CC BY
1
ON SOME PROPERTIES AND APPLICATIONS OF THE MODI-FRECHET DISTRIBUTION
Akhila P.1, Girish Babu M.2 •
Department of Statistics, Government Arts and Science College, Kozhikode, Kerala, India [email protected], [email protected] 2
Abstract
In this paper we introduce a novel expansion ofFrechet distribution from Modi family of probability distributions. The important statistical properties like moments, stochastic ordering, and entropy are studied in this paper. Two distinct characterizations of the proposed distribution are derived through the hazard rate function and truncated moments. The statistical inference about the parameters of the new distribution is studied using the method of maximum likelihood estimation. To study the flexibility and practical utility of the distribution, two real-life data sets from the reliability sector and from the biomedical field were analyzed. An extensive simulation study is also conducted to validate the accuracy and consistency of the estimation techniques.
Keywords: Characterization, Entropy, Frechet distribution, Hazard rate function, Maximum
Likelihood Estimation, Statistical modelling.
1. Introduction
The study of statistical distributions is crucial across various disciplines, like economics, engineering, and particularly in reliability analysis. The reliability sector focuses on modeling and understanding failure rates in systems, components, and products over time. These necessitating distributions are robust and versatile to capture the inherent complexities of these processes. This paper introduces a new distribution meticulously designed to meet these demands and to offer enhanced adaptability for reliability analysis.
In modern industries, accurate and reliable models are essential for predicting critical systems, machinery, and equipment lifespan and failure patterns. Traditional distributions, such as the Weibull distribution (see, [1] & [2]) and the exponential distribution (see [3]), have long been utilized in reliability studies due to their simplicity and ease of use. However, these models often fall short when modeling complex or non-standard failure rates. For instance, while the Weibull distribution is well-suited for systems with increasing or decreasing failure rates, it struggles with scenarios involving bathtub-shaped failure rates, which are common in electronic systems. Similarly, the exponential distribution assumes a constant failure rate, making it inadequate for mechanical systems that experience wear-out failures over time. Our proposed distribution overcomes these limitations by providing a more flexible framework that can adapt to a broader range of reliability scenarios, including those with non-monotonic hazard functions.
Moreover, this distribution has been applied to the biomedical field, specifically in analyzing infant mortality rates, where the precise modeling of survival times and risk factors is crucial.
Akhila P., Girish Babu M. PROPERTIES AND APPLICATIONS OF THE MODI-FRÉCHET DISTRIBUTION
Traditional statistical models can struggle with the intricacies of biomedical data, particularly in capturing the variability and heterogeneity inherent in patient population. By offering a more adaptable structure, our distribution enhances the accuracy and reliability of statistical modeling in both reliability and biomedical contexts, making it a valuable tool for researchers and practitioners alike.
René Fréchet developed the Fréchet distribution [4], recognized as the maximum value distribution, a concept further explored by Fisher and Tippet [5] and Gumbel [6]. This distribution has become widely used and studied across various fields due to experimental research from multiple disciplines. It is particularly significant in survival analysis and reliability studies, finding applications in engineering, social, physical, environmental, and life sciences. The cumulative distribution function (cdf) and probability density function (pdf) of Fréchet distribution are, respectively,
(x) = e-(x f,x > 0, (1)
and
ga,A(x) = AaAx-(A+1)e-(x) ,x > 0, (2)
where a > 0 is the scale parameter and A > 0 is the shape parameter.
For further reading, see Kotz and Nadarajah [7] and Mubarak [8]. The Fréchet distribution has been extensively generalized in the literature. Recent developments are; Slash-Exponential-Fréchet distribution by Gmez et al. [9], Cosine Fréchet Loss distribution by Abonongo et al. [10], Marshall-Olkin exponentiated Fréchet distribution [11], the inverted Gompertz-Fréchet distribution [12], Yun-Fréchet distribution [13], cubic transmuted Fréchet distribution [14], the generalized odd log-logistic Fréchet distribution [15], the novel Kumaraswamy power Fréchet distribution [16] and generalization of Fréchet distribution[17]. Harlow [18] demonstrated that the Fréchet distribution is crucial for modeling the statistical behavior of material properties in various engineering applications.
Modi et al. [19] proposed the Modi family of distributions with cdf T(x) and pdf t(x) as follows:
= (1 + fl?^, x > 0'a > 0, fi > 0, (3)
afi + S(x)
t(x) = + , x > 0, a > 0, fi > 0, (4)
W {afi + S(x))2 H
where S(x) is an arbitrary cdf of a continuous univariate distribution and s(x) is the corresponding pdf. Recent contributions to this family of distributions include Modi Exponential Distribution [19], Modi Weibull [20] and Modi Exponentiated Exponential Distribution [21]. In this paper we introduce a new distribution developed from this family of distributions, utilizing the Frechet distribution as the baseline distribution. Named the Modi-Frechet Distribution, this four-parameter distribution offers a superior fit compared to other competitive lifetime distributions.
The present paper is organized as follows: In Section 2 the model construction and basic statistical properties such as moments, stochastic ordering, and entropy are studied. Section 3 is devoted to characterizations of the distribution based on hazard function and truncated moments. In Section 4 parameters of the new distribution are derived using the maximum likelihood estimation method. A simulation study has been carried out in Section 5. The flexibility and utility of the proposed model are studied in Section 6 and conclusions are given in Section 7.
2. Modi Frechet Distribution
In this section, we develop a special distribution from Modi family, based on the Frechet distribution. The cdf and pdf of Modi Frechet distribution (MFD) are;
F(x)
(1 + aß )e-( X )A aß + e-( X )A
The corresponding pdf is given by;
f (x)
x > 0, a, ß, a, A > 0.
(1 + aß) (,AaßaAx-(A+1)e-(X)A) [aß + e-(x )A )2
x > 0, a, ß, a, A > 0.
(5)
(6)
(c) (d)
Figure 1: Plots of the pdfofthe MFD for various parameter values.
Fig. 1. shows the pdf can be unimodal, approximately normal, increasing-decreasing, and right-skewed.
The hazard function of MFD is;
h(x)
(1 + aß) (AaAx-(A+1)e-($)A)
[aß + e-(x)A) (l - e-(x)A) '
x > 0, a, ß, , A > 0.
(7)
(c) (d)
Figure 2: Plots of the hrf of the MFD for various parameter values.
Fig 2. shows decreasing, increasing-decreasing, constant, and unimodal behaviour of hazard function.
We derived the quantile function of MFD. The quantile function obtained using the inversion method is given as;
F-1 (y)
a
(log (1 + a? - y) - log {ya?))
1/A'
ye[0,1]
(8)
2.1. Moments
The mean, standard deviation, variance, skewness, and kurtosis for the MFD are computed using the raw moments. With the help of R software, we computed them using the standard definitions.
Table 1: Moment characteristics of the MFD for various parameter values.
Parameters a ^ 0.6 1 2 5
2 = 9 Mean 3.0120 4.2253 4.6556 4.6569
r = 4 Variance 0.1283 1.4521 2.1379 2.1402
A = 5 Skewness 6.8163 3.9213 3.5360 3.5351
Kurtosis 210.25 57.6110 48.1140 48.0920
2 = 5.5 Mean 2.2716 2.6963 2.8949 2.9019
r = 2.6 Variance 0.0909 0.3086 0.4320 0.4365
A = 6.5 Skewness 4.2027 2.9117 2.5991 2.5899
Kurtosis 49.9690 24.4780 20.4440 20.3350
2 = 2.5 Mean 3.1501 3.4990 3.7779 3.8626
r = 3.3 Variance 0.6722 0.1080 1.5023 1.6292
A = 4.8 Skewness 4.7686 4.1751 3.8589 3.7811
Kurtosis 93.3690 72.3430 62.8610 60.7000
2 = 1.2 Mean 0.8987 0.9206 0.9431 0.9596
r = 0.9 Variance 0.0143 0.0170 0.0198 0.0218
A = 9 Skewness 2.4785 2.3161 2.1727 2.0808
Kurtosis 16.4210 14.7680 13.4440 12.6580
The calculated values are presented in Table 1. It shows that the MFD is suitable for under-dispersed data. The skewness and kurtosis values show positive skewness and leptokurtic behaviour. As a increases both mean and variance are increasing while skewness and kurtosis values decreasing.
2.2. Stochastic Ordering
Stochastic ordering is a powerful tool to demonstrate the comparison of random variables in terms of statistical functions of distribution theory. Different types of orderings can also be defined based on the hazard rate, reverse hazard rate, or by applying transformations to the random variables, as discussed in [22]. Let X! and X2 be two random variables with parameters a1, 2,r, A and a2, 2,r, A, their respective density functions f1(x) and f2(x), the reliability functions be F1 (x) and F2(x), then we say X1 is smaller than X2 if
• Fi(x) < F2(x) for all x, Xi <st X2 (Stochastic order).
• jJx) — F^x) for all X1 <hr X2 (Hazard rate order).
• p(xc) — F^x) for all X1 <rh X2 (Reversed hazard rate order).
• ^/KxT is a monotonic decreasing function for all x,=^ X1 <lr X2 (Likelihood ratio order).
Suppose the densities of X1 and X2 be
f1 + 4) fAa^rAx-(A+1)e-(i)A)
f1 (x) = —-^-,-, x > 0, and
(*{ + e-( i )A) 2
(1 + /2) Aa22rAx-(A+1)e-(xA f2(x) = --—-2--, x > 0.
f (a2 + e-(xVs 2
respectively. Then,
case (i): When a is different.
m = (i + 4) (4 + e-(x)A) f2(x) 4 (i + + e-(z)A
For al < a2, (f^) < 0 which satisfies Xi <ir X2. case (ii): When fi is different.
f2(x) (l + )(afii + e-
z )A\2
For fil < fi2, < 0 which satisfies Xi <lr X2.
case (iii): When z is different.
fi(x) _ zt^ (a^ + e-(^)A) f2(x) zA e-(X)A (afi + e-(zl)t)2
For a < a2, ^f^x)) < 0 which satisfies X! <ir X2. case (iv): When A is different.
f1(x)_ A1 aA e-(?)A1 (a? + e-(a^
f2(x) A2 zA e-(Z)t2 (afi + e-(z)Ai
For Ai < A2, (£$)' < 0 which satisfies Xi <lr X2.
2.3. Entropy
Every statistical distribution inherently possesses some degree of uncertainty, and entropy serve as a quantifiable measure of this uncertainty. In modern statistical analysis, information measures like entropy plays a crucial role in addressing and understanding such uncertainties, making them vital tools for statisticians.
If X is a non-negative continuous random variable with pdf f (x), and cdf F(x) then the Renyi Entropy is defined by,
l
l- Jo
The Shannon entropy of X is defined as
l t'œ R H(x)_ Y—0logJ0 [f (x)]R dx. (9)
œ
S(x) _ - J f (x)ln [f (x)] dx. (10)
Using the pdf of MFD, we can write;
/ ) / ) fx-(A+1)e-(z)T
[f (x)]R _ (l + afi) (AafizA) -JL. (ii)
V 7 V 7 (*fi + e-(z)A)2R
2
2
2
Varentropy, the variance of Shannon information associated with a random variable X, was introduced by Song [23] as a measure of distribution shape, offering an alternative to kurtosis. This concept captures the variability of information content, also known as information varentropy, as discussed by Bobkov and Madiman [24]. Varentropy is significant in fields like information theory, computer science, and statistics, providing valuable insights into how information is distributed around the entropy of X.
Consider X as a continuous random variable with a density function f (x). The Shannon varentropy of X is then defined as follows:
y = y(X) := Var[h(X)] = J f (x) [lnf (x)]2 dx -
I f (x)lnf (x)dx
(12)
The calculated entropy values presented in Table 2 provide a detailed comparison of Shannon entropy, Rényi entropy, and varentropy across different parameter settings. As shown in the table, the Shannon entropy values are consistently negative, indicating the uncertainty associated with each parameter set. In contrast, Rényi entropy exhibits both positive and negative values, reflecting the variation in the information content under different parameter configurations. Varentropy values, which measure the dispersion of information content around the entropy, are consistently positive, with the magnitude decreasing as the parameter shape parameters fi and A increase. This comprehensive comparison highlights how each entropy measure captures distinct aspects of the information content and its variability.
Table 2: Entropy measures for different parameters
2
Parameters Shannon Entropy Renyi Entropy Varentropy
fi A 0.3 0.2 0.9 0.5 2.5 0.8 0.3 0.2 0.9 0.5 2.5 0.8 0.3 0.2 0.9 0.5 2.5 0.8
a = 0.9, a = 0.2 -44.2040 -12.9194 -7.8138 37.8881 25.5873 20.5589 40.1693 8.5762 4.3618
a = 1.2, a = 0.8 -45.2709 -13.9355 -9.3877 -19.0633 -13.1722 -11.1970 40.6525 8.7924 4.5635
a = 3, a = 1.2 -48.6200 -16.6447 -11.7576 -1.9420 -1.4060 -1.2323 42.0571 9.2918 4.7959
3. Characterization Results
Accurately characterizing probability distributions is pivotal across diverse fields, as it facilitates profound insights into complex phenomena. The characterization of continuous probability distributions has been extensively investigated, with seminal contributions from researchers including Glanzel [25, 26] and Hamedani [27], who have pioneered various techniques. In this section, we have rigorously established the characterizations of the MFD by examining its truncated moments and hazard function.
3.1. Characterization based on truncated moments
The characterization of the probability distributions through truncated moments was initially pioneered by Galambos and Kotz [28]. Building on this foundational work, numerous scholars have made significant contributions to the field. Among the most notable are Kotz and Shanbag [29], as well as Glanzel et al. [30] with further advancements by Glanzel [25, 31]. The characterization of the MFD using truncated moments is an extension of these efforts, specifically developed in accordance with Theorem 3.1 from [25] which is stated as follows,
Theorem 3.1. Let (Q, E, P) be a given probability space, and let D = [a, ft] be an interval for some a < b (a = to, ft = -to might as well be allowed). Let X : Q ^ D be a continuous random variable with
Akhila P., Girish Babu M. PROPERTIES AND APPLICATIONS OF THE MODI-FRECHET DISTRIBUTION
distribution function G(x) and let k1 and k2 be two real functions defined on D such that
E[k1(X)\X > x] = E[k2(X)\X > x]Z(x),x e D
is defined with some real function Z. Assume that ki, k2 e C1(D), Z e C2(D), and G(x) is a twice continuously differentiable and strictly monotone function on the set D. Finally, assume that the equation K2Z = Ki has no real solution in the interior ofD. Then G is uniquely determined by the functions ki, k2 and Z. In particular,
G(x) = f C
J a
where the function t is a solution of the differential equation t' = zKZ2-2Kl and C is a constant chosen to make JDdG = 1.
The above theorem has the advantage that the cdf G is not required to have a closed form and is given in terms of an integral whose integrand depends on the solution of a first-order differential equation, which can serve as a bridge between probability and differential equation.
Z(v)
Z(u)K2(u) - K1(u)
-t(V)
Proposition 3.1. Let X : Q ^ (0, to) be a continuous random variable, and let
K2(x) = [a? + e-( a) ) and k1 (x) = k2 (x)e-( a) for x > 0. The pdfofX is Eq.6 if and only if the
function Z defined in Theorem 3.1 has the form
Z(x) = 1 e-(a) ,x > 0. (13)
Proof. Let X have pdf Eq.6, then
(1 - G(x))E[k2(X)\X > x] = (1 + a?^a?e-(a f, x > 0,
(1 + aha?
(1 - G(x))E[K2(X)\X > x]=y '— e-2(i) , x > 0,
and then
Z(x)K2(x) - k1 (x) = -1 e-(x) (a? + e-(a) ^ < 0, for x > 0. Conversely, if Z is given as Eq.12, then
t'(x) = , Z(x>K2(x\ , = -AaAx-M, x > 0, Z(x)K2(x) - K1(x)
and hence,
t(x)=(a)A
or
e-T(x) = e-( a )A. Now, using Theorem 3.1, X has the pdf Eq.6.
Akhila P., Girish Babu M. PROPERTIES AND APPLICATIONS OF THE MODI-FRECHET DISTRIBUTION
3.2. Characterization based on hazard function
The hrf h(x) of a twice differentiable distribution function F(x) and its corresponding pdf f (x) satisfy the first-order differential equation:
$ = W-h(x). (14)
For many univariate continuous distributions, this is the sole characterization expressible in terms of the hazard function. Hamedani and Ahsanullah [32] provided characterizations of certain widely recognized distributions grounded in the hazard function. The following characterization introduces a non-trivial distintion for the MFD when ft = 1, diverging from the aforementioned trivial form.
Proposition 3.2. Let X : Q ^ (0, to) be a continuous random variable. The pdf of X is Eq.6 if and only if its hazard function h(x) satisfies the differential equation
xA+1 h' (x) + (A + 1)xA h(x) = d
dx
(1 + a)AcA e-( x)A _(a + e-(x)X)(1 - e-(x)X)_
Proof. When ft = 1, the pdf f (x) and hrf h(x) of X are respectively
and
Then we have
f (x) =
h(x)
(1 + a) (AacAx-(A+1)e-(f)A)
a + e (x)A~)
(1 + a) (a^x-^e-(x)A^
x > 0, a, cr, A > 0.
x > 0, a,c, A > 0.
(15)
(16)
(17)
f'(x) = - (A + 1) f(x) x
+ AcA x
-{A+1}_ 2A(x-(A+r)e-(x)A a + e-( x)A
Using Eq.14 we can write,
h' (x)+ h(x)
(A + 1) _ (1 + a)A2c2A^-2(A+1)e-(x)A (1 + a)2A2c2Ax-2^e-(x)
(a + e-(x)A) (1 - e-(x)A) (a + e-(x)A)2 (1 - e-(i)A) 2(1 + a)A2c2Ax-2(A+1) (e-(x)A)2 (a + e-(x)A)2 (1 - e-(x)A) '
which implies,
(18)
a+1i 1 ? \ n ^a,^ (1 + a)A2c2Ax-(A+1)(1 + a)X2c2Ax-(A+1) (e-(x)A)2 xA+1 h'(x) + (A + 1)xAh(x) = (-' ) (-+ v 7
(a + e-(x)A) (1 - e-(x)A) ^ + e-(x(1 - e-(x)A)2 (1 + a)A2c2Ax-(A+1) (e-(x)A)2
(a + e-(x)A) 2(1 - e-(x)A)
x
Now, Eq.15 holds, then
dx
xA+1h(x)
d_ dx
(1 + a) A(e
(a )A
a + e-(a1 - e-(?)
from which we obtain
h(x) =
(1 + a) (\aA^-(A+1)e-(ax)A)
(a + e-(a)A) (1 - e-(f)A) '
which is the hrf of MFD when ? = 1. ■
4. Maximum Likelihood Estimation
This section provides the parameter estimates for the MFD derived through the maximum likelihood method. This method is widely recognized as the predominant approach in statistical inference. The log-likelihood for Q = (a,?, a, A)T based on a given sample is given by;
logL(a, ?, A, a) =nlog(1 + a?) + nlog(A) + n?log(a) + nAlog(a)-
i=1
i=1
(A + 1) £ log(x) - £( a) - 2 E log
i=1
a? + e
-(a )A
(19)
To obtain the maximum likelihood estimators (MLE) of the MFD, we maximize the log-likelihood function. This is accomplished by taking the first derivative of the Eq.19 with respect to parameters a, ?, A and a.
d log L(a, ?, A, a) = n?a?-1 n?
da 1 + a? a
2E\ ('Y
'=1y, a? + - xi)
d log L(a, ?, A, a) na?loga n I a?loga
—? = T+y + "log*- 2 E ^—i)
3 ^ ^ = A + nloga-E logxi - £( a)(f)
\ a )A -t)A Jog (a)
< A) E
a? + e
-(a)
and
d log L(a,?, A,a) = nA E ZA\ ^
" a + E va) Ui
dA
< A) E
( (a)Ae-(a)A
-(a)A
a? + e
MLE Q = (a, ?, a, A) of Q = (a, ?, a, A) can be obtained by solving simultaneously the following normal equations.
dlogL da
0;
dlogL d?
0;
dlogL da
0;
dlogL dA
0.
d
A
n
n
n
-1
n
A
A
Table 3: Simulation results.
True value n Average Value MSE Bias
50 10.0852 1041.897 -4.0852
100 8.4113 188.1615 -2.4113
a=6 200 7.1525 52.4647 -1.1525
300 6.8824 42.855 -0.8824
500 6.4090 6.5330 -0.4090
50 3.4438 18.3209 -0.4438
100 3.4103 10.7163 -0.4103
P = 3 200 3.5286 30.133 -0.5286
300 3.4228 10.4221 -0.4228
500 3.3492 5.8894 -0.3492
50 1.7126 4.7213 -0.7126
100 1.3902 1.7355 -0.3902
c= 1 200 1.2486 0.6024 -0.2486
300 1.1863 0.3508 -0.1863
500 1.1088 0.1051 -0.1088
50 0.9450 0.0316 0.0550
100 0.9521 0.0186 0.0479
A=1 200 0.9621 0.0113 0.0379
300 0.9654 0.0082 0.0346
500 0.9777 0.0043 0.0223
5. Simulation Study
In this section, we assess the accuracy of parametric estimation through Monte Carlo simulation. Using the quantile function of MFD given in Eq.8, we generate samples of observations for sizes n = 50,100,200,300 and 500 with N = 1000 replications. Two sets of parameter values are considered; a = 6, ft = 3,r = 1, A = 1 and a = 1.2, ft = 2.5, r = 0.2, A = 0.5.
The numerical outcomes are evaluated using the R statistical programming language, leveraging the widely used optimization package 'optim'. The Average Value, Mean Square Error (MSE), and Average Bias are computed and displayed in Tables 3 and 4. The results indicate that as the sample size increases, the MSE decreases and the Average Value of each parameter converges to the initial parameter values. These findings demonstrate the accuracy and consistency of the estimation methods.
6. Applications
In this section, we fit the MFD model to a reliability data set to check the model's flexibility. The MFD was compared to that of Modi Exponentiated distribution (MED) by [19], Modi Exponentiated Exponential distribution (MEED) by [21] and Modi Weibull distribution (MWD) by [20]. The maximum likelihood method is employed to estimate the parameters for the candidate models. We evaluated different goodness-of-fit measures to illustrate the flexibility of the model. Specifically, -logL(negative log-likelihood function), W (Cramér-von Mises Statistic), A (Anderson-Darling Statistic) KiS (Kolmogorov"Smirnov Statistic), AIC (Akaike Information Criterion), CAIC (Akaike Information Criterion with correction), BIC (Bayesian Information Criterion) and HQIC (Hannan"Quinn Information Criterion). Where
Table 4: Simulation results.
True value n Average Value MSE Bias
50 4.2749 84.5601 -3.0749
100 3.6350 69.6801 -2.4350
a = 1.2 200 2.9488 58.9493 -1.7480
300 2.8059 100.5479 -1.6059
500 2.2570 19.7147 -1.0570
50 5.9501 103.8486 -3.4501
100 4.8420 37.1668 -2.3420
ß = 2.5 200 4.2936 36.3745 -1.7936
300 3.9396 19.2506 -1.4396
500 3.5822 10.1569 -1.0822
50 1.3159 45.1542 -1.1159
100 0.8870 12.4818 -0.6870
a = 0.2 200 0.5636 6.9543 -0.3636
300 0.3382 0.4328 -0.1382
500 0.3496 1.0409 -0.1496
50 0.5014 0.0112 -0.0014
100 0.4897 0.0086 0.0103
A = 0.5 200 0.4960 0.0056 0.0039
300 0.4972 0.0039 0.0028
500 0.4971 0.0034 0.0029
AIC = - 2logL + 2k,
OV-yj
CAIC = - 2logL +
(n - k - 1)' BIC = - 2logL + klog(n), HQIC = - 2logL + 2klog(log(n))
where L is the likelihood function, k is the number of parameters of the model and n is the sample size. By respecting the standards in the field, the best model corresponds to smaller -logL, KiS, AIC, CAIC, BIC, HQIC, and greater p-value. Here, we used the "AdequacyModel" package in R programming language to obtain the MLEs and goodness-of-fit tests of the given data sets.
Data Set I: This data represents the total time on test plot analysis for mechanical components of the RSG-GAS reactor [33]
2.160 0.746 0.402 0.954 0.491 6.560 4.992 0.347 0.150 0.358 0.101 1.359 3.465 1.060 0.614 1.921 4.082 0.199 0.605 0.273 0.070 0.062 5.320.
Data Set II: data set is the information of the infant mortality rate per 1,000 live births for a few chosen nations in 2021, as reported by https://data.worldbank.org/indicator/SP.DYN.IMRT.IN
56 10 22 3 69 6 7 11 4 4 19 13 7 27 12 3 4 11 84 27 25 6 35 14 11 12 6
Table 5: Basic statistical description of the dataset.
Size (n) Min. Max. Mean Median SD Skewness Kurtosis
23 0.06 6.56 1.58 0.61 1.93 1.36 3.54
27 3 84 18.81 11 20.51 1.95 3.05
Table 5 displays basic descriptive statistics of the datasets. Here, the distribution of the dataset shows a positive skewness and leptokurtic behaviour, which goes with the moment properties of this distribution. Figure 3 shows the boxplots and Figure 4 shows the TTT plots of the data set and it goes with the features of hrf of MFD.
Figure 3: The box plots of the first data set (left) and the second data set(right).
0.0 0.2 0.4 0.6 O.S 1.0 0.0 0.2 0.4 0.6 0.8 1.0
i/n i/n
Figure 4: The TTT plots of the first data set (left) and the second data set(right).
Table 6: The MLEs of the first data set.
Model MLEs -log L
MFD MWD MEED MED a = 3.1367, ß = 5.9797, a = 0.8032, A = 0.3838 a = 3.3513, ß = 0.9966, a = 0.7085, A = 0.9824 a = 4.4807, ß = 3.2477, a = 0.4016, A = 0.5374 a = 5.8525, a = 9.6862, A = 0.8800 33.0133 34.2910 33.4931 34.8765
Table 7: The goodness of fit statistics for the first data set.
Model W A AIC BIC CAIC HQIC K-S p value
MFD 0.0471 0.3874 74.0266 78.5686 76.2488 75.1689 0.0971 0.9670
MWD 0.0544 0.3702 76.5819 81.1239 78.8041 77.7242 0.1827 0.3799
MEED 0.0864 0.5451 75.7530 79.1595 77.0161 76.6097 0.1700 0.4687
MED 0.0795 0.5052 74.9862 79.5282 77.2085 76.1285 0.1374 0.7273
Table 6 shows the results of the MLEs and negative log-likelihood values. From Table 7 we can conclude that MFD provides the lowest W, A, AIC, BIC, CAIC, HQIC, K-S values, and the largest p-value. Therefore, MFD is chosen as the best fit for the data.
Histogram of x
(a) (b)
Figure 5: Fitted pdf (a) and cdf(b) of distributions to the first data set.
Figure 5 illustrates the fitted pdfs overlaid on the histogram and the corresponding cdfs for the dataset. The histogram indicates that the data distribution is unimodal and exhibits a pronounced positive skewness. The comparison of theoretical and empirical cdfs reveals that the MFD provides the closest fit to the empirical cdf, outperforming other distributions in terms of accuracy. To verify that the log-likelihood function behaves properly and that a distinct optimum has been attained, we plot the profiles of the log-likelihood function for the MF distribution under the first dataset and displayed in Figure 6.
Figure 6: Fitted profile of the log-likelihood function for the MLEs from the MFD based on the first data set.
The performance of MFD for the second data was also compared to that of MWD, MEED and MED. The MLEs and goodness-of-fit statistics for the second data set are presented in Tables 8 and 9.
Table 8: The MLEs of the second data set.
Model MLEs -log L
MFD MWD MEED MED a = 2.3680, ft = 9.4772, a = 1.2422, A = 8.0659 a = 5.5851, ft = 11.4085, a = 1.1231, A = 12.9991 a = 1.5881, ft = 0.7937, a = 0.0544, A = 1.5894 a = 9.4307, a = 16.8944, A = 0.0642 102.7194 109.7501 104.7283 106.7475
Table 9: The goodness of fit statistics for the second data set.
Model W A AIC BIC CAIC HQIC K-S p value
MFD 0.0459 0.3064 213.4387 218.6221 215.2569 214.9800 0.0992 0.9532
MWD 0.1477 0.9498 227.5002 232.6835 229.3183 229.0414 0.1752 0.3783
MEED 0.1274 0.8277 217.4565 222.6399 219.2747 218.9978 0.1706 0.4121
MED 0.1273 0.8277 219.4949 223.3825 220.5384 220.6509 0.1752 0.3790
Histogram of x
Figure 7: Fitted pdf (a) and cdf(b) of distributions to the second data set.
Figure 7 presents the fitted pdfs and corresponding cdfs for the second dataset. The histogram reveals a unimodal distribution with notable positive skewness. The plot exhibits that the cdf of MFD is very closer to the empirical cdf than others. To further validate the model, we examine the behavior of the log-likelihood function for the MFD. The profiles of the log-likelihood function are plotted for the second dataset, (see Fig.8) confirming the proper behavior of the function and the attainment of a distinct optimum.
0.2 0.4 0.6 0.8 1.0 1.2 0.000 0.002 0.004 0.006 0.008 0.010
a P
Figure 8: Fitted profile of the log-likelihood function for the MLEs from the MFD based on the second data set.
7. Conclusion
In this article, we proposed a new distribution based on the Modi family, namely MFD. Several statistical properties of the proposed distribution, such as moments, skewness, kurtosis, stochastic ordering, and entropy are evaluated. Two characterizations of the distribution are obtained using the hazard rate function and truncated moments. The simulation study showed the accuracy and consistency of the maximum likelihood estimation method. Two real-world data sets one from the reliability sector and the other from biomedical sector were used to demonstrate the flexibility of the proposed model. The MFD provided the best fit for the data compared to other sub-models in the family.
References
[1] Abernethy, R. B. The New Weibull Handbook: Reliability & Statistical Analysis for Predicting Life, Safety, Supportability, Risk, Cost and Warranty Claims; 5th ed., Dr. Robert B. Abernethy, North Palm Beach, 2004.
[2] Rinne H. The Weibull Distribution: A Handbook, CRC Press, Boca Raton, 2009.
[3] Meeker, W. Q., and Escobar, L. A. Statistical Methods for Reliability Data, Wiley-Interscience, 1998.
Akhila P., Girish Babu M.
PROPERTIES AND APPLICATIONS OF
THE MODI-FRÉCHET DISTRIBUTION
[4] Fréchet M.(1927). Sur la loi de probabilité de l'écart maximum. Ann. Soc. Polon. Math., 6:651-664.
[5] Fisher, R. A. and Tippett, L. H. C.(1928). Limiting forms of the frequency distribution of the largest or smallest member of a sample Mathematical Proceedings of the Cambridge Philosophical Society., 24(2):180-190.
[6] Gumbel, E. J. Statistics of Extremes, Columbia University Press, New York Chichester, West Sussex, 1958.
[7] Kotz S., and Nadarajah S. Extreme value distributions: theory and applications, Imperial College Press, London, 2000
[8] Mubarak M.(2012). Parameter estimation based on the Fréchet progressive type II censored data with binomial removals. Journal of Quality, Statistics and Reliability., 2012(1):1-5.
[9] Gómez, Y. M., and Barranco-Chamorro I., and Castillo, J. S., and Gómez, H. W.(2024). An Extension of the Fréchet Distribution and Applications. Axioms., 13(4):253.
[10] Abonongo J., and Mwaniki I., and Aduda J.(2024). Cosine Fréchet Loss Distribution: Properties, Actuarial Measures and Insurance Applications. Computational Journal of Mathematical and Statistical Sciences., 3(1):1-32.
[11] Aurise N. and Leo, O. O., and Euna N., and Alexis H., and Abdisalam, H. M.(2023). Marshall-Olkin Exponentiated Fréchet Distribution. Journal of Data Analysis and Information Processing., 11:262-292.
[12] Akarawak, E. E. E., and Sunday, J. A., and Mundher, A. K., and Adedayo, F. A., and Adeyinka,S. O., and Asimi, A. A.(2023). The inverted Gompertz-Fréchet distribution with applications. Scientific African., 21:e01769.
[13] Akhila P., and Babu, M. G., and Bakouch, H. S.(2023). A Versatile Probabilistic Model based on Yun-G Family of Distributions and its Applications in Engineering Sector. Journal of the Kerala Statistical Association., 34(1):52-83.
[14] Shalabi, R. M (2020). Cubic Transmuted Fréchet Distribution. Journal of Applied Sciences Research., 16(1):9-19.
[15] Abdelkhalek, and Rania, H. M. (2022). The Generalized Odd Log-Logistic Fréchet Distribution for Modeling Extreme Values. Pakistan Journal of Statistics and Operation Research., 8(3):649-674.
[16] Najwan A., and Aijaz A., and Muzamil J., and Ahmed, M. G., and Mohammed, A. M., and Eslam H., and Ehab, M. E., and Moyazzem, M. H. (2023). The novel Kumaraswamy power Fréchet distribution with data analysis related to diverse scientific areas. Alexandria Engineering Journal., 70(9):651-664.
[17] Jayakumar K., and Babu, M. G. (2019). A New Generalization of the Fréchet Distribution: Properties and Application. Statistica., 79(3):267-289
[18] Harlow, D. G. (2002). Applications of the Fréchet distribution function. International Journal of Material and Product technology, 5(17):482-495.
[19] Modi K., and Kumar D., and Singh Y. (2020). A new family of distribution with application on two real data sets on survival problem. Sci Technol Asia, 25(1):1-10.
[20] Kumawat H., and Modi K. and Nagar P. (2023). Modi-Weibull Distribution: Inferential and Simulation Study. Annals of Data Science.
[21] Ndayisaba, A. D., and Odongo, L. O. and Ngunyi A., (2023). The Modi Exponentiated Exponential Distribution. Journal of Data Analysis and Information Processing, 11(4):341-359.
[22] Shaked M., and Shanthikumar, J. G. Stochastic Orders, Springer, New York, 2007.
[23] Song, K. S. (2001). R©nyi information, loglikelihood and an intrinsic distribution measure. Journal of Statistical Planning and Inference, 93(1-2):51-69.
[24] Bobkov B., and Madiman M.(2011). Dimensional behaviour of entropy and information C. R. Acad. Sci. Paris, Ser. I, 349(3-4):201-204.
[25] Glanzel W.(1987). A characterization based on truncated moments and its application to some distribution families Mathematical Statistics and Probability Theory, B:75-84.
Akhila P., Girish Babu M.
PROPERTIES AND APPLICATIONS OF
THE MODI-FRECHET DISTRIBUTION
[26] Glänzel W.(1988). A characterization of the normal distribution. Studia Scientiarum Mathematicarum Hungarica, 2:89-91.
[27] Hamedani, G. G.(1993). Characterizations of Cauchy, normal, and uniform distributions Studia Scientiarum Mathematicarum Hungarica, 3:243-248.
[28] Galambos J., and Kotz S. Characterizations of probability distributions. A unified approach with an emphasis on exponential and related models, Lecture Notes in Mathematics, 675. Berlin, Germany, Springer, 1978.
[29] Kotz S., and Shanbhag, D. N.(1980). Some new approaches to probability distributions Advances in Applied Probability, 12(4):903-921.
[30] Glänzel, W., and Telcs, A., and Schubert, A.(1984). Characterization by truncated moments and its application to Pearson-type distributions Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 66:173-183.
[31] Glänzel W.(1990). Some consequences of a characterization theorem based on truncated moments Statistics, 21:613-618.
[32] Hamedani, G. G., and Ahsanullah M.(2005). Characterizations of univariate continuous distributions based on hazard function II Journal of Statistical Theory and Applications, 4:218-238.
[33] Suprawhardana, M. S., and Prayoto S., and Sangadji(1999). Total time on test plot analysis for mechanical components of the RSG-GAS reactor Atom Indonesia, 25(2):155-161.
