The Weibull family of lifetime distributions play a fundamental role in reliability engineering and life testing problems. This paper investigates the potential usefulness of transmuted new generalized Weibull (TNGW) distribution for modeling lifetime data. This distribution is an important competitive model that contains twenty-three lifetime distributions as special cases. We can obtain the TNGW distribution using the quadratic rank transmutation map (QRTM) technique. We derive the analytical shapes of the density and hazard functions for graphical illustrations. In addition, we explore some mathematical properties of the TNGW model including expressions for the quantile function, moments, entropies, mean deviation, Bonferroni and Lorenz curves and the moments of order statistics. The method of maximum likelihood is used to estimate the model parameters. Finally the applicability of the TNGW model is presented using nicotine in cigarettes data for illustration.
The probabilistic modeling approach is a traditional device to explain real world scenarios in many areas of research. Therefore, any probabilistic model defined on the positive real line can be considered as a lifetime model. The most popular model is the Weibull distribution that has been extensively used over the past decades to model lifetime data in reliability engineering, astronomy, medicine, psychology, botany, zoology, agriculture, fisheries and actuaries. The two-parameter Weibull distribution was introduced by Swedish physicist Waloddi Weibull in 1937 and has proved to be a versatile model with a wide range of applicability for analyzing lifetime data. The beauty of this model is the ability to provide reasonably accurate failure analysis and failure forecasts with extremely small samples, Abernethy (2000). Ever since, it has been extensively used for analyzing lifetime data. Many authors have introduced new distributions to model bathtub shaped instantaneous failure rates.
For example, Mudholkar and Srivastava (1993) proposed the exponentiated Weibull distribution to analyze failure data. Reliability analysis using an additive Weibull and extended Weibull models with bathtub-shaped failure rate functions were introduced by Xie and Lai (1996) and Xie
where
The four-parameter NGW distribution generalizes the eleven lifetime distributions with an increasing and decreasing bathtub shaped hazard rate function.
Let
This class of transmuted distributions received considerable attention after the recent work of Shaw and Buckley (2009), which provided greater flexibility of its tails and can be applied in many areas of reliability studies. A distribution can be made more flexible by adding another parameter. The QRTM technique is one way to do so. It shows promise by expanding the range of available tail properties, as can be seen in improved goodness-of-fit results, even after adjusting for the additional parameter via measures such as the AIC. Aryal and Tsokos (2011) used this technique to propose the transmuted Weibull distribution as well as derived some mathematical properties with application. Recently Khan and King (2013a, 2013b) introduced the transmuted modified Weibull, transmuted generalized inverse Weibull distributions and formulated some of its properties with application. Khan and King (2014), Khan
The structure of this paper is as follows. In Section 2, we present the analytical shapes of the density and hazard functions. Some statistical properties such as the quantile functions, moment estimation and moment generating function are addressed in Section 3. Entropies, mean deviation, Bonferroni and Lorenz curves are derived in Section 4. In Section 5, we derive the density function and moments of order statistics. Maximum likelihood estimates (MLE
A positive random variable
The cdf corresponding to (
Here
and
Figure 1 illustrates the shapes of the TNGW pdf with some selected choice of parameters. When the cdf of the TNGW distribution has zero value then it represents no failure components. Figure 2 illustrates the hazard function of the TNGW distribution with a different choice of parameters. The distribution has increasing, decreasing and constant behaviors for hazard rates. The TNGW distribution provides a wide usage of the Weibull family of lifetime distributions. The TNGW model can be used in more complex situations and provides more flexibility in real world scenarios. The TNGW distribution contains several well-known distributions as special cases several well-known distributions. Twenty-three distributions included as special cases of the TNGW distribution are displayed in Table 1.
This section presents the quantile analysis,
The quantile
The above
The
The
The
The
The
The
The
The
The
The
The
By substituting
The
using (
Using the binomial expansion the above equation reduces to
the above integral yields the following
The important features and characterizations of the TNGW distribution can be studied using
By definition
Using Taylor series expansions, the above integrals reduce to
the above integral yields the
The entropy of a random variable
where
the above integral reduces to
where
Finally, we obtain the TNGW Rényi entropy as
where
The
where
the above integral yields the TNGW
where
Table 4 lists the values of Rényi entropy of the TNGW distribution for selected values of the parameters. Table 5 lists the values of
If
The mean
The quantity
where
The density of the
where
where
The pdf of
where
and
Using (
where
Consider the random samples
The components of the score function can be obtained by differentiating (
and
respectively. We obtain the analytical expressions for the MLEs of a five-parameter score vector, yields the ML estimators
where
where −
We conducted the simulation study to evaluate the performance of MLEs with respect to the sample size for the TNGW distribution. By using
This section illustrates the usefulness of the TNGW distribution with nicotine cigarette data. The data set consists of 396 observations of nicotine content in milligrams in cigarettes for several cigarettes brands in 1995. The data was obtained from the Federal Trade Commission (FTC), an independent agency of the US government, whose main mission is the promotion of consumer protection. The report entitled “Tar, Nicotine and Carbon Monoxide of the Smoke of 1249 varieties of domestic cigarettes for the Year 1995” at FTC (1998) consists of data sets and information about the source of data, smoker behavior and beliefs about nicotine, tar and carbon monoxide contents in cigarettes. The model parameters are estimated using the method of maximum likelihood and five goodness-of-fit statistics are used to compare TNGW distribution with other five lifetime models, their associated density functions are given by
Kumaraswamy Weibull Poisson (KwWP) distribution with pdf
where
Transmuted additive Weibull (TAW) distribution with pdf
where
New generalized Weibull (NGW) distribution with pdf
where
Exponentiated Weibull (EW) distribution with pdf
where
Generalized power Weibull (GPW) distribution with pdf
where
The required numerical evaluations are implemented using R language. The MLEs of the parameters (with their standard errors) for the nicotine in cigarettes data are displayed in Table 7. Table 8 illustrates their corresponding values of the Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (CAIC), Anderson-Darling (
In this paper, we have constructed the five-parameter distribution (referred to as the TNGW distribution), which includes twenty-three lifetime distributions as special cases, as well as study its theoretical properties. The TNGW model is more flexible than the KwWP, TAW, NGW, EW, and GPW distributions proposed recently in the literature. We obtained the analytical shapes of density and hazard functions of the TNGW distribution. The TNGW distribution has increasing, decreasing and constant failure rate patterns for lifetime data. The flexibility and usefulness of the TNGW model is illustrated in an application to nicotine in cigarettes data using MLE.
Sub-models of the transmuted NGW distribution
Distribution | | | | | |
---|---|---|---|---|---|
TNGE | - | 1 | - | - | - |
TNGR | - | 2 | - | - | - |
TMW | - | - | - | 1 | - |
TMR | - | 2 | - | 1 | - |
TME | - | 1 | - | 1 | - |
TGW | 0 | - | - | - | - |
TGR | 0 | 2 | - | - | - |
TGE | 0 | 1 | - | - | - |
TW | 0 | - | - | 1 | - |
TR | 0 | 2 | - | 1 | - |
TE | 0 | 1 | - | 1 | - |
NGW | - | - | - | - | 0 |
NGR | - | 2 | - | - | 0 |
NGE | - | 1 | - | - | 0 |
MW | - | - | - | 1 | 0 |
MR | - | 2 | - | 1 | 0 |
ME | - | 1 | - | 1 | 0 |
GW | 0 | - | - | - | 0 |
GR | 0 | 2 | - | - | 0 |
GE | 0 | 1 | - | - | 0 |
W | 0 | - | - | 1 | 0 |
R | 0 | 2 | - | 1 | 0 |
E | 0 | 1 | - | 1 | 0 |
T = transmuted; M = modified; N = new; G = generalized; W = Weibull; R = Rayleigh; E = exponential.
Moments values of the TNGW distribution
( | | | | | |
---|---|---|---|---|---|
1, 0.5, 1, 1 | −1.0 | 0.7365 | 1.0861 | 2.4618 | 7.6323 |
−0.5 | 0.5954 | 0.8382 | 1.8687 | 5.7545 | |
0.5 | 0.5000 | 0.5555 | 1.0000 | 2.5185 | |
1.0 | 0.3333 | 0.2222 | 0.2222 | 0.2963 | |
1, 1, 1, 1 | −1.0 | 0.7500 | 0.8750 | 1.4062 | 2.9062 |
−0.5 | 0.6250 | 0.6875 | 1.0781 | 2.2031 | |
0.5 | 0.3750 | 0.3125 | 0.4218 | 0.7968 | |
1.0 | 0.2500 | 0.1250 | 0.0937 | 0.0938 | |
1, 2, 2, 2 | −1.0 | 0.7656 | 0.6588 | 0.6263 | 0.6499 |
−0.5 | 0.6846 | 0.5535 | 0.5066 | 0.5129 | |
0.5 | 0.5224 | 0.3429 | 0.2671 | 0.2391 | |
1.0 | 0.4414 | 0.2376 | 0.1474 | 0.1022 | |
2, 3, 2, 3 | −1.0 | 0.7331 | 0.5810 | 0.4918 | 0.4409 |
−0.5 | 0.6650 | 0.4975 | 0.4057 | 0.3546 | |
0.5 | 0.5288 | 0.3305 | 0.2336 | 0.1821 | |
1.0 | 0.4608 | 0.2470 | 0.1475 | 0.0957 |
Moments based measures of the TNGW distribution
( | Mean | Var | CV | CS | CK | |
---|---|---|---|---|---|---|
1, 0.5, 1, 1 | −1 | 0.7365 | 0.5436 | 1.0011 | 2.1481 | 10.258 |
−0 | 0.5954 | 0.4836 | 1.1681 | 2.3592 | 11.583 | |
0.5 | 0.5000 | 0.3055 | 1.1054 | 2.4681 | 12.475 | |
1.0 | 0.3333 | 0.1111 | 1.0001 | 2.0000 | 9.0027 | |
1, 1, 1, 1 | −1 | 0.7500 | 0.3125 | 0.7453 | 1.6096 | 7.0810 |
−0 | 0.6250 | 0.2968 | 0.8717 | 1.7144 | 7.5045 | |
0.5 | 0.3750 | 0.1718 | 1.1055 | 2.4658 | 12.472 | |
1.0 | 0.2500 | 0.0625 | 1.0000 | 1.9968 | 9.0256 | |
1, 2, 2, 2 | −1 | 0.7656 | 0.0726 | 0.3521 | 0.5448 | 3.4351 |
−0 | 0.6846 | 0.0848 | 0.4254 | 0.4669 | 3.2145 | |
0.5 | 0.5224 | 0.0699 | 0.5065 | 0.8010 | 3.8797 | |
1.0 | 0.4414 | 0.0427 | 0.4685 | 0.5392 | 3.1849 | |
2, 3, 2, 3 | −1 | 0.7331 | 0.0435 | 0.2847 | 0.2194 | 3.0218 |
−0 | 0.6650 | 0.0552 | 0.3535 | 0.1036 | 2.8772 | |
0.5 | 0.5288 | 0.0508 | 0.4265 | 0.4384 | 3.0592 | |
1.0 | 0.4608 | 0.0346 | 0.4040 | 0.2691 | 2.7051 |
CV = coefficient of variation; CS = coefficient of skewness; CK = coefficient of kurtosis.
Rényi entropy values for the TNGW distribution
( | |||||
---|---|---|---|---|---|
1, 1, 1, 1 | −1 | 0.2891 | 0.2627 | 0.3698 | 0.4703 |
−0 | 0.2833 | 0.2602 | 0.3697 | 0.4741 | |
0.5 | 0.0828 | 0.0325 | 0.0018 | −0.0410 | |
1.0 | −0.3010 | −0.3635 | −0.4014 | −0.4273 | |
1, 2, 2, 2 | −1 | −0.0321 | −0.0636 | −0.0827 | −0.0958 |
−0 | 0.0080 | −0.0224 | −0.0412 | −0.0541 | |
0.5 | −0.0553 | −0.0869 | −0.1062 | −0.1193 | |
1.0 | −0.1415 | −0.1697 | −0.1872 | −0.1993 | |
2, 3, 2, 3 | −1 | −0.1299 | −0.1605 | −0.1791 | −0.1918 |
−0 | −0.0745 | −0.1045 | −0.1230 | −0.1358 | |
0.5 | −0.0996 | −0.1280 | −0.1456 | −0.1578 | |
1.0 | −0.1732 | −0.1994 | −0.2158 | −0.2272 | |
2, 4, 2, 5 | −1 | −0.2356 | −0.2674 | −0.2867 | −0.2998 |
−0 | −0.1667 | −0.1999 | −0.2201 | −0.2339 | |
0.5 | −0.1502 | −0.1806 | −0.1994 | −0.2124 | |
1.0 | −0.2070 | −0.2359 | −0.2538 | −0.2662 |
( | |||||
---|---|---|---|---|---|
1, 1, 1, 1 | −1 | 0.3333 | 0.2334 | 0.1809 | 0.1484 |
−0 | 0.2500 | 0.1584 | 0.1107 | 0.0809 | |
0.5 | −0.4166 | −0.8916 | −1.7369 | −3.4587 | |
1.0 | −1.0000 | −2.1666 | −5.0000 | −12.550 | |
1, 2, 2, 2 | −1 | −0.0767 | −0.1703 | −0.2571 | −0.3545 |
−0 | 0.0183 | −0.0545 | −0.1096 | −0.1614 | |
0.5 | −0.1359 | −0.2463 | −0.3606 | −0.5000 | |
1.0 | −0.3850 | −0.5926 | −0.8816 | −1.3175 | |
2, 3, 2, 3 | −1 | −0.3488 | −0.5468 | −0.8150 | −1.2134 |
−0 | −0.1872 | −0.3089 | −0.4464 | −0.6238 | |
0.5 | −0.2578 | −0.4016 | −0.5779 | −0.8188 | |
1.0 | −0.4899 | −0.7526 | −1.1468 | −1.7771 | |
2, 4, 2, 5 | −1 | −0.7204 | −1.2129 | −2.0811 | −3.7062 |
−0 | −0.4680 | −0.7555 | −1.1914 | −1.9056 | |
0.5 | −0.4131 | −0.6487 | −0.9883 | −1.5179 | |
1.0 | −0.6107 | −0.9816 | −1.5909 | −2.6526 |
Monte Carlo simulation results based on MLE, S.E, Bias and MSE for the TNGW distribution
Parameter | Mean | S.E | Bias | MSE | |
---|---|---|---|---|---|
50 | 0.1009 | 0.7272 | −0.3991 | 0.6881 | |
1.6416 | 0.8418 | −0.3584 | 0.8371 | ||
1.5098 | 0.9281 | 0.5098 | 1.1213 | ||
0.4551 | 0.3787 | −0.0449 | 0.1454 | ||
0.4438 | 0.6967 | −0.0562 | 0.4885 | ||
100 | 0.6365 | 0.8028 | 0.1365 | 0.6631 | |
1.7085 | 0.5387 | −0.2915 | 0.3752 | ||
0.7689 | 0.7212 | −0.2311 | 0.5735 | ||
0.6048 | 0.1507 | 0.1048 | 0.0337 | ||
0.7595 | 0.3492 | 0.2595 | 0.1893 | ||
200 | 0.6783 | 0.8762 | 0.1783 | 0.7995 | |
1.7260 | 0.5125 | −0.2740 | 0.3377 | ||
1.3868 | 1.4812 | 0.3868 | 2.3435 | ||
0.4933 | 0.0795 | −0.0067 | 0.0064 | ||
0.0113 | 0.7808 | −0.4887 | 0.8485 | ||
400 | 0.4748 | 0.2600 | −0.0252 | 0.0682 | |
2.1514 | 0.2564 | 0.1514 | 0.0886 | ||
0.9169 | 0.2982 | −0.0831 | 0.0958 | ||
0.5194 | 0.0552 | 0.0194 | 0.0034 | ||
0.6142 | 0.2851 | 0.1142 | 0.0943 | ||
800 | 0.5961 | 0.2093 | 0.0961 | 0.0530 | |
2.3019 | 0.2544 | 0.3019 | 0.1558 | ||
0.8400 | 0.2618 | −0.1600 | 0.0941 | ||
0.5490 | 0.0383 | 0.0490 | 0.0038 | ||
0.6526 | 0.2559 | 0.1526 | 0.0887 |
MLE = maximum likelihood estimate; MSE = mean square error.
Estimates of the model parameters for the nicotine in cigarettes data
Distribution | Estimates | ||||
---|---|---|---|---|---|
GPW( | 2.6135 (0.1054) | 1.3514 (0.0422) | |||
EW( | 0.8851 (0.1742) | 1.4647 (0.2082) | 3.0386 (0.3637) | ||
NGW( | 1.6403 (0.5489) | 0.4030 (0.4101) | 1.6472 (0.1628) | 2.7252 (0.2035) | |
TAW( | 1.4979 (0.1349) | 1.1698 (0.6121) | 0.0762 (0.1016) | 3.0361 (0.1868) | 0.0113 (0.0252) |
KwWP( | 0.7905 (0.1439) | 0.2064 (0.0283) | 3.0436 (0.0228) | 0.0215 (0.4799) | 1.9227 (0.0396) |
TNGW( | 1.5793 (0.5532) | 0.3098 (0.3432) | 1.2887 (0.2270) | 2.9588 (0.2259) | 0.4920 (0.2617) |
Goodness-of-fit statistics for the nicotine in cigarettes data
Distribution | AIC | CAIC | K-S test [ | ||
---|---|---|---|---|---|
GPW( | 142.48 | 142.51 | 4.0559 | 0.7726 | 0.1243 [9.54E−6] |
EW( | 143.70 | 143.76 | 3.9705 | 0.7510 | 0.1194 [2.47E−5] |
NGW( | 140.39 | 140.49 | 3.6479 | 0.6904 | 0.1213 [1.71E−5] |
TAW( | 144.38 | 144.53 | 3.8011 | 0.7244 | 0.1249 [8.50E−6] |
KwWP( | 146.56 | 146.71 | 3.9465 | 0.7538 | 0.1250 [8.43E−6] |
TNGW( | 140.36 | 140.47 | 3.4984 | 0.6577 | 0.1176 [3.45E−5] |
AIC = Akaike information criterion, CAIC = consistent Akaike information criterion,