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FasnaK RT&A' No 4 (71)
A METHOD FOR GENERATING LIFETIME MODELS Volume 17, December 2022
A METHOD FOR GENERATING LIFETIME MODELS AND ITS APPLICATION TO REAL DATA
Fasna K
University of Calicut [email protected]
Abstract
In the present work, we are going to propose a new transformation called Beta transformation. The new model includes the exponential distribution as a special case and it is known as Beta transformed exponential(BTE) distribution. We have been obtained its various statistical properties such as moments, moment generating function, median, hazard rate function, entropies, and order statistics. Parameters of BTE distribution are estimated by the method of maximum likelihood, Cramer-von-Mises and method of least square. Monte Carlo simulation is performed in order to investigate the performance of these estimates. Finally, two data sets have been analyzed to show how the proposed model works in practice.
Keywords: Cramer-von-Mises method, Exponential distribution, Hazard rate function, Method of maximum likelihood,, Method of least squares.
1. Introduction
The development of new methods of expanding the existing distributions is quite rich in the literature of distribution theory. There are several methods to propose new distributions by the use of some baseline distribution in statistical literature. This has been done through different approaches.
In Statistical literature no. of transformations are available to produce new cumulative distribution function (cdf) corresponding to a given cdf. Suppose, we have a cdf F(x), then the associated proposed cdf will be Gi (x).
• The most popular among them is the power transformation initiated by Gupta et al. (1998) having the form
G1(x) = [F(x)]a; a > 0
• Quadratic rank transformation map (QRTM) proposed by Shaw and Buckley (2007) having the form
G2(x) = (1 + A)F(x) - AF2(x); | A |< 1
• DUS transformation proposed by Kumar et al. (2015) having the form
eF(x) - 1
G3 (x) = 1 ;e = exp(1)
• SS-transformation proposed by Kumar et al. (2015) having the form
n
G4 (x) = sin( 2 F(x))
Minimum Guarantee (MG)-distribution proposed by Kumar et al. (2017) having the form
G5 (x) = e1 F(x)
Log-transformation proposed by Maurya et al. (2016) and having the form
ln(2 - F(x))
G6(x) = 1 -
ln2
Transformation based on the generalization of Kumar et al. (2015) called GDUS transformation proposed by Maurya et al. (2017) having the form
eFa(x) — 1
G7(x) =--—; a > 0
e — 1
New transformation initiated by Kyurkchiev (2017) to develop a sigmoid family of functions for Verhulst Logistic function is
2F(x)
G9 (x)
1 + F(x)
• New trigonometry based transformation called PCM proposed by Kumar et al. (2021) and having the form
n
G10 (x) = tan( - F(x))
The lifetime of a system can be modeled with statistical distributions that can be used in modeling lifetime data; among them, the most popular are gamma and weibull distributions. The proposed model contains several lifetime distributions as its special cases that are very flexible and able to accommodate different types of data sets since the probability density function and hazard rate can take on different forms such as increasing, decreasing, and constant shapes, and the potentiality of this model has been tested statistically by using it to model some real life data set.
In this article, We have decided to propose a new transformation known as beta transformation
for x £ S is given below
G(x)= {A" - 'i-f(')] f ß > ^ 1 (1)
1F(X) if ß = 1
Where, G(x) and F(x) are the cdfs of the proposed transformation and baseline distribution. On differentiating (1) w.r.t. x, we get the probabilty density function (pdf) g(x) and is given by
g(x) = i ^ f (x)P—F(x) if t > 0, t = 1 (2)
g( ) \f (x) if t = 1 ()
For t = 1, g(x) is a weighted version of f (x), where the weight function
w(x) = t—F(x),
and g(x) can be written as
where constant c = E(w(X)), Here c = .
g(x) = f (x)w(x) .
The survival reliability function(sf) S(x) and the hazard rate function(hrf) h(x) are obtained as
^1-F(x)-1 •( a -I 1 S(x)={ tLM~ if p = 1 1 1 - F(x) if p = 1
(3)
and
Hx)= J f (x) if P = 1
m={ m if p=1
1-F(x)
(4)
Lifetime models are used to explain the life of a system or device. These models are used in reliability, engineering, biological field, insurance, etc. The motivations for introducing our beta transformation model is that it is efficient to analyze lifetime data and very easy method of inducting an additional parameter to a family of distributions functions. It improve the characteristics, bring more flexibility to the given family and provide better fits than the other models having the same or higher number of parameters. The proposed method is very interesting with a closed form for the cdf and capable of modeling heavy tailed data sets. The aim of this article is to introduce a transformation that yields new distributions by using a given baseline distribution. It contains only one new parameter other than the parameters involved in the baseline distribution. To illustrate the usefulness of this new transformation, We choose exponential as the baseline distributions in the present work.
The rest of this work is as follows. In Section 2, We introduce a special sub-case of (1), called a beta transformed exponential(BTE) distribution by considering exponential model as a parent distribution. Some mathematical properties are derived in Section 3. Certain characterizations of the proposed distribution are provided in Section 4. Estimation of parameter has been carried out in Section 5, Simulation study have been discussed in Section 5. Illustrate the flexibility of models using two real-life data sets discussed in Section 7. Finally, the article is concluded in Section 8.
2. Beta transformed exponential distribution
In this section, a sub model of the beta transformed family, called the beta transformed exponential (BTE) distribution is introduced. Let G(x; 9) be cdf of the exponential random variable given by G(x; 9) = 1 - e-9x; x, 9 > 0. Using this in equation(1), then the cdf of the BTE for x > 0 with the shape and scale parameters as p > 0 and 9 > 0 has the following form
G(x)
p-1 [1 - pe^x-1 ] if p = 1 1 - e-9x if p = 1
(5)
The pdf g(x) is given by
g(x) = /
jopogp e-x pe-9x if p = 1
e- x if p = 1
The survival reliability function S(x) and the hazard rate function(hrf) h(x) are obtained as
(6)
S(x) = J ^ if p = 1 if p = 1
-9x
(7)
and
e-9xpe if p = 1 h(x) = 1 pe 9x-1 ^ H =
9 if p = 1
We have the following results for a general distribution function F(x).
Table 1: Behavior of the hazard functions of the three distributions.
Parameter
Gamma
Weibull
BTE
t = 1 0 t > 1 Increasing from 0to0
t < 1 Decreasing from ro to0
0
Increasing from
0 to ro Decreasing from ro to 0
Decreasing from t0 log t to 0 t—1 to 0 Increasing from
to 0
d
• If f (x) is a decreasing function, and t > 1, then g(x) is a decreasing function.
• If f (x) is a decreasing function, and f (x) is log-convex, then for t > 1, the hazard rate function h(x) is a decreasing function.
It can be easily seen that f (x; t, 0) is a unimodal function with mode at (log(l°8P)).
Here note that,
limx^0 h(x) = ^'—lt ,and
limx^ro h(x) = 0.
We have the following cases:
• When t < 1, h(x) is an increasing function increases from 1t to 0
• When t > 1, h(x) is an decreasing function decreases from ^—ft to 0
• When t = 1, h(x) is a constant function.
By taking the second derivative of f (x; t, 0), it easily follows that the pdf of BTE(t, 0) is log-convex if t > 1 and log-concave if t < 1;
Table 1 provides the comparison of the hazard function of the BTE distribution with the corresponding hazard functions of Weibull and Gamma distributions. In all these cases the shape and scale parameters are assumed to be t and 0, respectively. It is clear from Table 1 that the hazard function of the BTE distribution is a decreasing or an increasing function depending on the shape parameter similarly as the Gamma and Weibull distributions, the ranges are quite different.
Figure 1 and 2 provides the plots of the pdf and hrf of the model for different values of t when 0 = 1
BTE distribution for t > 1, r is a decreasing function from 1 to 0, as t varies from 1 to ro. If X ~ BTE(t, 0), then BTE distribution has the following mixture representation:
l°g ß
X :7-1iogß (9)
X1 with probability if 1 X2 with probability if 1 — t—1,
where X! and X2 have the following pdfs:
fX1 (x) = 0e—0x; x > 0 (10)
fX2(x) = t — 1o—lg 10e—0x (te—0x — 1); x > 0, (11)
respectively. From (9), as t approaches 1, X behaves like an exponential distribution, and as t increases, it behaves like X2.
Figure 1: plot of pdf of distribution
3. The basic mathematical properties
This section provides some mathematical properties of proposed distribution.
3.1. Quantile function
The qth quantile xq of the BTE random variable is given by
- 1log
log(l - ^
log P
l +
(12)
3.2. Moments
In this subsection, we intend to derive the moments and the moment generating function of the BTE distribution. Let X follow (6), then, the rth moment of X is derived as
Vr
using(6)in(13), we get
Vr
r!
xrf (x; p, 9)dx, (ln p)k
er(p -1) kt! k!k
E
(l3)
(14)
Furthermore, a general expression for the moment generating function (mgf) of the BTE random variable X is given by
Mx (t)
9 EE (ln P)k+1
(P - 1) E k!
1
9 + 9k -1
; t <9
(15)
3.3. Sample Generation
The method to generate a sample is the inverse CDF transformation method. If X is U(0,1) with CDF F(x), then by the transformation, we generate the sample from the equation G(x) = U
x
q
CO
CO
in
btta=3.1 beta=D.E-beta=5 bEta = 10
Figure 2: plot of hrf of distribution
implies x = G 1 (U) of BTE distribution
-1 l°g
1 +
log(1 -
U(ß-1)
log ß
(16)
3.4. Order Statistics
Order statistics are used in applied fields of statistics such as reliability and lifetime testing.
Let X1,X2,...,Xn be a random sample from BTE(ß,d) . Also, let X(1),X(2),...,X(n), denote the corresponding order statistics. Then the pdf and cdf of kth order statistics, are given by
and
fx (x)
n!
[F(x)]k-1 [1 - F(x)]n-kf (x)
n— k
(k - 1)!(n - k)!
n!
g logß e-exße-ex m _ ß-F(x)iik-1
(k - 1)!(n - k)! ß - 1 e ß [ß - 1[1 ß JJ
1 - [ j^j [1 - ß-F(x
nk
Fx (x) = E
j=k
E
j=k
[F(x)]j[1 - F(x)]n-j
[i - ß-F(x)
1 - [ j-T [1 - ß-F(x)]]
n- j
(17)
(18)
respectively.
The pdf of the minimum and maximum of order statistics are obtained by putting X = X1 and
X = Xn respectively in equation (6).
)
x
n
3.5. Entropy
The entropy of a random variable measures the variation of the uncertainity. A large value of entropy indicates the greater uncertainty in the data. The concept of entropy is important in
different areas such as physics, probability and statistics, communication theory, and economics,
etc. Several measures of entropy have been studied and compared in the literature.
If X is an absolute continuous random variable with REX(p) for p > 0 and p = 1, is defined as
1
REx (P) = r^ !og 1 - p
f(x)pdx
(19)
From equation(19), we get
REX(p) = ^m + «20,
4. Characterization of Beta transformed exponential distribution
In this section, we present certain characterizations of the BTE distribution based on a simple relationship between two truncated moments. This characterization result employs a theorem due to Glanzel (1987), which stated as follows:
Theorem 1. Let (Q,F, P) be a given probability space and let H = [a, b] be an interval for some a < b (a = -to, b = to might as well be allowed). Let X : Q ^ H be a continuous random variable with the distribution function F and let q1 and q2 be two real functions defined on H such that
E[q2(X)|X > x] = E[q1(X)lX > x]n(x),xeH,
is defined with some real function q. Assume that q\, q2 are continuous functions, q has continuous derivative and F is twice continuously differentiable and strictly monotone function on the set H. Finally, assume that the equation = q2 has no real solution in the interior of H. Then F is uniquely determined by the functions qi, q2 and q, particularly
F(x) = Ja Cln MM- qM lexp(-s<"))d",
n ?1
uiuuuii ui nie uiiiciciiuai cv^uauun o — --
chosen to make fH dF = 1.
where the function s is a solution of the differential equation s = and C is a constant,
—9x
Proposition 1. Let X : Q ^ (0, œ) be a continuous random variable and let q1(x) = p-e and q2(x) = q1(x)e-9x for x > 0. The random variable X has pdf (6) if and only if the function n defined in Theorem 1 has the form
/ \ 9 + 1 _x n (x) = —x— e x, x > 0 0
Proof. Let X be a random variable with pdf (6), then (1 - F(x))E[q1 (X) | X > x] = p-pe-9x,x > 0 and
(1 - F(x))E[q2(X) | X > x] = ^,x > 0 and finally
n(x)q-i_(x) - q2(x) = > 0, forx > 0.
œ
œ
Conversely, if n is given as above, then
s'(x) = n'(x>(1^ = -(1 + 0),x > 0 nqi(x) - (2(x)
and hence
s(x) = -(1 + 0)x, x > 0, or e-s(x) = e(1+0)x, x > o. Now, in view of Theorem 1, X has density(6).
■
Corollary 1. Let X : Q ^ (0, œ) be a continuous random variable and let q1 (x) be as in
Proposition 1. The pdf of X is (6) if and only if there exist functions (2 and n defined in Theorem 1 satisfying the differential equation
n (x)(1(x) --(1 + 0),x > 0
№(x) — q2(x)
Remark 1. The general solution of the differential equation in Corollary 1 is
n(x) = e—(1+0)x y(1 + 0)[q1 (x)]—1 q2(x)e(1+0)xdx + D
where D is a constant. Note that a set of functions satisfying the above differential equation is given in Proposition 1 with D = 0. However, it should be also noted that there are other triplets (qi, q2, n) satisfying the conditions of Theorem 1.
5. Estimation and simulation
In this section, we use the method of maximum likelihood, method of Cramer-von-Mises and ordinary least square method for estimation of parameters of BTE distributions.
5.1. Method of Maximum Likelihood Estimation
This is an extensively used method initiated by C.F. Gauss and elaborative study initiated by Prof. R. A. Fisher to obtain the estimator of the unknown parameter of the distribution. If X1, X2..., Xn be a set of random observations from the population BTE(t, 0) distribution having pdf g(x; t, 0), then its log likelihood function will be as follows
log L = n log0 + n log — 0 ¿x; + ¿e—00xi log(t). (21)
Vt — V ¿=1 ¿=1
The likelihood equations are,
d log L = n(t — 1 — t logt) + 1 e—0x,. = 0 (22)
dt t(t — 1) logt + t 0, (22)
and
^ = n - t x. + :ç x.e-0"= 0. (23)
1=1 1 = 1
The MLE of t and 0 can be obtained by solving this nonlinear system of equations. It is usually more convenient to use nonlinear optimization algorithms such as the Newton-Raphson algorithm.
FasnaK RT&A No 4 (71) A METHOD FOR GENERATING LIFETIME MODELS_Volume 17 December 2022
5.2. Method of Cramer-von Mises
Cramer-von-Mises type minimum distance estimators are based on minimizing the distance between the theoretical and empirical cumulative distribution functions. Macdonald(1971) provided empirical evidence that the bias of these estimators is smaller than the bias of other minimum distance estimators. The Cramer-von-Mises estimators, tcME and 0CME are the values of t and 0 minimizing
2i — 1
1
c(t,0 ) = 12- + E
12n i=1
F(ti I t, 0) —
2n
Differentiating the above equation partially, with respect to the parameters t and 0 respectively and equating them to zero, we get the normal equations. Since the normal equations are nonlinear, we can use iterative method to obtain the solution.
5.3. Method of Least-Square Estimation
The least square estimators were proposed by Swain et al. (1988) to estimate the parameters of Beta distributions. Here, we apply the same technique for the BTE distribution. The least square estimators of the unknown parameters t and 0 of BTE distribution can be obtained by minimizing
n T i
E r(fi't,0) - -+T
with respect to unknown parameters t and 0.
5.4. Simulation study
We conduct Monte Carlo simulation studies to compare the performance of the estimators discussed in the previous sections and the process is repeated 1000 times. We evaluate the performance of the estimators based on bias and mean squared error. Methods are compared for sample sizes n = 500,700 and 1000.
For each estimate we calculate the mean-squared error. The statistics are obtained using the following formulae.
MSE(t) = n En=1 (t — t)2 MSE(0) = n En=1 (0 — 0)2
The estimates, and the mean square errors (MSE) of the parameter estimates for the Maximum likelihood estimation procedure, method of Cramer-von-Mises and method of least squares are presented in Tables 1-3.
From Tables , we note that the maximum likelihood method performs well for estimating the model parameters. Also, as the sample size increases, the MSEs of the average estimates of maximum likelihood estimates decrease as expected. The following observations can be drawn from the Tables 1-3.
1. All the estimators show the property of consistency, i.e. the MSE decreases as the sample size increases.
2. The MSE of t decreases with an increasing n for all the method of estimations.
3. The MSE of 0 decreases with an increasing n for all the method of estimations.
4. The MSE of t and 0 generally increases with an increasing beta and theta for any given n in all methods of estimation.
5. In terms of MSE, all the methods of estimation produce smaller MSE for t compared to that of
Table 2: Simulation result for f = 0.5 and 6 = 0.1.
n Est. MLE CVM LSE
500 f 6 MSE(f) MSE(6) 0.5173 0.1004 5.247x10-5 0.0209 0.5087 0.1004 7.3129x10-5 0.0244 0.5245 0.1021 7.4119 x10-5 0.0311
700 P 6 MSE( /3) MSE(6) 0.5171 0.1000 3.702 x10-5 0.0150 0.5222 0.1003 7.0427x10-5 0.0220 0.5306 0.0994 5.4085 x10-5 0.0260
1000 /3 6 MSE( 3) MSE(6) 0.5013 0.1004 2.495 x10-5 0.0093 0.5062 0.1005 4.0516x10-5 0.0129 0.5845 0.0972 3.9591 x10-5 0.0187
Table 3: Simulation result for f = 0.9 and 6 = 0.5.
n Est. MLE CVM LSE
f 0.9315 0.9527 0.9941
500 6 0.5044 0.5049 0.4966
MSE(f) MSE(6) 0.0019 0.0943 0.0029 0.1899 0.0025 0.1144
f 0.9196 0.9489 0.9046
700 6 0.5018 0.5044 0.5011
MSE(f) MSE(6) 0.0013 0.0664 0.0025 0.1047 0.0016 0.0556
f 0.9260 0.9310 0.9120
1000 6 0.5004 0.4948 0.5034
MSE(f) MSE(6) 0.0009 0.0436 0.0014 0.0503 0.0014 0.0484
6. Applications
In this section, we consider two real life data sets to illustrate the importance of the proposed distribution. The model parameters are estimated by the method of maximum likelihood and compare the fit of the BTE distribution with the following distributions: KuE,EW,W and E models.
(a) Kumaraswamy Exponential (KuE) distribution having pdf
f (x; 6, f, c) = 6fce-cx (1 - ecx )6-1[1 - (1 - e-cx )6 ]f-1; x > 0,6, f, c > 0. (24)
(b) Exponentiated Weibull (EW) distribution having pdf
f (x; 6, f, c) = 6f9 cx6-1e-(fx)6 (1 - e-(fx)B )c-1; x > 0,6, f, c > 0. (25)
(c)Weibull (W) distribution having pdf
f(x; 6, f) = f6fxf-1e(-6x)f; x > 0,6, f > 0. (26)
Table 4: Simulation result for ß = 1.5 and 6 = 1.
n Est. MLE CVM LSE
ß 1.5861 1.7673 1.6562
500 6 1.004 0.9999 1.0142
MSE(ß) MSE(6) 0.0101 0.3336 0.0235 0.9141 0.0218 0.8282
ß 1.5470 1.5931 1.5058
700 6 1.005 1.0114 1.0263
MSE(ß) mse(6) 0.0075 0.2346 0.0128 0.4939 0.0174 0.2132
ß 1.5278 1.5810 1.5941
1000 6 1.003 0.9929 1.0107
MSE(ß) mse(6) 0.0048 0.1388 0.0052 0.1505 0.0152 0.3750
Table 5: The descriptive statistics of Data set.
Min 1st Q Median Mean 3rd Q Max
0.30 17.50 40.00 46.33 60.00 154.00
(d) Exponential (E) distribution having pdf
f (x; 0) = 0e—ex; x > 0, 0 > 0 (27)
The values of the log-likelihood functions— ln(L), AIC(Akaike Information Criterion), AICC(Akaike Information Criterion with correction) and BIC(Bayesian Information Criterion) are calculated for the five distributions in order to verify which distribution fits better to data. The better distribution corresponds to smaller — ln(L), AIC, AICC and BIC values. Here, AiC = —2 ln(L) + 2k, A1CC = —2ln(L) + (n—i) and BIC = — 2ln(L) + k ln(n); where L is the likelihood function evaluated at the maximum likelihood estimates, k is the number of parameters and n is the sample size. The K-S distance Dn = SMpx|F(x) — Fn(x)|,where,Fn(x) is the empirical distribution. Kolmogorov-Smirnov (K-S) statistic is computed to compare the fitted models. The required computations are carried out in the R-language introduced by R Development Core Team (2019).
6.1. Data set 1
The first real data set represents the survival times of 121 patients with breast cancer obtained from a large hospital in a period from 1929 to 1938 taken from Lee (1992). The data are: (0.3, 0.3, 4.0, 5.0, 5.6, 6.2, 6.3, 6.6, 6.8, 7.4, 7.5, 8.4, 8.4,10.3,11.0,11.8,12.2, 12.3,13.5,14.4, 14.4,14.8, 15.5, 15.7, 16.2, 16.3, 16.5, 16.8, 17.2, 17.3, 17.5, 17.9, 19.8, 20.4, 20.9, 21.0, 21.0, 21.1, 23.0, 23.4, 23.6, 24.0, 24.0, 27.9, 28.2, 29.1, 30.0, 31.0, 31.0, 32.0, 35.0, 35.0, 37.0, 37.0, 37.0, 38.0, 38.0, 38.0, 39.0, 39.0, 40.0, 40.0, 40.0, 41.0, 41.0, 41.0, 42.0, 43.0, 43.0, 43.0, 44.0, 45.0, 45.0, 46.0, 46.0, 47.0, 48.0, 49.0, 51.0, 51.0, 51.0, 52.0, 54.0, 55.0, 56.0, 57.0, 58.0, 59.0, 60.0, 60.0, 60.0, 61.0, 62.0, 65.0, 65.0, 67.0, 67.0, 68.0, 69.0, 78.0, 80.0,83.0, 88.0, 89.0, 90.0, 93.0, 96.0, 103.0, 105.0, 109.0, 109.0,111.0,115.0,117.0,125.0, 126.0,127.0,129.0,129.0,139.0, 154.0). The data is skewed-to-the right with skewness =1.0432 and kurtosis =0.4021
The descriptive statistics of the above data set are given in Table 4. The values in Table 5 shows that the BTE distribution leads to a better fit to the other four models.
Figure 3, shows the fitted density curves, Empirical and the fitted cumulative distribution functions for the data set 1.
Table 6: Maximum likelihood parameter estimates and goodness of fit for various models fitted for the Data set.
Model parameter estimates log L AIC AICC BIC K-S p-value
BTE p = 0.131 6 = 0.033 -579.155 1162.309 1162.411 1167.901 0.0534 0.8802
KuE 6 = 1.651 p = 0.098 c = 0.231 -583.314 1172.63 1172.83 1181.02 0.1152 0.0803
EW 6 = 1.393 p = 0.017 c = 0.798 -579.879 1165.76 1165.96 1174.15 0.0664 0.6606
W p = 1.306 6 = 0.019 -580.024 1164.05 1164.15 1169.64 0.0588 0.7967
E 6 = 0.022 -585.128 1172.26 1172.29 1175.05 0.1206 0.0594
Figure 3. Histogram with fitted pdf's (left) and Empirical cdf with fitted cdf's (right) for the data set 1.
6.2. Data set 2
Here we consider the data set of the life of fatigue of Kelvar 373/epoxy that are subject to constant pressure at the 90% stress level until all had failed. The data sets are taken from Andrews and Herzberg (1985). The data are:
(0.0251, 0.6751,1.0483, 1.4880,1.8808, 2.2460, 3.4846, 0.0886, 0.6753, 1.0596,1.5728,1.8878, 2.2878, 3.7433, 0.0891, 0.7696, 1.0773, 1.5733,1.8881, 2.3203, 3.7455, 0.2501, 0.8375, 1.1733 1.7083, 1.9316, 2.3470, 3.9143, 0.3113, 0.8391, 1.2570,1.7263, 1.9558, 2.3513, 4.8073, 0.3451, 0.8425,1.2766, 1.7460, 2.0048, 2.4951, 5.4005, 0.4763, 0.8645,1.2985, 1.7630, 2.0408, 2.5260, 5.4435, 0.5650, 0.8851, 1.3211, 1.7746, 2.0903, 2.9941, 5.5295, 0.5671, 0.9113, 1.3503, 1.8275, 2.1093, 3.0256, 6.5541, 0.6566, 0.9120, 1.3551, 1.8375, 2.1330, 3.2678, 9.0960, 0.6748, 0.9836, 1.4595, 1.8503, 2.2100, 3.4045). The data is skewed-to-the right with skewness =1.9794 and kurtosis =5.160
The descriptive statistics of the above data set are given in Table 6. The values in Table 7 shows that the BTE distribution leads to a better fit to the other four models.
Figure 4, shows the fitted density curves, Empirical and the fitted cumulative distribution functions for the data set 2.
Table 7: The descriptive statistics of Data set.
Min 1st Q Median Mean 3rd Q Max
0.025 0.905 1.736 1.959 2.296 9.096
FasnaK RT&A No 4 (71) A METHOD FOR GENERATING LIFETIME MODELS_Volume 17, December 2022
Table 8: Max/mum likelihood parameter estimates and goodness o/fit/or various models fitted for the Data set.
Model parameter estimates log L AIC AICC BIC K-S p-value
BTE ft = 0.070 9 = 0.873 -121.410 246.820 246.984 251.481 0.099 0.4167
EW 9 = 1.101 ft = 0.609 c = 1.443 -122.166 250.332 250.665 257.324 0.0992 0.4160
W ft = 1.326 9 = 0.469 -122.526 249.052 249.219 253.714 0.1098 0.2968
E 9 = 0.510 -127.114 256.228 256.282 258.559 0.5120 0.0266
(c) Fitted pdf plots of Data set 2
(d) Empirical and the fitted cumulative distribution functions for the data set 2
Figure 4. Histogram withfitted pd/'s (left) and Empirical cd/withfitted cd/'s (right)/or the data set 2.
7. Concluding remarks
In this paper,we have proposed beta transformation in order to get a transformed distribution of some available baseline distribution. Beta transformation of exp(0) distribution has been considered to check its application to the real problem called the Beta transformed (BTE) distribution. In the present work, we have provide expressions for the quantiles,moments,moment generating function,hazard rates,entropies and order statistics. The model parameters are estimated by maximum likelihood, Cramer-von Mises and least squares method. We have performed an extensive simulation study to compare these methods. We have compared estimators with respect to mean-squared error. The simulation results show that maximum likelihood estimators is the best performing estimator in terms of MSE. The next best performing estimator is the least square estimator followed by the Cramer-von Mises estimator.
Two real data sets are analyzed to show the importance and flexibility of this distribution.
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