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Transmuted new generalized Weibull distribution for lifetime modeling
Communications for Statistical Applications and Methods 2016;23:363-383
Published online September 30, 2016
© 2016 Korean Statistical Society.

Muhammad Shuaib Khan1,a, Robert Kinga, and Irene Lena Hudsona

aSchool of Mathematical and Physical Sciences, The University of Newcastle, Australia
Correspondence to: School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia. E-mail: Shuaib.stat@gmail.com
Received April 10, 2016; Revised July 10, 2016; Accepted September 9, 2016.
 Abstract

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.

Keywords : new generalized Weibull distribution, moment estimation, entropies, order statistics, maximum likelihood estimation
1. Introduction

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 et al. (2002). Nadarajah and Kotz (2006) introduced four exponentiated type distributions: exponentiated gamma, exponentiated Weibull, exponentiated Gumbel, and exponentiated Fréchet distributions. Sarhan and Zaindin (2009) introduced modified Weibull distribution. Cordeiro et al. (2013) introduced beta exponentiated Weibull distribution and studied various statistical properties with applications. Cordeiro et al. (2014) proposed an increasing five-parameter Kumaraswamy modified Weibull distribution, decreasing bathtub shaped and unimodal hazard rate functions and various statistical properties with applications. A beta modified Weibull distribution was introduced by Silva et al. (2010), which includes seventeen distributions as special cases and the study of various structural properties with application. Recently Zaindin and Sarhan (2011) proposed a new generalized Weibull (NGW) distribution. The cumulative distribution function (cdf) of the NGW distribution is given by (for x > 0)

G(x)=[1-exp {-αx-ηxθ}]φ,

where θ, φ > 0 are the shape parameters and α, η > 0 are the scale parameters. The probability density function corresponding to (1.1) is given by

g(x)=φ(α+ηθxθ-1)exp {-αx-ηxθ}[1-exp {-αx-ηxθ}]φ-1.

The four-parameter NGW distribution generalizes the eleven lifetime distributions with an increasing and decreasing bathtub shaped hazard rate function.

Let F1 and F2 be the cdfs of two lifetime distributions with a common sample space, then we can define the pair of general rank transmutation proposed by Shaw and Buckley (2009) as GR12(u)=F2(F1-1(u)) and GR21(u)=F1(F2-1(u)). The functions GR12 (u) and GR21 (u) both map the unit interval I = [0, 1] into itself, and are mutual inverses under suitable assumptions. Naturally, they satisfy GRij (0) = 0 and GRij (1) = 1 (for i, j = 1, 2). The quadratic rank transmutation map (QRTM) is defined by GR12 (u) = u + λu(1 − u), |λ| ≤ 1 from which it follows F2(x) = (1 + λ) F1(x) − λ F1(x)2 and the corresponding probability density function (pdf) is f2(x) = f1(x){(1 + λ) − 2λ F1(x)}, where f1(x) and f2(x) are the pdfs corresponding to the cdfs of F1(x) and F2(x) respectively. For more details about the QRTM approach and some general results, see Shaw and Buckley (2009), Bourguignon et al. (2016).

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 et al. (2014) studied the flexibility of transmuted inverse Weibull distributions and studied various structural properties with an application to survival data. Khan and King (2015) recently proposed the transmuted modified inverse Rayleigh distribution and formulated some of its properties with application to real data. Elbatal and Aryal (2013) proposed the transmuted additive Weibull distribution with application to reliability data. Merovci (2013) proposed the transmuted Rayleigh distribution. Tian et al. (2014) introduced and studied the transmuted linear exponential distribution with application to reliability data. Sharma et al. (2014) have also proposed a transmuted inverse Rayleigh distribution with application to survival data that represents the primary motivation for studying the transmuted new generalized Weibull (TNGW) distribution for modeling survival data and also investigate the shapes, skewness, kurtosis and tail variations using simulation. The TNGW distribution is also called the transmuted exponentiated modified Weibull distribution recently proposed by Ashour and Eltehiwy (2013). We can obtain the TNGW distribution using the QRTM technique by taking G(x) to be the cdf of the NGW distribution as the baseline model. This research investigates the potential usefulness of the TNGW distribution by adding the transmuted parameter λ that offers more flexibility to model lifetime data. We study several mathematical properties of the five-parameter TNGW distribution with application to nicotine in cigarettes data.

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 (MLEs) of the unknown parameters are discussed in Section 6. A simulation study is also conducted to examine the bias and mean square error of maximum likelihood estimators in Section 6. Section 7 addresses the potentiality of the TNGW model by means of the nicotine in cigarettes data and associated inferences, followed by concluding remarks.

2. Transmuted new generalized Weibull distribution

A positive random variable X has TNGW distribution with parameters α, θ, η, φ > 0 and the transmuting parameter |λ| ≤ 1, x > 0. Using the QRTM technique, we can obtain five-parameter TNGW distribution defined by

f(x)=φ(α+ηθxθ-1)exp {-αx-ηxθ}v2(x)[1-exp {-αx-ηxθ}]1-φ,vg(x)={1+λ-gλ[1-exp {-αx-ηxθ}]φ},         g=1,2.

The cdf corresponding to (2.1) is given by

F(x)=[1-exp {-αx-ηxθ}]φv1(x).

Here vg(x) is used to simplify the presentation of this density function. The parameter α is the location parameter, the parameter η controls the scale of the distribution, whereas the parameters θ and φ control the shape of the distribution respectively. The parameter λ is a kind of transmuting parameter that provides more flexibility in the TNGW model. The TNGW distribution is the extended form of the NGW distribution and reduces to the base model when the parameter λ = 0. If X is a random variable with density function (2.1), we write X ~ TNGW(x; α, θ, η, φ, λ). The associated reliability and hazard rate functions follow from (2.1) and (2.3) are defined by

R(x)=1-[1-exp {-αx-ηxθ}]φv1(x),

and

h(x)=φ(α+ηθxθ-1)exp {-αx-ηxθ}v2(x)[1-exp {-αx-ηxθ}]1-φ[1-[1-exp {-αx-ηxθ}]φv1(x)].

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.

3. Moments and quantiles

This section presents the quantile analysis, kth moment and moment generating function of the TNGW distribution.

3.1. Quantile and median

The quantile xq of the TNGW distribution is the real solution of the following equation

ηxqθ+αxq+ln {1-((1+λ)-(1+λ)2-4λq2λ)1φ}=0.

The above equation (3.1) has no closed form solution in xq, in general the TNGW model contains twenty-three quantile models as special cases. For the non-linear structure, we can obtain these quantile models by using numerical method such as Newton-Raphson technique. Among twenty-three quantile models, we formulate the mathematical expressions for eleven quantile models for the transmuted distributions in closed form solution as special cases by substituting the parametric values in equation (3.1) as given by

U(φ,λ,q)=ln {1-((1+λ)-(1+λ)2-4λq2λ)1φ}.
  • The qth quantile of the TNGE(x; α, η, φ, λ) by substituting θ = 1

    xq=-1α+ηU(φ,λ,q).

  • The qth quantile of the TNGR(x; α, η, φ, λ) by substituting θ = 2

    xq=-α+α2-4ηU(φ,λ,q)2η.

  • The qth quantile of the TMW(x; α, θ, η, λ) by substituting φ = 1

    ηxqθ+αxq+U(1,λ,q)=0.

  • The qth quantile of the TMR(x; α, η, λ) by substituting φ = 1, θ = 2

    xq=-α+α2-4ηU(1,λ,q)2η.

  • The qth quantile of the TME(x; α, φ, λ) by substituting φ = 1, θ = 1

    xq=-1α+ηU(1,λ,q).

  • The qth quantile of the TGW(x; θ, η, φ, λ) by substituting α = 0

    xq={-1ηU(φ,λ,q)}1θ.

  • The qth quantile of the TGR(x; η, φ, λ) by substituting α = 0, θ = 2

    xq=-1ηU(φ,λ,q).

  • The qth quantile of the TGE(x; η, φ, λ) by substituting α = 0, θ = 1

    xq=-1ηU(φ,λ,q).

  • The qth quantile of the TW(x; θ, η, λ) by substituting α = 0, φ = 1

    xq=(U(1,λ,q)η)1θ.

  • The qth quantile of the TR(x; η, λ) by putting α = 0, φ = 1, θ = 2

    xq=U(1,λ,q)η.

  • The qth quantile of the TE(x; η, λ) by putting α = 0, φ = 1, θ = 1

    xq=U(1,λ,q)η.

By substituting q = 0.5 in (3.1) we obtain the median of the TNGW distribution. Figure 3 shows the median life to illustrate the effect of transmuting parameter λ for some selected choice of parameters. Figure 3 also shows the B-life (or percentile life) of the TNGW distribution as a function of the shape parameter φ. We evaluate the performance of Bowley skewness and Moors kurtosis using the measure based on quantiles as a function of the transmuting parameter λ. Graphical representations of the Bowley skewness and Moors kurtosis show the three phases of these plots and are displayed in Figure 4, respectively.

3.2. Moments

Theorem 1

If X has the TNGW(x; α, θ, η, φ, λ) with |λ| ≤ 1, then the kthmoment of X is

μ´k=(1+λ)m,n=0(φ-1m)ηnφ(-1)m+nn!(m+1)-nψ(α,θ,η,m,n,k)-2λm,n=0(2φ-1m)ηnφ(-1)m+nn!(m+1)-nψ(α,θ,η,m,n,k),

where

ψ(α,θ,η,m,n,k)=αΓ(k+θn+1)(α(m+1))k+θn+1+θηΓ(k+θn+θ)(α(m+1))k+θn+θ.
Proof

The kth moment of the TNGW distribution is as follows

μ´k=0xkφ(α+ηθxθ-1)exp {-αx-ηxθ}v2(x)[1-exp {-αx-ηxθ}]1-φdx

using (2.1) and (2.2) the above integral can be written as

μ´k=(1+λ)0xkφ(α+ηθxθ-1)exp {-αx-ηxθ}[1-exp {-αx-ηxθ}]1-φdx-2λ0xkφ(α+ηθxθ-1)exp {-αx-ηxθ}[1-exp {-αx-ηxθ}]1-2φdx.

Using the binomial expansion the above equation reduces to

μ´k=(1+λ)m=0(φ-1m)φα(-1)m0xkexp {-αx(m+1)-ηxθ(m+1)}dx+(1+λ)m=0(φ-1m)φηθ(-1)m0xk+θ+1exp {-αx(m+1)-ηxθ(m+1)}dx-2λm=0(2φ-1m)φα(-1)m0xkexp {-αx(m+1)-ηxθ(m+1)}dx-2λm=0(2φ-1m)φηθ(-1)m0xk+θ+1exp {-αx(m+1)-ηxθ(m+1)}dx

the above integral yields the following kth moment,

μ´k=(1+λ)m,n=0(φ-1m)ηnφ(-1)m+nn!(m+1)-nψ(α,θ,η,m,n,k)-2λm,n=0(2φ-1m)ηnφ(-1)m+nn!(m+1)-nψ(α,θ,η,m,n,k).

The important features and characterizations of the TNGW distribution can be studied using equation (3.3). The values of the first four ordinary moments for some selected choices of parameters are shown in Table 2. Using ordinary moments, we obtained the mean, variance, coefficient of variation, coefficient of skewness and coefficient of kurtosis that are displayed in Table 3. The results in these tables show that as the shape parameters increases as the values of skewness and kurtosis decreases. These values can be determined numerically by using R and SAS languages.

Theorem 2

If X has the TNGW(x; α, θ, η, φ, λ) with |λ| ≤ 1, then the moment generating function (mgf ) of X is given by

Mx(t)=(1+λ)r=0p,q=0(φ-1p)ηpφtr(-1)p+qq!r!(p+1)-nω(α,θ,η,p,q,r)-2λr=0p,q=0(2φ-1p)ηpφ(-1)p+qq!r!(p+1)-nω(α,θ,η,p,q,r),

where

ω(α,θ,η,p,q,r)=αΓ(r+θq+1)(α(p+1))r+θq+1+θηΓ(r+θq+θ)(α(p+1))r+θq+θ.
Proof

By definition

Mx(t)=(1+λ)0etxφ(α+ηθxθ-1)exp {-αx-ηxθ}[1-exp {-αx-ηxθ}]1-φdx-2λ0etxφ(α+ηθxθ-1)exp {-αx-ηxθ}[1-exp {-αx-ηxθ}]1-2φdx.

Using Taylor series expansions, the above integrals reduce to

Mx(t)=(1+λ)r=0p=0(φ-1p)trφα(-1)pr!0xrexp {-αx(p+1)-ηxθ(p+1)}dx+(1+λ)r=0p=0(φ-1m)trφηθ(-1)pr!0xr+θ+1exp {-αx(p+1)-ηxθ(p+1)}dx-2λr=0p=0(2φ-1p)trφα(-1)pr!0xrexp {-αx(p+1)-ηxθ(p+1)}dx-2λr=0p=0(2φ-1p)trφηθ(-1)pr!0xr+θ+1exp {-αx(p+1)-ηxθ(p+1)}dx

the above integral yields the Mx(t) as

Mx(t)=(1+λ)r=0p,q=0(φ-1p)ηpφtr(-1)p+qq!r!(p+1)-nω(α,θ,η,p,q,r)-2λr=0p,q=0(2φ-1p)ηpφ(-1)p+qq!r!(p+1)-nω(α,θ,η,p,q,r).
4. Entropy and mean deviation

The entropy of a random variable X with probability density from the TNGW(x; α, θ, η, φ, λ) is a measure of variation of the uncertainty. The Rényi entropy approaches the Shannon entropy when ρ → 1. The Rényi (1961) introduced the entropy denoted as, IR(ρ), for X is a measure of variation of uncertainty and is defined as

IR(ρ)=11-ρlog {0f(x)ρdx},

where ρ > 0 and ρ ≠ 1. The integral in IR(ρ) for the TNGW(x; α, θ, η, φ, λ) can be defined by substituting (2.1) and (2.2) in (4.1) as

IR(ρ)=11-ρlog{0φρ(α+ηθxθ-1)ρexp {-αρx-ηpxθ}v2(x)ρ[1-exp {-αx-ηxθ}]ρ(1-φ)dx},

the above integral reduces to

IR(ρ)=11-ρlog {i,j,k=0zφ,ρ,λ,i,j0xk(θ-1)exp {-αx(ρ+j)-ηxθ(p+j)}dx},

where

zφ,ρ,λ,i,j=φρ(ρi)(ρk)(ρ(φ-1)+φij)(ηθα)k(2λ1+λ)i(-1)i+j(1+λ)ραρ.

Finally, we obtain the TNGW Rényi entropy as

IR(ρ)=ρ1-ρlog α+ρ1-ρlog φ+ρ1-ρlog(1+λ)+11-ρlog {i,j=0k,m=0(ρi)(ρk)(ρ(φ-1)+φij)(2λ1+λ)i(-1)i+j+mm!Vj,k,m},

where

Vj,k,m=ηm(ρ+j)mα(ρ+j)k(θ-1)-θm+1(ηθα)kΓ(k(θ-1)-θm-1).

The β-or (q-entropy) was introduced by Havrda and Charvát (1967), and is defined as

IH(β)=1β-1{1-0f(x)βdx},

where β > 0 and β ≠ 1. Suppose X has the TNGW distribution then by substituting (2.1) and (2.2) in (4.2), we obtain

IH(β)=1β-1{1-0φβ(α+ηθxθ-1)βexp {-αβx-ηβxθ}v2(x)β[1-exp {-αx-ηxθ}]β(1-φ)dx},

the above integral yields the TNGW β-entropy as

IH(β)=1β-1{1-i,j=0k,m=0ξφ,β,λ,i,j(-1)mηm(β+j)mm!α(β+j)k(θ-1)-θm+1Γ(k(θ-1)-θm-1)},

where

ξφ,β,λ,i,j=(φα)β(βi)(βk)(β(φ-1)+φij)(ηθα)k(2λ1+λ)i(-1)i+j(1+λ)β.

Table 4 lists the values of Rényi entropy of the TNGW distribution for selected values of the parameters. Table 5 lists the values of β-entropy of the TNGW distribution for selected values of the parameters.

If X has the TNGW distribution, then we can derive the mean deviation about the mean and about the median from the following equations

δ1=0|x-μ|f(x)dx         and         δ2=0|x-M|f(x)dx.

The mean μ is given in equation (3.3) and the median M is obtained from equation (3.1). These measures are calculated using the relationships:

δ1=2[μF(μ)-ψ(μ)]         and         δ2=μ-2ψ(M).

The quantity ψ(q) is used to determine the Bonferroni and Lorenz curves, which are very useful in econometrics and finance, reliability and survival analysis, demography, insurance and medical sciences. For a given probability p, they can be constructed from (3.1) and B(P) = ψ(q)/Pμ and L(P) = ψ(q).

ψ(q)=(1+λ)i,j=0(φ-1i)ηjφ(-1)i(i+1)jj!(α(i+1))θj+2Uα,θ,η,i,j-2λi,j=0(2φ-1i)ηjφ(-1)i(i+1)jj!(α(i+1))θj+2Uα,θ,η,i,j,

where

Uα,θ,η,i,j=αγ(θj+2,αq(i+1))+θη[α(i+1)]θ-1γ(θj+θ+1,αq(i+1)).
5. Order statistics

The density of the rth order statistic X(r) of a random sample drawn from the TNGW distribution with |λ| ≤ 1, with the density function of X(r) is given by

fr:n(x)=(F(x))r-1(1-F(x))n-rf(x)B(r,n-r+1),         x>0,

where B(r, nr + 1) is the Beta function and it is a normalizing constant, by substituting (2.1) and (2.3) in (5.1), we obtain

fr:n(x)=n(n-1r-1)i=0n-r(n-ri)(-1)iVr:i(x),

where

Vr:i(x)=φ(α+ηθxθ-1)exp {-αx-ηxθ}v1(x)r+i-1v2(x)[1-exp {-αx-ηxθ}]1-φ(r+i).

The pdf of rth order statistics of the TNGW distribution reduces to the combining terms as

fr:n(x)=n(n-1r-1)i=0n-rj,k=0(-1)i+j+kUi,j,kzk(x),

where

Ui,j,k=(n-ri)(r+i-1j)(φ(r+i+j)-1k)(1+λ)r+i-1(λ1+λ)j,

and

zk(x)=φ(α+ηθxθ-1)exp {-αx(k+1)-ηxθ(k+1)}v2(x).

Using (5.2), the Sth moment of the rth order statistics X(r) is given by

μsn:r=i=0n-rj=0ψi,j{(1+λ)k,m=0(-1)k+mηmτk,mci,j,k,0(k+1)-mm!-2λk,m=0(-1)k+mηmτk,mci,j,k,1(k+1)-mm!},

where ci,j,k,g=(φ(r+i+j+g)-1k), g = 0, 1

ψi,j=n(n-1r-1)(n-ri)(r+i-1j)(1+λ)r+i-1(λ1+λ)j,τk,m=αΓ(S+θm+1)(α(k+1))S+θm+1+ηθΓ(S+θm+θ)(α(k+1))S+θm+θ.
6. Maximum likelihood estimation

Consider the random samples x1, x2, . . . , xn consisting of n observations from the TNGW distribution and Θ = (α, θ, η, φ, λ)T be the parameter vector. The log-likelihood function of (2.1) is given by

log L=nlog φ+i=1nlog (α+ηθxiθ-1)-αi=1nxi-ηi=1nxiθ+(φ-1)i=1nlog {1-exp (-αxi-ηxiθ)}+i=1nlog {1+λ-2λ[1-exp {-αxi-ηxiθ}]φ}.

The components of the score function can be obtained by differentiating (6.1) with respect to α, θ, η, φ and λ, then equating it to zero, we obtain

log Lα=i=1n(α+ηθxiθ-1)-1+(φ-1)i=1nxiexp (-αxi-ηxiθ){1-exp (-αxi-ηxiθ)}-i=1nxi-i=1n2λφxi[1-exp {-αxi-ηxiθ}]φ-1exp {-αxi-ηxiθ}{1+λ-2λ[1-exp {-αxi-ηxiθ}]φ},log Lθ=i=1nη(α+ηθxiθ-1)-1xiθ-1(θ(θ-1)xi+1)+η(φ-1)i=1nexp (-αxi-ηxiθ)xiθlog xi{1-exp (-αxi-ηxiθ)}-ηi=1nxiθlog xi-i=1n2ληφ[1-exp {-αxi-ηxiθ}]φ-1exp {-αxi-ηxiθ}xiθlog xi{1+λ-2λ[1-exp {-αxi-ηxiθ}]φ},log Lη=i=1nθxiθ-1(α+ηθxiθ-1)-1+(φ-1)i=1nxiθexp (-αxi-ηxiθ){1-exp (-αxi-ηxiθ)}-i=1nxiθ-i=1n2λφxiθ[1-exp {-αxi-ηxiθ}]φ-1exp {-αxi-ηxiθ}{1+λ-2λ[1-exp {-αxi-ηxiθ}]φ},log Lφ=nφ+i=1n{1-exp (-αxi-ηxiθ)}-2λi=1n[1-exp {-αxi-ηxiθ}]φlog [1-exp {-αxi-ηxiθ}]{1+λ-2λ[1-exp {-αxi-ηxiθ}]φ},

and

log Lλ=i=1n1-2[1-exp {-αxi-ηxiθ}]φ{1+λ-2λ[1-exp {-αxi-ηxiθ}]φ},

respectively. We obtain the analytical expressions for the MLEs of a five-parameter score vector, yields the ML estimators α̂,θ̂, η̂, φ̂ and λ̂ of the TNGW distribution. These parameters can be estimated by using the BFGS method in R package “Adequacy Model” (http://www.r-project.org). We required the observed information matrix for the interval estimation and hypothesis testing. All the second order derivatives exist for the five-parameter TNGW distribution. Thus we have the observed information matrix as

(α^,θ^,η^,φ^,λ^)T~N5{(α,θ,η,φ,λ)T,K(Θ)-1},

where K(Θ)−1 is the variance covariance matrix of the unknown parameters having the components Kϑiϑj = 2 logL/∂ϑiϑj, i, j = 1, 2, 3, 4, 5. The asymptotic multivariate normal N5{0, K(Θ)−1} distribution can be used to construct the confidence intervals for each parameters ϑ. An approximate 100(1 − γ)% asymptotic confidence intervals (ACI) for each parameter α, θ, η, φ and λ can be determined as

ACIr=(ϑ^r-Zγ2-kr,r,ϑ^r+Zγ2-kr,r),

where −kr,r represents the elements of the observed information matrix and Zγ/2 is the upper γth percentile of the standard normal distribution.

We conducted the simulation study to evaluate the performance of MLEs with respect to the sample size for the TNGW distribution. By using equation (3.1), we generated samples from the TNGW distribution for different sizes n = 50, 100, 200, 400, 800 for the fixed choice of the parameters α = 0.5, θ = 2, η = 1, φ = 0.5 and λ = 0.5. In the context of the computational complexities, the quantile model involves non-linear equation and needs to be calculated by some iterative process. The simulation process is repeated 500 times using the BFGS optimization method in R. We fitted the TNGW distribution for these samples using the maximum likelihood method. Table 6 describes the results for five different parameter values α, θ, η, φ, λ with their corresponding standard errors, bias and mean square error (MSE). Table 4 reports that the simulated results are quite promising with respect to the increasing sample size. The histogram of two simulated data sets for some selected values of parameters based on 1,000 observations are displayed in Figure 5.

7. Application

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

    f(x)=abcλβcxc-1[1-exp (-(xβ)c)]a-1[1-(1-exp (-(βx)c))a]b-1×exp [-λ{1-(1-(1-exp (-(βx)c))a)b}-(βx)c](1-exp(-λ)),

    where a, b, c > 0 are the shape parameters and β, λ > 0 are the scale parameters of the KwWP distribution proposed by Ramos et al. (2015).

  • Transmuted additive Weibull (TAW) distribution with pdf

    f(x)=(αθxθ-1+ηβxβ-1)exp (-αxθ-ηxβ){1-λ+2λexp (-αxθ-ηxβ)},

    where α, η > 0 are the scale parameters, β, θ > 0 are the shape parameters and λ is the transmuted parameter of the TAW distribution proposed by Elbatal and Aryal (2013).

  • New generalized Weibull (NGW) distribution with pdf

    f(x)=φ(α+ηθxθ-1)exp {-αx-ηxθ}[1-exp {-αx-ηxθ}]φ-1,

    where α, η > 0 are the scale parameters and θ, φ > 0 are the shape parameter of the NGW distribution proposed by Zaindin and Sarhan (2011).

  • Exponentiated Weibull (EW) distribution with pdf

    f(x)=φηθxθ-1exp (-ηxθ){1-exp (-ηxθ)}φ-1,

    where η > 0 is the scale parameter and φ, θ > 0 are the shape parameters of the EW distribution introduced by Mudholkar and Srivastava (1993).

  • Generalized power Weibull (GPW) distribution with pdf

    f(x)=αβxα-1(1+xα)β-1exp (1-(1+xα)β),

    where α, β > 0 are the shape parameters of the GPW distribution proposed by Nikulin and Haghighi (2006).

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 (A*), the Cramér-von Mises (W*) and the K-S test goodness-of-fit statistics to verify which model fits better. The results suggest that the TNGW distribution has the smallest values of these statistics; therefore, the TNGW model can be chosen as the best model among the six fitted models. Figure 6 shows the density functions with histogram and empirical fitted plots of the six distributions. From the visualization of density functions and Table 8 indicate that the TNGW distribution provides a better fit than the other five distributions. Therefore, the TNGW distribution can be chosen as the best model for nicotine in cigarettes data in terms of model fitting. The estimated fitted survival function and P-P plot of the TNGW model for the nicotine in cigarettes data are displayed in Figure 7. These plots suggest that the TNGW distribution could be chosen as the best model because it does fit better to the P-P plot and survival curve for the nicotine in cigarettes data; therefore, we conclude that the TNGW model has better relationship for the nicotine in cigarettes data.

8. Conclusion

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.

Figures
Fig. 1. Plots of the TNGW pdf for some selected values of the parameters.
Fig. 2. Plots of the TNGW hazard function for some selected values of the parameters.
Fig. 3. Median and B-life of the TNGW distribution.
Fig. 4. Skewness and kurtosis of the TNGW for different values of .
Fig. 5. Plots of the TNGW densities for simulated data sets.
Fig. 6. Estimated densities of the TNGW, KWWP, TAW, NGW, GPW and EW models for the nicotine in cigarettes data.
Fig. 7. Estimated fitted survival function and P-P plot of the TNGW model for the nicotine in cigarettes data.
TABLES

Table 1

Sub-models of the transmuted NGW distribution

 Distribution αθηφλ
TNGE-1---
TNGR-2---
TMW---1-
TMR-2-1-
TME-1-1-
TGW0----
TGR02---
TGE01---
TW0--1-
TR02-1-
TE01-1-
NGW----0
NGR-2--0
NGE-1--0
MW---10
MR-2-10
ME-1-10
GW0---0
GR02--0
GE01--0
W0--10
R02-10
E01-10

T = transmuted; M = modified; N = new; G = generalized; W = Weibull; R = Rayleigh; E = exponential.


Table 2

Moments values of the TNGW distribution

 (α, θ, η, φ) λμ́1μ́2μ́3μ́4
1, 0.5, 1, 1−1.0  0.7365  1.0861  2.4618  7.6323 
−0.5 0.59540.83821.86875.7545
0.5 0.50000.55551.00002.5185
1.0 0.33330.22220.22220.2963

1, 1, 1, 1−1.0 0.75000.87501.40622.9062
 −0.5 0.62500.68751.07812.2031
0.5 0.37500.31250.42180.7968
1.0 0.25000.12500.09370.0938

1, 2, 2, 2−1.0 0.76560.65880.62630.6499
−0.5 0.68460.55350.50660.5129
0.5 0.52240.34290.26710.2391
1.0 0.44140.23760.14740.1022

2, 3, 2, 3−1.0 0.73310.58100.49180.4409
−0.5 0.66500.49750.40570.3546
0.5 0.52880.33050.23360.1821
1.0 0.46080.24700.14750.0957

Table 3

Moments based measures of the TNGW distribution

 (α, θ, η, φ) λMeanVarCVCSCK
1, 0.5, 1, 1 −1.0  0.7365  0.5436  1.0011  2.1481  10.258 
−0.5 0.59540.48361.16812.359211.583
0.5 0.50000.30551.10542.468112.475
1.0 0.33330.11111.00012.00009.0027

1, 1, 1, 1−1.0 0.75000.31250.74531.60967.0810
−0.5 0.62500.29680.87171.71447.5045
0.5 0.37500.17181.10552.465812.472
1.0 0.25000.06251.00001.99689.0256

1, 2, 2, 2−1.0 0.76560.07260.35210.54483.4351
−0.5 0.68460.08480.42540.46693.2145
0.5 0.52240.06990.50650.80103.8797
1.0 0.44140.04270.46850.53923.1849

2, 3, 2, 3−1.0 0.73310.04350.28470.21943.0218
−0.5 0.66500.05520.35350.10362.8772
0.5 0.52880.05080.42650.43843.0592
1.0 0.46080.03460.40400.26912.7051

CV = coefficient of variation; CS = coefficient of skewness; CK = coefficient of kurtosis.


Table 4

Rényi entropy values for the TNGW distribution

 (α, θ, η, φ) λρ = 2ρ = 3ρ = 4ρ = 5
1, 1, 1, 1 −1.0  0.2891  0.2627  0.3698  0.4703 
−0.5 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 −0.0321 −0.0636 −0.0827 −0.0958 
−0.5 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 −0.1299 −0.1605 −0.1791 −0.1918 
−0.5 −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 −0.2356 −0.2674 −0.2867 −0.2998 
−0.5 −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 

Table 5

β-entropy values for the TNGW distribution

 (α, θ, η, φ) λβ = 2β = 3β = 4β = 5
1, 1, 1, 1 −1.0  0.3333  0.2334  0.1809  0.1484 
 −0.5 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 −0.0767 −0.1703 −0.2571 −0.3545 
−0.5 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 −0.3488 −0.5468 −0.8150 −1.2134 
−0.5 −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 −0.7204 −1.2129 −2.0811 −3.7062 
−0.5 −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 

Table 6

Monte Carlo simulation results based on MLE, S.E, Bias and MSE for the TNGW distribution

n Parameter  Mean S.E  Bias  MSE 
 50 α0.10090.7272 −0.3991 0.6881
θ1.64160.8418 −0.3584 0.8371
η1.50980.92810.5098 1.1213
φ0.45510.3787−0.0449 0.1454
λ0.44380.6967−0.0562 0.4885

100α0.63650.80280.1365 0.6631
θ1.70850.5387−0.2915 0.3752
η0.76890.7212−0.2311 0.5735
φ0.60480.15070.1048 0.0337
λ0.75950.34920.2595 0.1893

200α0.67830.87620.1783 0.7995
θ1.72600.5125−0.2740 0.3377
η1.38681.48120.3868 2.3435
φ0.49330.0795−0.0067 0.0064
λ0.01130.7808−0.4887 0.8485

400α0.47480.2600−0.0252 0.0682
θ2.15140.25640.1514 0.0886
η0.91690.2982−0.0831 0.0958
φ0.51940.05520.0194 0.0034
λ0.61420.28510.1142 0.0943

800α0.59610.20930.0961 0.0530
θ2.30190.25440.3019 0.1558
η0.84000.2618−0.1600 0.0941
φ0.54900.03830.0490 0.0038
λ0.65260.25590.1526 0.0887

MLE = maximum likelihood estimate; MSE = mean square error.


Table 7

Estimates of the model parameters for the nicotine in cigarettes data

DistributionEstimates
GPW(α, β)2.6135 (0.1054)1.3514 (0.0422)
EW(φ, η, θ)0.8851 (0.1742)1.4647 (0.2082)3.0386 (0.3637)
NGW(φ, α, η, θ)1.6403 (0.5489)0.4030 (0.4101)1.6472 (0.1628)2.7252 (0.2035)
TAW(α, β, η, θ, λ)1.4979 (0.1349)1.1698 (0.6121)0.0762 (0.1016)3.0361 (0.1868)0.0113 (0.0252)
KwWP(a, b, c, λ, β)0.7905 (0.1439)0.2064 (0.0283)3.0436 (0.0228)0.0215 (0.4799)1.9227 (0.0396)
TNGW(φ, α, η, θ, λ) 1.5793 (0.5532) 0.3098 (0.3432) 1.2887 (0.2270) 2.9588 (0.2259) 0.4920 (0.2617)

Table 8

Goodness-of-fit statistics for the nicotine in cigarettes data

DistributionAICCAICA*W* K-S test [p-value] 
GPW(α, β) 142.48  142.51  4.0559  0.7726  0.1243 [9.54E−6] 
EW(φ, η, θ)143.70143.763.97050.75100.1194 [2.47E−5]
NGW(φ, α, η, θ)140.39140.493.64790.69040.1213 [1.71E−5]
TAW(α, β, η, θ, λ)144.38144.533.80110.72440.1249 [8.50E−6]
KwWP(a, b, c, λ, β)146.56146.713.94650.75380.1250 [8.43E−6]
TNGW(φ, α, η, θ, λ) 140.36140.473.49840.65770.1176 [3.45E−5]

AIC = Akaike information criterion, CAIC = consistent Akaike information criterion, A* = Anderson-Darling, W* = Cramér-von Mises.


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