CC BY
54
On the Minimum of Exponential and Teissier
Distributions
VISHWA PRAKASH JHA1 AND V. KUMARAN2
•
1,2Department of Mathematics, National Institute of Technology, Tiruchirappalli 620015, India
2 Corresponding author kumaran@nitt.edu
1 vpjha.nitt@gmail.com Abstract
In reliability theory minimum of two random variables has a significant meaning, and models with increasing failure rates play a vital role. Motivated by these facts, in this article, a two-parameter lifetime distribution with an increasing failure rate is constructed by considering the method of a minimum of two independent random variables following the exponential and Teissier distributions and studied in detail. Several exciting features, such as moments, quantiles, Bonferroni and Lorenz curves, entropies, stress-strength reliability, moments of a residual lifetime, and order statistics, are derived for the proposed distribution. For the estimation purpose, eight different techniques have been used, including maximum likelihood, ordinary least square, weighted least square, Cramer-von Mises, maximum product spacing, Anderson-Darling, right-tailed Anderson-Darling, and bootstrapping (parametric and nonparametric). The performance of these estimators is compared using three real datasets. The exact Fisher information matrix elements are derived, and confidence intervals based on the information matrix and bootstrapping techniques are constructed. A simulation study is carried out to see the efficiency of the maximum likelihood in terms of mean square error and bias. Negative log-likelihood, Akaike information criteria, Bayesian information criteria, Consistent Akaike information criteria, and Hannan-Quinn information criteria are the goodness-of-fit statistics employed. Furthermore, other nonparametric test statistics such as Kolmogorov-Smirnov, Anderson-Darling, and Cramer-von Mises are used for model selection. Moreover, three real datasets related to epidemiology, seismology, and reliability are modeled and compared with exponential, exponentiated exponential, Lindley, exponentiated Lindley, Rayleigh, exponentiated Rayleigh, Gompertz, exponentiated Gompertz, Weibull, and exponentiated Weibull distributions to demonstrate how the suggested model performs in practice. And it is observed that the proposed distribution provides a better fit among all considered models, according to most of the test statistics. The proposed lifetime distribution is unimodal and capable of modeling positive datasets with an increasing failure rate which contains Gompertz one-parameter model as a particular case. It is a simple model with only two parameters resulting from expressions for different characteristics that are analytically tractable. So, it is expected that it will be helpful in various disciplines where such types of data exist, such as reliability, lifetime modeling, and survival analysis.
Keywords: Probability distribution, Moments, Information Matrix, Maximum likelihood estimator, Bootstrap, Simulations.
1. Introduction
To model, the frequency of mortality associated with aging alone, Teissier [1] developed an increasing failure rate distribution, known as the Teissier distribution (TD). Muth [2] pointed out that the TD has a heavier tail than some classical distributions like the gamma, Weibull, and log-normal distributions. The TD was used by Rinne [3] to model the lifetime of a real dataset related to used motor cars. Leemis and McQueston [4] established a univariate distributional
relationship, in which this model was reconsidered and renamed the "Muth distribution." Some statistical features of Muth distribution were thoroughly investigated by Jodra et al. [5]. Irshad et al. [6, 7] studied inference and some other extensions of the Muth distribution. Saroj et al. [12] introduced inverse muth distribution.
Recently, exponentiated Teissier distribution (ETD) has been given by Sharma et al. [11]. Eghw-erido [12], and Poonia and Aazad [13] applied the alpha power transform (APT) technique of Mahdavi and Kundu [14] on TD and ETD, respectively. The exponential distribution (ED) is a well-known classical distribution with some distinguishing features such as a constant hazard rate and memorylessness. Related references on exponential distributions can be found in the literature, for example, see Gupta and Kundu [15], Nadarajah and Haghighi [16] and Mahdavi and Kundu [14]. Gompertz proposed that human mortality increases exponentially with age. Makeham extended Gompertz's suggestion of competing risks by adding one and two parameters to the standard Gompertz distribution known as Gompertz Makeham-I (GMD-I) and Gompertz Makeham-II (GMD-II) distributions, respectively. Chapter 10 of Marshall and Olkin [17] provides a comprehensive review of the Gompertz and all extensions made by Makeham. According to chapter 10 of Marshall and Olkin [17], GMD-I has three cases. The cdf of the second case of GMD-I is given as
F(x; 0, p, £) = 1 - e-£(e0x+p0x-1) 0 > 0,-1 < p < 0, £ > 0. (1)
However, TD is a particular case of the second case of GMD-I when £ = 1 and p = -1. The case of p = — 1 is under communication. Recently, many lifetime distributions has been developed in reliability theory, Deepthy and Sebastian [9] developed Burr III Modified Weibull Distribution, Manoharan and Kavya [13] extended Lomax distribution to construct a reliability model. Some probable scenarios that arise in real-life applications because of the distribution of the minimum of two random variables are fascinating, see chapters 5 and 17 of Marshall and Olkin [17].
Suppose X! and X2 are two independent random variables follow Teissier and exponential distribution with parameters 0 and 0A respectively. The cumulative density functions (cdfs) of Xi and X2 are given by (for x > 0, 0 > 0, A > 0)
Fx1 (x; 0) = 1 - e0x-e0x+1, Fx2 (x; 0, A) = 1 - e-0Ax, (2)
respectively. Suppose X=Minimum{X1,X2}. The cdf, pdf, survival function, hazard rate function (hrf), cumulative hazard rate and reversed hazard rate of METD are given by (for
x > 0, 0 > 0, A > 0)
F(x; 0, A) = 1 - e1+0x-e0x-0Ax, (3)
f (x; 0, A) = 0 (A - 1 + e0x) e1+0x-e0x-0Ax, (4)
S(x; 0, A) = e1+0x-e0x-0Ax, h(x; 0, A) = 0 (A - 1 + e0x) , (5)
0 (a — 1 + e0x^ e0x+1
H(x; 0, A) = (e0x + 0Ax - 0x - 1), r(x; 0, A) = ' (6)
Interestingly, the pdf of METD can be obtained from second case of GMD-I as a special case by substituting £ = 1 and p = A - 1 in Eqn.(1), in this case 0 < A < 1. Unfortunately, this case has not received much attention in the literature. However, the METD model work for A > 1 also. It should be noted that Gompertz's one-parameter distribution is a particular case of METD when A = 1 in METD.
The rest of the article is organized as follows. In Section 2, some statistical properties of the METD have been derived. Section 3 deals with the estimation of the parameters of METD. In Section 4, simulation is carried out. In section 5 three applications are presented to show that the proposed distribution can be used quite effectively in analyzing the real-life datasets. Finally, section 6 provides some conclusions.
2. Statistical Properties
In this section some basic features of the proposed distribustions such as shape analysis of pdf and hrf of METD, moments, quantiles, Bonferroni and Lorenz Curve, Renyi entropy, stress-strength reliability (ssr), moments of residual life function and order statistics are studied.
2.1. Shape of pdf and hrf
The pdf of METD is log-concave as
. . - [(e9x - 1)2 + 2A(e9x - 1) + (A - 1/2)2 + 3/4] (log f (x)) =-(e°x + A - 1)2-, (7)
is negative for all x > 0, 9 > 0, A > 0. Moreover, limx^0+ f (x) = 9A and limx^M f (x) = 0. The pdf is decreasing for A > 1 and having a unique mode at 1 log (i (3 - 2A + V5 - 4A)) for
0 < A < 1, see Fig.1(a). Also (h(x))' = 92e9x > 0 hrf is exponentially increasing.
2.2. Moment generating function and moments
By using u = e9x, the moment generating function (mgf) of METD can be written as
MX(t) = e[El- 9 -!(1) - (1 - A)E0A- 9 (1)], (8)
where
E' (z) = f (¡+1)1™(logu)le-zuu-s ds, (9)
1 > -1, s € R and r(.) is the gamma function. Moreover, the rth derivative of MX(t) at t = 0 also the rth moment about origin, can be given as
E(Xr) = MXr)(0) = e9-rr(r + 1)(eA-X(1) - (1 - A)EA(1)). (10)
Using the moments, mean, variance, skewness and excess kurtosis of the METD can be calculated and shown in Fig.1.
2.3. Quantile function
By inverting the cdf of METD, the quantile function of METD(9, A) can be expressed as
1 log f(A - 1)W-Je A tt
£ p = <
if Л < 1
1
I 1 1 X , 1 log [1 - log(1 - p)] if Л = 1 (11)
1 log \(Л - 1 if Л > 1 0
where p G (0,1), W(.) represent the Lambert-W function( see Jodra [18]) and W_i(.) is the negative branch of the Lambert-W function, p = 0.25,0.50,0.75 corresponds to the first, second(median) and third quartiles.
2.4. Bonferroni and Lorenz Curve
The Bonferroni curve, Lorenz curve and Gini coefficient are defined as B(p) = py J0p F-1 (t) dt, L(p) = i f0p F_1 (t) dt, and G = 1 _ 2 /J L(p) dp, respectively, where 0 < p < 1, y = /J F-1(t) dt and F_1 (.) is the quantile function. Lorenz curve for METD is given as
L(p) = , J i0,Ip,1,A _ M) _ (1 _ A)J 0IVXA,1) (12)
L(p) 1 J(0,0,1, A _ 1,1) _ (1 _ A)J(0,0,1, A, 1) , ( )
where J(0, t, r, s, z) = p^+j J0 (log u)re_z"u_s du.
J = 0.0 J = 0.2 J = 0.4 J = 0.6 J = 1.0 J = 1.4 J = 1.6 J = 1.8
0.5 1 5 10
A
50 100 0.01 0.05 010 0 50 1 5 10
A
(a)
(b)
(c)
(d)
Figure 1: Plots of (a) Probability density function, (b) Mean, median, and mode, and (c) Mean, variance, skewness, and excess kurtosis, (d) Gini coefficient, for the METD when d = 1.
0.0 0.5
20 2.5
2.5. Renyi entropy
The Renyi entropy of a non-negative continuous random variable X with pdf f (x) is a measure of variation and is defined as IR(7) = 1-7 log /o° f (x)7dx, where 7 > 0 and 7 = 1. If X ~METD(0, A) then the Renyi entropy of X is given by (using u = ex)
1
Ir(7) = y^y log [e707-1 H(y,y(A - 1) + 1, A - 1,7)], (13)
where H(z, s, c, p) = /1° u-se-zu (c + u)p du, c > -1.
2.6. Stress strength reliability
Suppose Xi and X2 are two random variables from the METD family such that Xj ~ METD(0i, Ai) and X2 ~ METD(02, A2). The ssr of METD is specified as
TO
R = P(Xi > X2) = j fXi (x)Fx2 (x) dx. (14)
using u = e6x, the expression of R can be written as
R = 1 - e2 (Q (0,0A2 - 9 + Ay - 1) + (Ay - 1) Q (0,0 (A2 - 1) + Ay)) , (15)
where Q(0,s) = u-se-(u+u0) du and 0 = 01!0\-
2.7. Moments of residual life function
The rth moment of residual life of a random variable X with pdf f (x) and cdf F(x) is given by
1 c to
mr(t) = E[(X - t)r|X > t] = y - F^t)Jt (x - t)rf (x) dx, (16)
where r = 1,2,3,....
The mean and variance of residual life may be expressed as m(t) = m1 (t) - t, and V(t) = m2(t) -(m1 (t))2 where mr(t) = ^ xrf (x) dx. By using u = e0x and defining J(0, t,r,s,z) = r(r+i) fe7t (log u)re-zuu-s du , the numerator of mr(t) can be written as
/TO
xrf (x) dx = er(r + 1)0-r[j(0, t, r, A - 1,1) + (A - 1) J(0, t, r, A,1)]. (17)
V.P. Jha,V. Kumaran RT&A, No 4 (71)
On the Minimum of Exponential and Teissier Distributions Volume 17, December 2022
2.8. Order statistics
Assume that Xi, X2,..., Xn is a random sample of size n drawn from a population with the pdf
f (x) and the related order statistics X(1), X(2),..., X(n). The pdf of rth order statistics of METD is given as
fX(r)(x) = B(r,n ! r + 1) (A - 1 + e9X) (1 - ^(x))r-i ($(x))n-r+1, (18)
where $(x) = e1-9Ax+9x-e9x, B(a, b) = f01 ta-1 (1 - t)b-1 dt, x > 0, 9 > 0, p > 0, r < n. The mth moment of rth order statistics X(r) is derived as
n—r i+r-1
E[Xm . = r(m + 1) n£r . )i+u+i(i + r - A (n - Л
E[X(r)] = B(r, n - r +1) £ £ (-1) e I j )[ i) (19)
E^(j + 1) + (A - 1^+! (j + 1) ,
where E^(z) given in Eqn. (9) and q = (j + 1)(A - 1).
3. Estimation of Parameters
For the estimation purpose, eight different techniques have been used, including maximum likelihood (MLE), ordinary least square (OLS), weighted least square (WLS), Cramer-von Mises (CVM), maximum product spacing (MPS), Anderson-Darling (AD), right-tailed Anderson-Darling (RTAD), and bootstrapping (parametric and nonparametric). For the OLS and MPS techniques, see Swain et al. [19], and Cheng and Amin [20], respectively. A general theory of various estimation techniques can be found in Sharma et al. [11] and Dey et al. [?]. For parametric and nonparametric bootstrap estimation, 1000 samples were generated according to the algorithm given by Kharazmi et al. [21]. The discussed estimation procedures are applied to two real datasets, and the results are shown in section 5.
3.1. Method of the Maximum likelihood
The maximum likelihood (MLE) approach is the most extensively used approach for parameter estimation. This approach has numerous flexible properties, including consistency, asymptotic efficiency, and invariance. Suppose x1, x2,..., xn be a sample of size n from the METD. For the vector of parameters © = (9, A), the log-likelihood function of METD is given as
n n
1(©) = n log9 + £ [1 - e9xi + 9(1 - A)xi] + £ log [A - 1 + e9xi]. (20)
i=1 i=1
The MLEs of the parameters have been calculated numerically using Mathematica 12.3. After differentiation of the log-likelihood function with respect to the parameters 9 and A, the components of the score vector U(©) can be expressed as
n n x.e9xi n n 1 n
U" = n + £ A-r+e^ + £x(1 - A - e9xi), ua = £ a-Tre»x- 9£x- (21)
The MLEs of the parameters can be obtained by setting these equations to zero and solving them. Another advantage of the MLE approach is that it is useful to construct approximated confidence interval (ACI) of the parameters, see Lawless [?]. The exact 2 x 2 information matrix
1(©) required for interval estimate of METD parameters defined as
!(©)= I9,9 rA .The
V U,9 U,A )
members of the I(©|x) for the METD are given as
19,9 = -E(^©M) = 9-2[2e{(A - 1)(EA-1 (1) - /^(A - 1)) + EA-2(1)} + 1], (22)
1e,A
u,e —
_e( a2 mx) ^ — i
V dddA
(1) + (A - 1)E\(1) + J\_!(A - 1))
and
— -E(^ï— 'J'" A - 1),
where the integral Els (z) is defined in Eqn.(9) and the integral J(k)
1 î>œ e uu s log(u)
_r _
r(r+1) J1 k+u
The multivariate normal distribution N2 (0,I (©)-1) may be used to generate confidence intervals for model parameters under usual regularity conditions.
(23)
(24) du.
--«-- A = 0.5 A = 1.0 —T-- A = 1.5 1.0 0.8 s- 0.6 A= 0.5 A = 1.0 A = 1.5
• S 0.4 0.2 0.0 1 ▼
- A = 0.5
-A = 1.0
0 200 400 600
0 200 400
0 200 400
0 200 400 600 800
Figure 2: The MSE and bias of the parameters for the simulated samples.
d
4. Simulation
To investigate the performance of the MLEs for the METD parameters, several simulations are explored for different sample sizes. The samples are generated from Eqn. (11), using the inverse cdf technique. The parameters values are taken as 9 = 1 and A = 0.5,1.0 and 1.5 and the sample sizes are selected as n = 20,40,60,80,100,140,180,220,260,320,380,440,500,580,660,740,820 and 900. Each sample size is repeated 1000 times, biases and mean squared errors are calculated. Fig.2 displays the results of the simulation. The trends in Fig.2 reveal that as the sample size increases, the MSEs and bias of the MLEs decay toward zero, as expected by first-order asymptotic theory.
5. Data Analysis
Three real datasets are being used to demonstrate the practical significance of the METD model. The first dataset was downloaded from the webportal of the World Health Organization (https://covid19.who.int/) on October 12, 2021, which denotes the daily number of deaths in South Africa due to the novel coronavirus from May 11, 2020, to June 28, 2020. Currently, the first dataset is slightly modified by the WHO. The second dataset is related to seismology and is taken from the Wolfram data repository, which indicates the earthquake waiting times in days. The third dataset is taken from Murthy et al. [25], which is about aircraft windshield data and demonstrates the service times of windshields that had not failed at the time of observation.
• Dataset I
8, 12, 1, 13, 19, 9, 14, 3, 22, 26, 27, 30, 28, 10, 22, 52, 43, 28, 25, 34, 32, 40, 22, 50, 37, 56, 60, 44, 46, 82, 82, 48, 74, 70, 69, 57, 88, 57, 49, 63, 94, 46, 53, 61, 111,103, 87, 48, 73.
• Dataset II
840, 1901, 40, 139, 246,157, 695, 1336, 780, 1617, 145, 294, 335, 203, 638, 44, 562, 1354, 436, 937, 33, 721, 454, 30, 735, 121, 76, 36, 384, 38, 150, 710, 667, 129, 365, 280, 46, 40, 9, 92, 434, 402, 556, 209, 82, 736, 194, 99, 599, 220, 584, 759, 304, 83, 887, 319, 375, 832, 263, 460, 567, 328.
• Dataset III
0.046, 1.436, 2.592, 0.140, 1.492, 2.600, 0.150, 1.580, 2.670, 0.248, 1.719, 2.717, 0.280, 1.794, 2.819, 0.313, 1.915, 2.820, 0.389, 1.920, 2.878, 0.487, 1.963, 2.950, 0.622, 1.978, 3.003, 0.900, 2.053, 3.102, 0.952, 2.065, 3.304, 0.996, 2.117, 3.483, 1.003, 2.137, 3.500, 1.010, 2.141, 3.622, 1.085, 2.163, 3.665, 1.092, 2.183, 3.695, 1.152, 2.240, 4.015, 1.183, 2.341, 4.628, 1.244, 2.435, 4.806,1.249, 2.464, 4.881, 1.262, 2.543, 5.140.
Table 1 give a brief summary of the datasets and fitted METD. The datasets are starting from non-zero and right-skewed. Therefore, METD may be the right choice to model these datasets. Parametric and non-parametric bootstrap techniques also have been applied for estimation purpose by adopting the methodology of Efron and LePage [22]. According to the described methodology in the estimation section, the exact information matrix for the datasets I, II and
. : f 14517.2 77.4383 \ f 3.0789 x 106 484.359 \ d f 30.5202 3.54375 IIIaregivenas^ 77.4383 0.7927 484.359 0.0799 / V 3.54375 0.787313
respectively. Inverse of the exact information matrix for the datasets I, II and III are given
f 0.0001 -0.0140 e f 6.8421 x 10-6 -0.0414 e , f 0.0686361 -0.308936 e as\ -0.0140 2.6341 /V -0.0414 263.358 / V -0.308936 2.66068 y),respec" tively. The interval estimates of the parameters based on the expected information matrix are given as: (i) For the dataset I, S G (0.0134,0.0201), A G (0.0253,0.9342). (ii) For the dataset II, S G (0,0.0012), A G (0,7.2586) and (iii) For the dataset III, S G (0.3018,0.4312), A G (0.0812,0.8867). The point and interval estimates of the parameters based on the parametric bootstrap are given as: (i) For the dataset I, S = 0.0170, A = 0.4903,S G (0.0137,0.0206), A G (0.1577,1.0143). (ii) For the dataset II, S = 0.0007, A = 3.1266, S G (0.0003,0.0012), A G (1.4779,6.6621) and (iii) For the dataset III, S = 0.3709, A = 0.4771, S G (0.3096,0.4411), A G (0.1274,0.9156) . The point and interval estimates of the parameters based on the non-parametric bootstrap are given as: (i) For the dataset I, S = 0.0169, A = 0.4885, S G (0.0141,0.0201), A G (0.1315,0.9469). (ii) For the dataset II, S = 0.0007, A = 3.1244, S G (0.0003,0.0012), A G (1.4249,6.0172) and (iii) For the dataset III, S = 0.3696, A = 0.4913, S G (0.3115,0.4367), A G (0.1812,0.9456) All the seven different estimation
Table 1: Summary of the datasets andfitted METD.
Size Min Q.25 Median Q.75 Max Mean Skew Ex-Ku
Data I 49 1 25 46 61 111 45.46 0.42 -0.52
METD - 0 23.99 43.26 64.22 œ 45.47 0.41 -0.41
Data II 62 9 129 328 667 1901 437.21 1.49 2.52
METD - 0 136.87 323.22 624.12 œ 437.20 1.48 2.57
Data III 63 0.04 1.09 2.06 2.81 5.14 2.08 0.43 -0.26
METD - 0 1.09 1.98 2.94 œ 2.08 0.41 -0.41
approaches, as given in the estimation section, have been applied to estimate the parameters of METD for the both datasets, and results are shown in Table 2 with different test statistics and ranking based on Kolmogorov-Smirnov (KS) test. From Table 2, it may be concluded that according to the KS test, the maximum likelihood (ML) and CVM are the best estimator among considered estimation procedures for dataset I and II respectively, whereas MPS is the worst techniques among all considered methods for the both datasets. For third dataset CVM is most effective estimator and AD is not much efficient among all estimators.
The METD is compared with the following distributions: exponential distribution (ED), exponentiated exponential distribution (EED) of Kundu and Gupta [15], Teissier distribution (TD) of Teissier [1], exponentiated Teissier distribution (ETD) of Sharma et al. [11], Rayleigh distribution (RD), exponentiated Rayleigh distribution (ERD), Lindley distribution, exponentiated Lindley distribution, Gompertz Distribution (GOD), exponentiated Gompertz Distribution (EGOD) of El-Gohary et al. [24], Weibull distribution (WD), and exponentiated Weibull distribution (EWD) of Mudholkar and Srivastava [23]. Several goodness-of-fit (gof) statistics are used for model selection, including negative log-likelihood (NLL), Akaike (AIC), Bayesian (BIC), and Consistent
Table 2: Estimation of parameters by different techniques and various test statistics with ranking(r).
Method 0 A NLL OLSS WLSS CVMS MPSS ADS RTADS KS p-Value(KS) r
Dataset I
ML 0.0168 0.4797 227.402 0.0175 6.0462 0.0178 -4.0207 0.1138 0.0556 0.0475 0.9996 1
OLS 0.0162 0.5409 227.453 0.0159 5.0371 0.0190 -4.0177 0.1182 0.0548 0.0507 0.9989 4
WLS 0.0160 0.5599 227.496 0.0161 4.9512 0.0202 -4.0172 0.1262 0.0593 0.0518 0.9984 6
CVM 0.0168 0.4873 227.403 0.0172 5.9974 0.0177 -4.0206 0.1136 0.0556 0.0498 0.9991 3
MPS 0.0157 0.5926 227.587 0.0171 5.1453 0.0228 -4.0168 0.1450 0.0706 0.0551 0.9964 7
AD 0.0166 0.5003 227.408 0.0166 5.5836 0.0178 -4.0195 0.1122 0.0536 0.0489 0.9993 2
RTAD 0.0165 0.5208 227.421 0.0161 5.3265 0.0182 -4.0187 0.1137 0.0530 0.0513 0.9987 5
Dataset II
ML 0.0006 3.2178 438.650 0.0343 17.6435 0.0335 -4.5439 0.2896 0.1375 0.0610 0.9642 6
OLS 0.0005 3.8710 438.714 0.0331 17.7131 0.0338 -4.5406 0.2960 0.1444 0.0582 0.9764 2
WLS 0.0005 3.8398 438.690 0.0338 17.4445 0.0341 -4.5409 0.2905 0.1404 0.0596 0.9708 4
CVM 0.0006 2.9352 438.662 0.0340 17.9474 0.0329 -4.5459 0.2940 0.1378 0.0577 0.9784 1
MPS 0.0002 8.7240 438.942 0.0364 18.2424 0.0386 -4.5389 0.3158 0.1631 0.0612 0.9631 7
AD 0.0006 3.4156 438.657 0.0339 17.5170 0.0336 -4.5426 0.2889 0.1376 0.0600 0.9687 5
RTAD 0.0006 3.1760 438.651 0.0338 17.6555 0.0330 -4.5439 0.2900 0.1370 0.0588 0.9742 3
Dataset III
ML 0.3665 0.4840 98.1613 0.0347 16.0498 0.0379 -4.6152 0.2623 0.1334 0.0681 0.9127 4
OLS 0.3799 0.4184 98.2516 0.0322 16.8590 0.0325 -4.6204 0.2722 0.1394 0.0637 0.9458 2
WLS 0.3682 0.4751 98.1627 0.0341 16.0314 0.0370 -4.6156 0.2612 0.1329 0.0673 0.9188 3
CVM 0.3888 0.3811 98.4149 0.0332 18.4910 0.0315 -4.6256 0.3026 0.1577 0.0625 0.9534 1
MPS 0.3480 0.5702 98.3114 0.0442 18.2744 0.0514 -4.6127 0.3125 0.1647 0.0709 0.8870 6
AD 0.3660 0.3162 99.2066 0.1814 67.5575 0.1848 -4.6290 1.0657 0.5165 0.0959 0.5740 7
RTAD 0.3695 0.4734 98.1657 0.0340 16.0688 0.0367 -4.6160 0.2611 0.1325 0.0687 0.9074 5
Akaike (CAIC), and Hannan-Quinn (HQIC) information criteria. Other robust test statistics, including Kolmogorov-Smirnov (KS), Anderson-Darling (AD), and Cramer-von Mises (CVM), as well as the p-value of KS, are examined for model selection in addition to these information metrics. The model with the lowest test statistics value and the highest p-value was chosen as the best model among all competitors. Table 3 shows all the relevant gof statistics of the fitted distributions for both datasets. For dataset I, according to AD, CVM, and KS tests, METD achieved the first rank, whereas according to AIC, BIC, CAIC, and HQIC, METD achieved the second rank, and ETD achieved the first rank among all competitor models. According to NLL, EGOD, ETD, and EWD have better ranks than METD. Therefore, as METD is a simple model in comparison with exponentiated models and has a significant p-value, it is concluded that METD may be a good choice to model the first dataset.
METD ranked first in NLL, AD, CVM, and KS tests for dataset II, while METD ranked second in AIC, BIC, CAIC, and HQIC tests, and ED ranked first among all competitors models. ED has no significant p-value. Therefore, it is concluded that METD provides a reasonable fit for the second dataset in comparison with all competitor distributions. For third dataset, METD is consistently achieved best rank according to all gof test statistics. Only it is second under NLL and KS test. According to the AIC and KS test, the top four models are selected for both datasets. Histograms of datasets with pdfs of distributions, are displayed in Fig. 3. Once again, these plots confirm the conclusion that the METD is an appropriate model for these datasets.
6. Conclusion
The METD is a two-parameter distribution proposed in this study. The hazard rate function of the METD is exponentially increasing and the probability density function is unimodal (0 < A < 1) and decreasing (A > 1). Two different datasets of different characteristics (unimodal and decreasing) are provided to show the practical significance of the present distribution. Furthermore, the proposed distribution can be used as an alternative to some well-known distributions such as exponential, Lindley, Rayleigh, Gompertz, Weibull, and their exponentiated models, and it is expected that it will provide a better fit for similar datasets than the models discussed in this paper. The METD demonstrated in this study shows its ability to model
0 20
40 60
x
(a)
1000 x
(c)
80 100 120 0 20 40
ED GOD WD METD
2000 0
(b)
(d)
80 100 120
■ ERD
■ GOD
- EWD
- METD
0.00.............................. 0.00^—-
0 1 2 3 4 5 0 1 2 3 4 5
(e)
(f)
Figure 3: Histogram with some better fitted pdfs/or dataset I dataset II (c),(d) and dataset III (e),(f where
better models according to AIC in first column and better models according to KS test in second column.
0.0020
0.0015
0.0005
0.0000
500
1500
500
1000
1500
2000
0.30
0.20
0.15
0.10
0.05
Table 3: Variows goodness-of-fit statistics, the number of parameters (NP) with respective ranking r of all fitted distributions for the first and second datasets.
Distribution NP NLL r AIC r BIC r CAIC r HQIC r AD r CVM r KS PV(KS) r
Dataset I
ED 1 236.035 13 474.070 13 475.962 13 476.962 13 474.788 13 2.8483 12 0.5423 13 0.1999 0.0344 13
EED 2 231.116 11 466.232 11 470.015 11 472.015 11 467.667 11 0.6772 10 0.1161 11 0.1207 0.4377 11
TD 1 233.764 12 469.528 12 471.42 12 472.42 12 470.246 12 3.0822 13 0.3962 12 0.1780 0.0787 12
ETD 2 227.391 2 458.782 1 462.566 1 464.566 1 460.218 1 0.1155 2 0.0180 2 0.0532 0.9977 5
RD 1 230.394 10 462.787 9 464.679 6 465.679 5 463.505 9 0.8837 11 0.0876 8 0.1026 0.6419 8
ERD 2 228.079 6 460.157 4 463.941 4 465.941 6 461.593 4 0.2162 6 0.0378 6 0.0838 0.8529 6
LD 1 230.188 9 462.376 8 464.268 5 465.268 4 463.094 8 0.5572 9 0.0998 10 0.1155 0.4933 10
ELD 2 230.165 8 464.330 10 468.114 10 470.114 10 465.765 10 0.5232 8 0.0911 9 0.1132 0.5191 9
GOD 2 227.437 5 458.875 3 462.659 3 464.659 3 460.310 3 0.1311 5 0.0204 5 0.0506 0.9989 4
EGOD 3 227.383 1 460.766 5 466.442 8 469.442 8 462.919 5 0.1163 3 0.0181 3 0.0479 0.9995 2
WD 2 228.807 7 461.614 7 465.397 7 467.397 7 463.049 7 0.2932 7 0.0442 7 0.0874 0.8158 7
EWD 3 227.401 3 460.803 6 466.478 9 469.478 9 462.956 6 0.1204 4 0.0187 4 0.0484 0.9994 3
METD 2 227.402 4 458.805 2 462.588 2 464.588 2 460.240 2 0.1138 1 0.0178 1 0.0475 0.9996 1
Dataset II
ED 1 438.986 7 879.971 1 882.098 1 883.098 1 880.806 1 0.3666 7 0.0534 7 0.0744 0.8562 8
EED 2 438.748 5 881.496 5 885.750 5 887.750 5 883.166 5 0.3574 6 0.0486 6 0.0738 0.8627 7
TD 1 487.937 13 977.874 13 980.001 13 981.001 13 978.709 13 32.2274 13 3.8607 13 0.3786 2.67 x 10-8 13
ETD 2 442.409 10 888.818 9 893.073 9 895.073 9 890.489 9 1.2232 10 0.1945 10 0.1106 0.4035 10
RD 1 464.709 12 931.417 12 933.544 12 934.544 12 932.252 12 13.7411 12 1.4724 12 0.2574 0.0004 12
ERD 2 439.801 8 883.602 8 887.857 7 889.857 7 885.273 7 0.4831 8 0.0589 8 0.0658 0.9347 3
LD 1 446.268 11 894.536 11 896.663 11 897.663 10 895.371 11 3.6521 11 0.3507 11 0.1594 0.0765 11
ELD 2 438.755 6 881.510 6 885.764 6 887.764 6 883.180 6 0.3317 4 0.0421 4 0.0684 0.9138 5
GOD 2 438.656 2 881.312 3 885.566 3 887.566 3 882.982 3 0.2905 2 0.0337 2 0.0614 0.9622 2
EGOD 3 441.424 9 888.847 10 895.229 10 898.229 11 891.353 10 0.8585 9 0.1241 9 0.0848 0.7306 9
WD 2 438.703 4 881.406 4 885.660 4 887.660 4 883.076 4 0.3406 5 0.0440 5 0.0700 0.9006 6
EWD 3 438.692 3 883.384 7 889.766 8 892.766 8 885.890 8 0.3230 3 0.0403 3 0.0669 0.9261 4
METD 2 438.650 1 881.301 2 885.555 2 887.555 2 882.971 2 0.2896 1 0.0335 1 0.0610 0.9641 1
Dataset III
ED 1 109.299 13 220.597 13 222.740 13 223.740 13 221.440 13 3.8816 12 0.7789 13 0.2077 0.0074 13
EED 2 103.547 10 211.093 10 215.380 11 217.380 11 212.779 11 1.3151 10 0.2329 10 0.1437 0.1339 10
TD 1 106.974 12 215.947 12 218.090 12 219.090 12 216.790 12 3.9012 13 0.4427 12 0.1705 0.0453 12
ETD 2 98.288 4 200.577 3 204.863 3 206.863 3 202.263 3 0.3049 4 0.0464 3 0.0761 0.8311 5
RD 1 102.492 9 206.984 8 209.127 8 210.127 5 207.827 8 1.2469 9 0.0841 6 0.0958 0.5754 6
ERD 2 99.198 6 202.397 4 206.683 4 208.683 4 204.083 4 0.5116 6 0.0903 7 0.1067 0.4383 7
LD 1 104.578 11 211.156 11 213.299 10 214.299 10 211.999 10 2.1351 11 0.4159 11 0.1564 0.0821 11
ELD 2 101.888 8 207.776 9 212.063 9 214.063 9 209.462 9 0.9654 8 0.1682 9 0.1300 0.2170 9
GOD 2 98.276 3 200.553 2 204.840 2 206.840 2 202.239 2 0.3033 3 0.0466 4 0.0679 0.9143 1
EGOD 3 98.231 2 202.463 5 208.893 5 211.893 7 204.992 5 0.2890 2 0.0428 2 0.0694 0.9009 3
WD 2 100.318 7 204.635 7 208.922 6 210.922 6 206.321 7 0.6425 7 0.0929 8 0.1086 0.4164 8
EWD 3 98.327 5 202.654 6 209.084 7 212.084 8 205.183 6 0.3105 5 0.0473 5 0.0760 0.8325 4
METD 2 98.161 1 200.323 1 204.609 1 206.609 1 202.008 1 0.2623 1 0.0379 1 0.0681 0.9127 2
Covid-19, earthquake waiting times and service times data appropriately.
Acknowledgments
The authors would like to thank the editor for providing some helpful remarks which definitely improved the presentation of the manuscript. V.P. Jha is thankful to the Ministry of Human Resources Development of India and NIT Tiruchirappalli for the Institute scholarship.
References
[1] Teissier, G. (1934). Recherches sur le vieillissement et sur les lois de la mortalite. II. Essai d'interpretation generale des courbes de survie. Annales de physiologie et de physicochimie biologique, 10, 260-284.
[2] Muth, E. J. (1977). Reliability models with positive memory derived from the mean residual life function. Theory and applications of reliability.
[3] Rinne, H. (1981). Estimating the lifetime distribution of private motor-cars using prices of used cars: The Teissier model. Statistiks Zwischen Theorie und Praxis, 172-184.
[4] Leemis, L. M., & McQueston, J. T. (2008). Univariate distribution relationships. The American Statistician, 62(1), 45-53.
[5] Jodra Esteban, P., Jimenez Gamero, M. D., & Alba Fernandez, M. V. (2015). On the Muth Distribution. Mathematical Modelling and Analysis, 20 (3), 291-310.
[6] Irshad, M. R., Maya, R., & Arun, S. P. (2021). Muth distribution and estimation of a parameter using order statistics. Statistica, 81(1), 93-119.
[7] Maya, R., Jodra, P., Irshad, M. R., & Krishna, A. (2022). The unit Muth distribution: statistical properties and applications. Ricerche di Matematica, 1-24.
[8] Saroj, A., Sonker, P. K., & Kumar, M. (2022). Statistical Properties and Application of a Transformed Lifetime Distribution: Inverse Muth Distribution. Reliability: Theory & Applications, 17(1 (67)), 178-193.
[9] Deepthy, G. S., & Sebastian, N. (2022). A New Life Time Distribution: Burr III Modified Weibull Distribution And Its Application in Burn in Process. Reliability: Theory & Applications, 17(1 (67)), 76-86.
[10] Manoharan, M., & Kavya, P. (2022). A New Reliability Model and Applications. Reliability: Theory & Applications, 17(1 (67)), 65-75.
[11] Sharma, V. K., Singh, S. V., & Shekhawat, K. (2022). Exponentiated Teissier distribution with increasing, decreasing and bathtub hazard functions. Journal of Applied Statistics, 49(2), 371-393.
[12] Eghwerido, J. T. (2021). The alpha power Teissier distribution and its applications. Afrika Statistika, 16(2), 2733-2747.
[13] Poonia, N., & Azad, S. (2022). Alpha power exponentiated Teissier distribution with application to climate datasets. Theoretical and Applied Climatology, 1-15.
[14] Mahdavi, A., & Kundu, D. (2017). A new method for generating distributions with an application to exponential distribution. Communications in Statistics-Theory and Methods, 46(13), 6543-6557.
[15] Kundu, D., & Gupta, R.D. (1999). Generalized exponential distribution. Aust. NZJ. Stat. 41(1), 173-188.
[16] Nadarajah, S., & Haghighi, F. (2011). An extension of the exponential distribution. Statistics, 45(6), 543-558.
[17] Marshall, A. W., & Olkin, I. (2007). Life distributions (Vol. 13). Springer, New York.
[18] Jodra, P. (2009). A closed-form expression for the quantile function of the Gom-pertz-Makeham distribution. Mathematics and Computers in Simulation, 79(10), 3069-3075.
[19] Swain, J. J., Venkatraman, S., & Wilson, J. R. (1988). Least-squares estimation of distribution functions in Johnson's translation system. Journal of Statistical Computation and Simulation, 29(4), 271-297.
[20] Cheng, R. C. H., & Amin, N. A. K. (1983). Estimating parameters in continuous univariate distributions with a shifted origin. Journal of the Royal Statistical Society: Series B (Methodological), 45(3), 394-403.
[21] Kharazmi, O., Saadatinik, A., & Jahangard, S. (2019). Odd hyperbolic cosine exponential-exponential (OHC-EE) distribution. Annals of Data Science, 6(4), 765-785.
[22] Efron, B., & LePage, R. (1992). Introduction to bootstrap (pp. 3-10). Wiley & Sons, New York.
[23] Mudholkar, G. S., & Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-rate data. IEEE transactions on reliability, 42(2), 299-302.
[24] El-Gohary, A., Alshamrani, A., & Al-Otaibi, A. N. (2013). The generalized Gompertz distribution. Applied mathematical modelling, 37(1-2), 13-24.
[25] Murthy, D. P., Xie, M., & Jiang, R. (2004). Weibull models. John Wiley & Sons.
