool for statistical modeling and analysis, ensuring accurate and meaningful
insights across diverse data scenarios.
Let us suposse F(x; а, в) be cdf of a random variable x with а, в parameters, then the cumula-
tive distribution function of Ejaz distribution is described as.
F(x; а, в) = 1 - e а
(евх-1)
2 - е-а(евх-1)
x > 0, а, в > 0
(1)
0 1
2 3
4 5
0 1 2 3 4 5
X
X
Figure 1: The cdf plots ofEjaz distribution for distinct parameter values.
The corresponding probability density function is described as
f (x; а, в)
2авe-а(eвx-1)+/}x 1 - e-^-1)
x > 0, а, в > 0
(2)
192
Aijaz Ahmad, M. A. Lone and Afaq A. Rather
EJAZ DISTRIBUTION A NEW TWO PARAMETRIC
DISTRIBUTION FOR MODELLING DATA
RT&A, No 1 (77)
Volume 19, March 2024
Here we examine the validity of pdf
TO
f (x; a,e)dx =1
On substituting ee% — 1
= J™ 2аве—я(евх—1)+ex (1 — е—я(евх—1)
z, so that o < z < to we have
[• TO
=2x e—az (1 — e—az) dz
--2a = 1
2x
dx
a = 0.6,b = 0.5
a = 0.8, b = 1.2
a = 1.2, b = 0.8
a = 0.5, b = 0.7
a = 0.3, b = 1.4
a = 0.5, b = 0.6
a = 0.7, b = 0.8
a = 0.9, b = 0.7
a = 0.3, b = 0.7
x
x
Figure 2: The pdf plots ofEjaz distribution for distinct parameter values.
2. Moments
To understand and characterize the properties of the formulated distribution, we perform a
moment analysis about the origin. This analysis allows us to derive essential statistical measures
such as skewness, kurtosis, and other relevant properties. By examining these moments, we gain
valuable insights into the distribution's shape, central tendency, and the presence of any outliers
or heavy tails, aiding in its comprehensive statistical characterization and interpretation.
Suppose x denotes a random variable follows Ejaz distribution. Then kth moment about origin
denoted as pk can be obtained as
Hk
E(xk)= xkf (x; afi)dx
TO
=2хв xke—x(eex—1)+ex
0
TO
1 — e—a(eex—1)
dx
Making substitution eex=z so that 1 < z < to, we have
Hk = 2fk{ex ^ (l°g(z))ke—azdz — e2x ^ (l°g(z))k e—2xzdzJ
Applying integro-Exponential function by Milgram [16].
1 c to
E (W) = j+j (l°g(t)yt—Se—xtdt
193
Aijaz Ahmad, M. A. Lone and Afaq A. Rather
EJAZ DISTRIBUTION A NEW TWO PARAMETRIC
DISTRIBUTION FOR MODELLING DATA
RT&A, No 1 (77)
Volume 19, March 2024
/ 2aeaГ(k + 1) / k/ \ xr\
Fk =-------p----" lE0(a) — e E0(2a)
Substituting k = 1,2,3,4 we obtain first four moments of the distribution about origin. The
variance a2, skewness ув", kurtosis p2 , coefficient of variation (C.V) and index of dispersion 7.
Let x be a random variable follows Ejaz distribution. Then the moment generating function of
the distribution denoted by MX(t) is given by
MX (t) =E(etx) = J etxf (x; a, p)dx
те tk те tk
E k! xkf (x;a, p) = E k! E(xk)
k=0 k! k=0 k!
“ tk 2aeaГ (k + 1) / ,
: E k!------------1 [Ek0(a) - eaE§(2a)
k=0 k' p
3. Reliability indicators
This section is focused on researching and developing distinct ageing indicators for the formulated
distribution.
3.1. Survival function
Let us suppose x be a continuous random variable with cdf F(x).Then its Survival function which
is also known as reliability function is stated as
f M
S(x) = pr (X > x) = f (x)dx = 1 - F(x)
x
Therefore, the survival function for Ejaz distribution is given by
S(x; a, P) =1 — F(x; a, p)
=e-a(epx —1) Л — £—a(epx —1)\ (3)
3.2. Hazard rate function
The hazard rate function of a random variable x is denoted as
f (x; a, p)
h(x; a, p)
S(x; a, p)
(4)
using equation (1) and (3) in equation (4), then the hazard rate function of Ejaz distribution is
given as
h(x; a, p) =
2apepx (1 — e—a(epx—1))
(2 — e—a(ePx—1) j
194
Aijaz Ahmad, M. A. Lone and Afaq A. Rather
EJAZ DISTRIBUTION A NEW TWO PARAMETRIC
DISTRIBUTION FOR MODELLING DATA
RT&A, No 1 (77)
Volume 19, March 2024
0 1 2 3 4 5
X
X
Figure 3: The hrf plots ofEjaz distribution for distinct parameter values.
3.3. Cumulative hazard rate function
The cumulative hazard rate function of a random variable x is given as
H(x, а, в) = - ln[F(x; а,в)]
(5)
using equation (1) in equation (5), then we obtain cumulative hazard rate function of Ejaz
distribution as
H(x; а, в) = a (eex - l) - log (2 - e-а(гвх-1))
(6)
3.4. Reverse Hazard rate function
The reverse hazard rate function of random variable x is described as
r(x; a, в)= Шов
using equation (1) and (2) in equation (6), then the reverse hazard rate function of Ejaz
distribution is given as
2aee-x(eex-1)+ex ^1 - e-x(eSix-1))
r(x; а, в) =
1 - g-x(ebx-1) ^2 — g-x(ePx-1)^
4. Order Statistics
Let us suppose x1, x2,..., xn be random samples of size n from Ejaz distribution with pdf f (x) and
cdf F(x). Then the probability density function of the kth order statistics is given as
fx (k) =-
n!
f (x) [F(x)]k-1 [1 - F(x)]
(k - 1)!(n - 1)!
Using equation (1) and (2) in equation (7), we have
-\n—k
(7)
fx (k)
(k - 1)!(n - 1)!
2aвe-a(eвx-1)+ex 1 - e-x(eex-1) 1 - e-x(eex-1) 2 - e-x(eex-1)
k-1
e
-а(ebx-1) 2 g—x(eвx-1)
in—k
n
195
Aijaz Ahmad, M. A. Lone and Afaq A. Rather
EJAZ DISTRIBUTION A NEW TWO PARAMETRIC
DISTRIBUTION FOR MODELLING DATA
RT&A, No 1 (77)
Volume 19, March 2024
The pdf of the first order statistics X1 of Ejaz distribution is given by
fx(1) =2жве-х(евх-1)+вх (1 - е-х(евх-1)J \е-х(евх-1) (2 - е-х(евх-1)J
The pdf of the first order statistics Xn of Ejaz distribution is given by
-| n— 1
fx (n) =2паве-х(евх-1)+вх (1 - е-х(евх-1)) [1 - е-х(евх-1) (2 - е-х(евх-1)
n-1
5. Maximum Likelihood Estimation
Let the random samples x1, x2, x3,..., xn are drawn from Ejaz distribution. The likelihood function
of n observations is given as
L = П {2ф-х(евх-1)+вх (1 - е-х(евх-1)))
i=1
The log-likelihood function is given as
l =nlog(2) + nlog(a) + nlog(e) - х [евх - 1^ + fix + ^ log ^1 - е-х(е^‘-1) j (8)
i=n
The partial derivatives of the log-likelihood function with respect to х and в are given as
(9)
dl 1 &x. л ” (е?х -1) е-х(евх-1)
=-=- - евх + 1 + Ty ’
да n ^
=1 1 - е-х(евх-1)
dl n в А х.евх.е-х(^х'-1)
™ = д - хх.ерх + xi - х ^-------------, вх. ^
дв в “! 1- е-х(евх.-1)
(10)
For interval estimation and hypothesis tests on the model parameters, an information matrix
is required. The 2 by 2 observed matrix is
1
1(ф) = —
n
e(Ш) e (dM)
( дх2 J ( дхдр J
E ( d2 logl\ E f d2 logl
E \ E \ ~df~
The elements of above information matrix can be obtain by differentiating equations (9) and
(10) again partially. Under standard regularity conditions when n ^ to the distribution of ф can
be approximated by a multivariate normal N(0,I(ф)-1) distribution to construct approximate
confidence interval for the parameters. Hence the approximate 100(1 - Z)% confidence interval
for х and в are respectively given by
± Z z_\J 1ак(ф) and в ± Z z j I- (ф)
6. Simulation Analysis
The bias, variance and MSE were all addressed to simulation analysis. From Ejaz distribution
taking N=500 with samples of size n=25, 50, 150, 200, 250 and 400. For various parameter
combinations, simulation results have been achieved. The bias, variance and MSE values are
calculated and presented in table 1 and 2. As the sample size increases, this becomes apparent
that these estimates are relatively consistent and approximate the actual values of parameters.
Interestingly, with all parameter combinations, the bias and MSE reduce as the sample size
increases.
196
Aijaz Ahmad, M. A. Lone and Afaq A. Rather
EJAZ DISTRIBUTION A NEW TWO PARAMETRIC
DISTRIBUTION FOR MODELLING DATA
RT&A, No 1 (77)
Volume 19, March 2024
Table 1: Bias, variance and their corresponding MSE'sfor different parameter values a = 1.2, в = 0.8
Sample size Parameters Bias Variance MSE
25 a 0.01130 0.00432 0.01473
в 0.01251 0.00154 0.00165
50 a 0.00314 0.00413 0.00514
в 0.00103 0.00071 0.00061
150 a -0.00021 0.00301 0.00201
в 0.00406 0.00049 0.00051
200 a -0.00201 0.00156 0.00205
в 0.00237 0.00027 0.00028
250 a 0.00120 0.00206 0.00203
в 0.00255 0.00025 0.00022
300 a 0.00177 0.00203 0.00201
в 0.00066 0.00021 0.00020
Table 2: Bias, variance and their corresponding MSE'sfor different parameter values a = 2.2, в = 1.5
Sample size Parameters Bias Variance MSE
25 a 0.01230 0.03553 0.03573
в 0.02003 0.01832 0.01031
50 a 0.01214 0.01105 0.01132
в 0.01121 0.00506 0.00420
150 a 0.00672 0.00668 0.00607
в 0.00146 0.00224 0.00216
200 a 0.00265 0.00416 0.00506
в 0.01076 0.00232 0.00214
250 a 0.0027 0.00360 0.00361
в 0.00208 0.00145 0.00145
300 a 0.00150 0.00301 0.00211
в 0.00063 0.00130 0.00130
7. Data Aanalysis
This subsection evaluates a real-world data sets to demonstrate the Ejaz distribution's applicability
and effectiveness. The Ejaz distribution (ED) adaptability is determined by comparing its efficacy
to the following conventional distributions.
1:- Weibull distribution having pdf
f (x; a, в) = aexe-1e-a^; x > 0, a, в > 0
2:- Frechet distribution having pdf
f (x; a, в) = aвx-в-1 e-ax ^; x > 0, a, в > 0
3:- Inverse Burr distribution having pdf
f (x; a, в) = aв (1 - x-a) в 1; x > 0, a, в > 0
4:- Lomax distribution having pdf
f (x; a,в) = aв (1 + ax)^-1; x > 0, a,в > 0
197
Aijaz Ahmad, M. A. Lone and Afaq A. Rather
EJAZ DISTRIBUTION A NEW TWO PARAMETRIC
DISTRIBUTION FOR MODELLING DATA
RT&A, No 1 (77)
Volume 19, March 2024
5:- Exponentiated Rayleigh distribution having pdf
f (x; a, в) = 2aftxe ax2 ^1 - e ax2^ ; x > 0, а, в > 0
6:- Lindley distribution having pdf
f (x; a)
2
а
(1 + a)
(1 + x) e-ax;
x > 0, а > 0
7:- Inverse Rayleigh distribution having pdf
2a -2
f (x; a,в) = —гe-ax ; x > 0, a > 0
x3
To compare the versatility of the explored distribution, we consider the criteria like AIC
(Akaike information criterion), CAIC (Consistent Akaike information criterion), BIC (Bayesian
information criterion) and HQIC (Hannan-Quinn information criterion). Distribution having
lesser AIC, CAIC, BIC and HQIC values is considered better.
AIC = - 2l + 2 p, AICC = -2l + 2pm/(m - p - 1), BIC = - 2l + p(log(m))
HQIC = -2l + 2plog(log(m)), K.S = max1<j<m ^F(xj) - ^~т~, ~ - F(x;-)^
Where 1V denotes the log-likelihood function,'p'is the number of parameters and'm'is the
sample size.
Data set 1: The followig observation are due to Caramanis et al and Mazmumdar and Gaver
[12], where they compare the two distinct algorithms called SC16 and P3 for estimating unit
capacity factors. The values resulted from the algorith SC16 are 2.01, 6.32, 3.52, 2.15, 5.42, 2.04,
2.77, 2.26, 1.95, 1.00, 2.45, 0.74, 0.98, 1.27, 2.77, 3.68,1.18,1.09,1.60, 0.57, 3.33, 0.91, 7.14, 2.08, 3.85,
1.99, 7.76, 2.52,1.57, 4.67, 4.22, 1.92, 1.59, 4.08, 2.02, 0.84,6.85, 2.18, 2.04, 1.05, 2.91, 1.37, 2.43, 2.28,
3.74, 1.30, 1.59, 1.83, 3.85, 6.30, 4.83, 0.50, 3.40, 2.33,4.25, 3.49, 2.12, 0.83, 0.54, 3.23, 4.50, 0.71, 0.48,
2.30, 7.73.
Data set 2: The followig observation are due to Caramanis et al and Mazmumdar and Gaver
[12], where they compare the two distinct algorithms called SC16 and P3 for estimating unit
capacity factors. The values resulted from the algorith SC16 are 0.1, 0.33, 0.44, 0.56, 0.59, 0.59,
0.72, 0.74, 0.92, 0.93,0.96, 1, 1, 1.02, 1.05, 1.07, 1.07, 1.08, 1.08, 1.08, 1.09, 1.12, 1.13,1.15, 1.16, 1.2,
1.21, 1.22, 1.22, 1.24, 1.3, 1.34, 1.36, 1.39, 1.44, 1.46,1.53, 1.59, 1.6, 1.63, 1.68, 1.71, 1.72, 1.76, 1.83,
1.95, 1.96, 1.97,2.02, 2.13, 2.15, 2.16, 2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53, 2.54, 2.78,2.93, 3.27, 3.42,
3.47, 3.61, 4.02, 4.32, 4.58, 5.55, 2.54, 0.77.
The ML estimates with corresponding standard errors in parenthesis of the unknown parame-
ters are presented in Table 3 and Table 5. Also the comparison statistics, AIC, BIC, CAIC, HQIC
and the goodness-of-fit statistic for the data sets are displayed in Table 4 and Table 6.
It is observed from the findings that ED provides best fit than other competative models based
on the measures of statistics, AIC, BIC, AICC, HQIC and K-S statistic. Along with p-values of
each model.
198
fitted density
0.0 0.1 0.2 0.3 0.4
Aijaz Ahmad, M. A. Lone and Afaq A. Rather
EJAZ DISTRIBUTION A NEW TWO PARAMETRIC
DISTRIBUTION FOR MODELLING DATA
RT&A, No 1 (77)
Volume 19, March 2024
Table 3: The ML Estimates (standard error in parenthesis) for data set 1
Model a в
ED 3.45342 0.19311
WD (1.92236) 0.16787 (0.08456) 1.59666
FD (0.04105) 1.82550 (0.15017) 1.42975
IBD (0.22717) 1.79634 (0.12938) 2.85966
LXD (0.15843) 0.00769 (0.36236) 48.2182
ERD (0.00464) 0.07499 (29.232) 0.73015
LD (0.01347) 0.59651 (0.11406)
IRD (0.05424) 1.78914 (0.22191)
Table 4: Comparison criterion and goodness-of-fit statistics for data set 1
Model -2l AIC AICC BIC HQIC K.S statistic p-value
ED 239.11 243.11 243.30 247.45 244.82 0.07732 0.8319
WD 240.85 244.85 245.04 249.20 246.56 0.0955 0.5927
FD 250.87 254.87 255.07 259.22 256.59 0.1491 0.1111
IBD 245.36 249.36 249.56 253.71 251.08 0.12373 0.2726
LXD 130.57 265.14 265.33 269.49 266.86 1.00 2.2e-16
ERD 243.000 247.00 247.19 251.34 248.71 0.12333 0.2762
LD 249.59 251.59 251.65 253.77 252.45 0.11653 0.3406
IRD 267.49 269.49 269.56 271.67 270.35 0.27703 9.293e-05
Estimated pdf of the fitted model for data set 1
Empirical cdf versus fitted cdf for data set 1
Estimated cdf
Fitted hazard rate function for data set 1
x
Figure 4: Fitted pdf, cdf and hrf for data set 1.
199
fitted density
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Aijaz Ahmad, M. A. Lone and Afaq A. Rather
EJAZ DISTRIBUTION A NEW TWO PARAMETRIC
DISTRIBUTION FOR MODELLING DATA
RT&A, No 1 (77)
Volume 19, March 2024
Table 5: The ML Estimates (standard error in parenthesis) for data set 2
Model a в
ED 3.45342 0.19311
WD (1.92236) 0.29347 (0.08456) 1.796236
FD (0.05540) 1.04750 (0.15662) 1.17538
IBD (0.13022) 2.31897 (0.08496) 1.85769
LXD (0.21444) 0.00841 (0.21925) 68.4375
ERD (0.00579) 0.22565 (47.3451) 0.90717
LD (0.03649) 0.87441 (0.14049)
IRD (0.07718) 0.45560 (0.05369)
Table 6: Comparison criterion and goodness-of-fit statistics for data set 2
Model -2l AIC AICC BIC HQIC K.S statistic p-value
ED 191.80 195.80 195.97 200.35 197.61 0.06414 0.6053
WD 192.05 196.05 196.22 200.60 197.86 0.098266 0.4902
FD 234.65 238.65 238.82 243.20 240.46 0.18994 0.01109
IBD 195.21 199.21 199.38 201.02 203.76 0.10925 0.3565
LXD 225.58 229.58 229.75 234.13 231.39 0.28959 1.139e-05
ERD 193.26 197.26 197.44 201.82 199.08 0.10202 0.4419
LD 213.05 215.05 215.10 217.32 215.95 0.2356 0.000671
IRD 323.71 325.71 325.77 327.99 326.61 0.4674 4.352e-14
Estimated pdf of the fitted model for data set 2
Empirical cdf versus fitted cdf for data set 2
Estimated cdf
Fitted hazard rate function for data set 2
x
Figure 5: Fitted pdf, cdf and hrf for data set 2.
200
Aijaz Ahmad, M. A. Lone and Afaq A. Rather
EJAZ DISTRIBUTION A NEW TWO PARAMETRIC
DISTRIBUTION FOR MODELLING DATA
RT&A, No 1 (77)
Volume 19, March 2024
8. Conclusion
In this research paper, we introduce a novel two-parameter lifetime distribution, named the "Ejaz
distribution." We delve into various mathematical properties associated with this distribution,
including its shape, moments, hazard rate, and order statistics. Furthermore, we discuss the
utilization of the maximum likelihood estimation method for estimating the distribution's parame-
ters. To illustrate the practical effectiveness and superiority of the Ejaz distribution in comparison
to existing alternatives such as the Weibull, Frechet, Inverse Burr, Lomax, Exponentiated Rayleigh,
Lindley, and inverse Rayleigh distributions, we conduct goodness-of-fit tests employing criteria
such as the Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC),
and Bayesian Information Criterion (BIC) on real-life lifetime datasets. Additionally, we perform
a simulation analysis, which reveals an intriguing trend: as the sample size increases, there is a
reduction in bias and mean squared error (MSE) across all parameter combinations.
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