A new class of probability distributions via cosine and

Communications in Statistics - Simulation and Computation

ISSN: 0361-0918 (Print) 1532-4141 (Online) Journal homepage: http://www.tandfonline.com/loi/lssp20

A new class of probability distributions via cosine and sine functions with applications Christophe Chesneau, Hassan S. Bakouch & Tassaddaq Hussain To cite this article: Christophe Chesneau, Hassan S. Bakouch & Tassaddaq Hussain (2018): A new class of probability distributions via cosine and sine functions with applications, Communications in Statistics - Simulation and Computation, DOI: 10.1080/03610918.2018.1440303 To link to this article: https://doi.org/10.1080/03610918.2018.1440303

Published online: 23 Feb 2018.

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COMMUNICATIONS IN STATISTICS—SIMULATION AND COMPUTATION , VOL. , NO. , – https://doi.org/./..

A new class of probability distributions via cosine and sine functions with applications Christophe Chesneaua , Hassan S. Bakouchb , and Tassaddaq Hussainc a LMNO, University of Caen, France; b Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt; c Mirpur University of Science and Technology (MUST), Mirpur, (AJK), Pakistan

ABSTRACT

ARTICLE HISTORY

In this paper, we introduce a new class of (probability) distributions, based on a cosine-sine transformation, obtained by compounding a baseline distribution with cosine and sine functions. Some of its properties are explored. A special focus is given to a particular cosine-sine transformation using the exponential distribution as baseline. Estimations of parameters of a particular cosine-sine exponential distribution are performed via the maximum likelihood estimation method. A simulation study investigates the performances of these estimates. Applications are given for four real data sets, showing a better fit in comparison to some existing distributions based on some goodness-of-fit tests.

Received  December  Accepted  February  KEYWORDS

Cosine-sine transformation; Estimation; Goodness-of-ﬁt statistics; Hazard rate function; Statistical distributions MATHEMATICS SUBJECT CLASSIFICATION

 MSC; E; E; F

1. Introduction The statistical literature contains a plethora of (probability) distributions for modeling different real life random phenomenon in several areas such as engineering, actuarial, medical sciences, demography, economics, finance and insurance. Since no particular distribution is appropriate for modeling every phenomenon, the list of new distributions with a high degree of flexibility is growing every year. Some of the recent univariate continuous distributions can be found in Hamedani (2016), Hamedani and Safavimanesh (2017), Hamedani (2017) and Tahir and Cordeiro (2016). As a matter of fact, the statistical literature has a lack of distributions which are based on trigonometric functions; most of them are based on algebraic functions. Among the distributions using trigonometric distributions, let us mention the von Mises distribution (see Evans, Hastings, and Peacock 2000, ”von Mises Distribution” Ch. 41, pp., 189-191), the distributions introduced by Fisher (1993) to analyze circular data, the beta-type distributions using some trigonometric functions proposed by Nadarajah and Kotz (2006), the circular Cauchy distribution introduced by Kent and Tyler (1988) and the sine square distribution explored by Al-Faris and Khan (2008). However, the increased interest of data analysis, and directional data analysis in particular, motivates the development of new approaches. The one explored by Kumar, Singh, and Singh (2015b) provides a modern alternative. They introduced a new class of distributions obtained by compounding a baseline distribution with the sine function called the SS transformation. For any cumulative distribution function (cdf) F (x), it is defined by G(x) = sin( π2 F (x)). Taking F (x) as the cdf of an exponential distribution, Kumar, Singh, and Singh (2015b) proposes a new distribution CONTACT Christophe Chesneau [email protected] LMNO, University of Caen, France. Color versions of one or more of the ﬁgures in the article can be found online at www.tandfonline.com/lssp. ©  Taylor & Francis Group, LLC

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to solve a parametric estimation problem inherent to bladder cancer patients data, and it is shown that it has a better fit compared to several well-known distributions. Based on the motivations above, we propose a generalization of the SS transformation, using both cosine and sine functions, which can give many of new trigonometric distributions, as shall be pointed out. Several tuning parameters are introduced to increase the degree of flexibility of the original SS transformation. We exhibit the associated probability density function (pdf). By considering two different sets of parameters, we introduce two particular CS transformations, called the CS1 and CS2 transformations. New distributions are derived by considering some well-known distributions. For the case using the exponential distribution, we discuss a parametric estimation via the maximum likelihood estimation method. A simulation study is performed to evaluate the performances of these estimates using different sample sizes and two criteria: the average bias and the average mean square error. Then we show the applicability of a specific cosine-sine exponential distribution by considering four data sets. They are fitted to our distribution and other recent distributions based on some wellknown goodness-of-fit statistics. Our distribution demonstrates a better flexibility among the compared distributions. The rest of the paper is organized as follows: Section 2 presents the general form of our cosine-sine transformation. New distributions are derived in Sections 3 and 4 using particular parametrization. In Section 5, the performance of the estimates of the parameters of two types of cosine-sine exponential distributions is investigated using a simulation study. In Section 6, four real data sets are fitted to a specific cosine-sine exponential distribution, with comparison to other recent distributions based on some goodness-of-fit statistics.

2. CS transformation In the present study, using a cdf F (x), we introduce the following cosine-sine transformation: (α + γ ) sin π2 F (x) , G(x) = α + β cos π2 F (x) + γ sin π2 F (x) + θ cos π2 F (x) sin π2 F (x)

x ∈ R,

(1)

where α ≥ 0, β ≥ 0, γ ≥ 0, θ ≥ 0 are parameters satisfying α + γ > 0 and α + β > 0. Remark that the last term in the denominator can be expressed as : cos( π2 F (x)) sin( π2 F (x)) = 12 sin(πF (x)). We will call the transformation (1) as the CS transformation for frequently used purposes. Also, note that the function G(x) given by (1) is a proper cdf and the proof of this is available upon request from the authors. Note that, taking α > 0, β = 0, γ = 0 and θ = 0, G(x) becomes the SS transformation of Kumar, Singh, and Singh (2015b). Further specific choices of the parameters, yielding new (α+γ ) F (x), when F (x) → 0, showing cdfs, which will be studied later. Observe that G(x) ∼ π2(α+β ) a first look to the flexibility of the CS distribution. The pdf associated to G(x) is given by 3 f (x) π (α + γ ) β + α cos π2 F (x) + θ sin π2 F (x) g(x) = x ∈ R. π π π 2 , 2 α + β cos 2 F (x) + γ sin 2 F (x) + θ cos 2 F (x) sin π2 F (x) From (1) and existing transformations, one can construct a wide variety of distributions. For instance, from a cdf H(x), one can use the cdf F (x) into (1) using r the dilation transformation : F (x) = H(αx), α > 0, r the power transformation introduced by Gupta and Kundu (1999) : F (x) = (H(x))α , α > 0,

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COMMUNICATIONS IN STATISTICS—SIMULATION AND COMPUTATION

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r the DUS transformation introduced by Kumar, Singh, and Singh (2015b) : F (x) = 1 (eH(x) e−1

− 1),

r the quadratic rank transmutation map (QRTM) introduced by Shaw and Buckley (2009), yielding the transformation : F (x) = (1 + λ)H(x) − λ(H(x))2 , λ ∈ [−1, 1]. All these possibilities give new families of distributions using cosine and sine functions, with a potential interest in probability and statistics. In the rest of the study, we focus our attention on two special cases of our CS transformation, both using a simplified parametrization for (α, β, γ , θ ).

3. The CS1 transformation An interesting special case of our CS transformation is with the parametrization α > 0, β = 0, γ = 0 and θ ≥ 0. Hence the cdf G(x) given by (1) only depends on the parameters α and θ. Using trigonometric formulas, we have 2α sin π2 F (x) α sin π2 F (x) = , x ∈ R. (2) G(x) = 2α + θ sin (πF (x)) α + θ cos π2 F (x) sin π2 F (x) We have more flexibility in our transformation due to the additional parameter θ, opening new perspective in parametric estimation for instance. We will call the transformation (2) as the CS1 transformation for frequently used purposes. Using again trigonometric formulas, the associated pdf is 3 f (x) 2πα α cos π2 F (x) + θ sin π2 F (x) , x ∈ R. (3) g(x) = (2α + θ sin (πF (x)))2 Table 1 presents new cdfs characterizing new distributions arising from the CS1 transformation by considering well-known cdfs for F (x). Proposition 1 below presents a decomposition of the expectation of a function of a random variable having the cdf given by (2). Proposition 1. Let X be a random variable having the cdf given by (2) and d(x) be a function on R. Then we have E(d(X )) =

+∞ k

bk, u ,

k=0 =0

where bk, and

u =

+∞ −∞

θk k = 2πα (k + 1) (−1) (2α + θ )k+2

π 3 π F (x) + θ sin F (x) α cos (sin (πF (x))) d(x) f (x)dx, 2 2

provided that u exist. The proof of Proposition 1 is given in the Appendix. Naturally, E(d(X )) can be useful to determine moments (by taking d(x) = xr , r ≥ 0), characteristic function (by taking d(x) = eitx , t ∈ R) etc.

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Table . Some cdfs using the CS transformation. cdf F (x)

Distribution Uniform(a, b)

x−a b−a 1[a,b] (x) + 1(b,+∞) (x)

Triangular(a)

(1 −

cdf G(x) of the CS transformation (x−a) 2α sin( π2(b−a) ) 1[a,b] (x) + 1(b,+∞) (x) 2α+θ sin( π (x−a) b−a ) 2

(x−a)2 a2

)1[0,a] (x) + 1(a,+∞) (x)

(1 − e− λ )1[0,+∞) (x)

Rayleigh(σ 2 )

(1 − e

2 − x2 2σ

2α+θ

2a 2 sin( π (x−a) a2

)

1[0,a] (x) + 1(a,+∞) (x)

−x

2α cos( π2 e λ ) 1[0,+∞) (x) −x 2α+θ sin(π e λ )

x

Exponential(λ)

2α cos( π (x−a) ) 2

2 − x

2α cos( π2 e 2σ 2 )

)1[0,+∞) (x)

1[0,+∞) (x)

2 − x

2α+θ sin(π e 2σ 2 )

Normal(μ, σ 2 )

(x) =

Gumbel(μ, β )

e−e

x

−∞

2

− (t−μ) √ 1 e 2σ 2 2π σ 2

dt

μ−x β

2α sin( π2 (x)) 2α+θ sin(π (x)) 2α sin( π2 e−e 2α+θ

μ−x β

)

μ−x β sin(π e−e

)

π x−μ ) 2(1+e− s ) π sin( x−μ ) 1+e− s

2α sin(

Logistic(μ, s)

1

x−μ

2α+θ

1+e− s

Cauchy(x0 , a)

1 π

Arcsine

2 π

Pareto(xm , k)

(1 − (

arctan(

x−x0 a

√

)+

1 2

√ arcsin( x)1[0,1] (x) + 1]1,+∞) (x) xm k x ) )1[xm ,+∞) (x)

x−x

x−x

2α(sin( 21 arctan( a 0 ))+cos( 21 arctan( a 0 ))) x−x 2α+θ cos(arctan( a 0 ))

√ α √ x 1 (x) + 1]1,+∞) (x) α+θ x(1−x) [0,1] x 2α cos( π2 ( xm )k )

1 (x) x 2α+θ sin(π ( xm )k ) [xm ,+∞)

1 Note that u can be expressed as u = 0 (α cos( π2 x) + θ (sin( π2 x))3 )(sin(πx)) d(F −1 (x)) dx. The complexity in the calculus of u mainly depends on the nature of d(x), F (x) and f (x). Introduction of the CS1E (α, θ, λ) distribution. Let us now focus our attention on the CS1 transformation in the case where F (x) is the cdf of the exponential distribution with parameter λ > 0. We call this distribution as the CS1E (α, θ, λ) distribution. As noted in Table 1, using trigonometric formulas, the associated cdf is π x x 1 − e− λ 2α cos π2 e− λ 2 G(x) = x 1[0,+∞) (x) = x 1[0,+∞) (x). 2α + θ sin π 1 − e− λ 2α + θ sin πe− λ 2α sin

(4)

Note that, taking θ = 0 and λ = ν1 , we obtain the SSE (ν ) distribution introduced by Kumar, Singh, and Singh (2015b). Using the same mathematical arguments, the associated pdf is given by x x x 3 e− λ 2πα α sin π2 e− λ + θ cos π2 e− λ 1[0,+∞) (x). g(x) = x 2 λ 2α + θ sin πe− λ

(5)

To show the flexibility of this distribution, we provide some graphs. Some associated pdfs are presented in Figure 1, where various shapes are observed and showing symmetries and skewness. Applicability of the CS1E (α, θ, λ) distribution will be explored in Section 6.

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Figure . Some pdfs g(x) = g(x, α, θ , λ) () associated to the distribution CS1E (α, θ , λ).

4. The CS2 transformation Let us now consider another particular case of the CS transformation. Taking α = 0, β > 0, γ > 0 and θ = 0, G(x) becomes the following function : γ sin π2 F (x) , G(x) = β cos π2 F (x) + γ sin π2 F (x)

x ∈ R.

(6)

In this case, observe that the associated pdf has the form: g(x) =

πγ β f (x) π 2 , 2 β cos 2 F (x) + γ sin π2 F (x)

x ∈ R.

(7)

Table 2 presents new cdfs characterizing new distributions arising from the CS2 transformation by considering well-known cdfs for F (x). Introduction of the CS2E (β, γ, λ) distribution. Let us now focus our attention on the CS2 transformation in the case where F (x) is the cdf of the exponential distribution with x parameter λ > 0 : F (x) = (1 − e− λ )1[0,+∞) (x). We call this distribution as the CS2E (β, γ , λ) distribution for frequently used purposed. As noted in Table 2, using trigonometric formula, we obtain π x 1 − e− λ 2 G(x) = x x 1[0,+∞) (x) β cos π2 1 − e− λ + γ sin π2 1 − e− λ x γ cos π2 e− λ = x x 1[0,+∞) (x). β sin π2 e− λ + γ cos π2 e− λ γ sin

(8)

Similar mathematical arguments yields the associated pdf: x

πγ βe− λ g(x) = 1 (x). π −x x 2 [0,+∞) 2λ β sin 2 e λ + γ cos π2 e− λ

(9)

In order to illustrate the nature of the CS2E (β, γ , λ) distribution, graph of some pdfs are presented in Figure 2 which shows various shapes of those functions.

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Table . Some cdfs using the CS transformation. cdf F (x)

Distribution Uniform(a, b)

x−a b−a 1[a,b] (x) + 1(b,+∞) (x)

Triangular(a)

(1 −

cdf G(x) of the CS transformation (x−a) γ sin( π2(b−a) ) (x−a) (x−a) 1[a,b] (x) + 1(b,+∞) (x) β cos( π2(b−a) )+γ sin( π2(b−a) ) 2

(x−a)2 a2

)1[0,a] (x) + 1(a,+∞) (x)

γ cos( π (x−a) ) 2 2 β sin( π (x−a) 2a2

(1 − e− λ )1[0,+∞) (x)

Rayleigh(σ 2 )

(1 − e

Normal(μ, σ 2 )

(x) =

Gumbel(μ, β )

e−e

2 − x2 2σ

−∞

1[0,a] (x) + 1(a,+∞) (x)

2a

x

−x β sin( π2 e λ

1[0,+∞) (x) −x )+γ cos( π2 e λ ) 2 − x

γ cos( π2 e 2σ 2 )

)1[0,+∞) (x)

x

2

)+γ cos( π (x−a) ) 2

− γ cos( π2 e λ )

x

Exponential(λ)

2a

2 − x

2 − x

β sin( π2 e 2σ 2 )+γ cos( π2 e 2σ 2 ) 2

− (t−μ) √ 1 e 2σ 2 2π σ 2

dt

μ−x β

1[0,+∞) (x)

γ sin( π2 (x)) β cos( π2 (x))+γ sin( π2 (x)) γ sin( π2 e−e

μ−x β β cos( π2 e−e

μ−x β

)

)+γ sin( π2 e−e

μ−x β

)

π x−μ ) 2(1+e− s ) π π )+γ sin( x−μ x−μ ) 2(1+e− s ) 2(1+e− s )

γ sin(

Logistic(μ, s)

1

β cos(

x−μ 1+e− s

Cauchy(x0 , a)

1 π

Arcsine

2 π

Pareto(xm , k)

(1 − (

arctan(

x−x0 a

x−x

)+

1 2

√ arcsin( x)1[0,1] (x) + 1]1,+∞) (x) xm k x ) )1[xm ,+∞) (x)

x−x

γ (sin( 21 arctan( a 0 ))+cos( 21 arctan( a 0 ))) x−x x−x (γ −β ) sin( 21 arctan( a 0 ))+(γ +β ) cos( 21 arctan( a 0 )) √ √ γ x √ 1 (x) + 1]1,+∞) (x) β 1−x+γ x [0,1] x

γ cos( π2 ( xm )k ) 1 (x) x x β sin( π2 ( xm )k )+γ cos( π2 ( xm )k ) [xm ,+∞)

5. Maximum likelihood estimation and simulation 5.1. Maximum likelihood estimation Let X1 , X2 , . . . , Xn be a random sample from the CS1E (α, θ, λ) distribution with parameter vector = (α, θ, λ) and x1 , x2 , . . . , xn be the observed values, then the joint probability

Figure . Some pdfs g(x) = g(x, β, γ , λ) () associated to the distribution CS2E (β, γ , λ).

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function of X1 , X2 , . . . , Xn as a log-likelihood function can be expressed as n n π xi 3 1 π xi 2πα ln α sin xi − + e− λ + θ cos e− λ ( ) = n ln λ 2 2 λ i=1 i=1 −2

n

xi ln 2α + θ sin(πe− λ ) .

i=1

The maximum likelihood estimators of can be obtained by solving the nonlinear nor∂ ( ) = 0. These equations cannot be solved analytically, so they are solved mal equations ∂ numerically. 5.2. Simulation study A general form to generate a random variable X from the CS1E (α, θ, λ) and CS2E (β, γ , λ) distributions is to generate the values x from the proposed models with parameters θ, α, λ and β, γ , λ by using the Mathematica 8.0 computational package. A simulation analysis of the CS1 and CS2 distributions was carried out by generating 1000 samples for each of triplet (θ, α, λ) and (β, γ , λ) with n = 25, 50, 100. The analysis computes the following values: • Average bias of the simulated estimates: 1 ( − ), n i=1 n

ˆ are the MLEs of the CS1E (α, θ, λ) distribution and = (β, ˆ γˆ , λ) ˆ where = (α, ˆ θˆ, λ) are the MLEs of the CS2E (β, γ , λ) distribution, respectively. r Average mean square error (MSE) of the simulated estimates: 1 ( − )2 . n i=1 n

Table 3 concerns the CS1E (α, θ, λ) distribution and it shows the average bias and average MSE of the estimates for different values of α, θ, λ. On the other side, Table 4 portrays the average bias and average MSE of estimates of the CS2E (β, γ , λ) parameters. During the simulation study, it is noticed that the MLE γˆ fluctuates between 0.0009 and 6 while the βˆ varies ˆ remains unchanged. From Tables 3 and 4, we conclude that from 0.25 to 35. However, the λ the bias takes negative and positive signs, and it approaches to value zero for both signs while the MSE decreases as the sample size increases. Table . Average bias and MSE values from simulation of the CS1E (α, θ , λ) distribution. Parameter α = 0.7221, θ = 0.3498, λ = 0.4123 α = 0.5212, θ = 1.2131, λ = 1.5317 α = 1.3247, θ = 3.7904, λ = 2.3683

Sample Size

Bias(α) ˆ

ˆ Bias(θ)

ˆ) Bias(λ

MSE(α) ˆ

ˆ MSE(θ)

ˆ) MSE(λ

n = 25 n = 50 n = 100 n = 25 n = 50 n = 100 n = 25 n = 50 n = 100

−. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −.

. −. −. −. −. −. −. −. −.

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

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C. CHESNEAU ET AL.

Table . Average bias and MSE values from simulation of the CS2E (β, γ , λ) distribution. Parameter

Sample Size

Bias(γˆ )

ˆ Bias(β)

ˆ) Bias(λ

MSE(γˆ )

ˆ MSE(β)

ˆ) MSE(λ

n = 25 n = 50 n = 100 n = 25 n = 50 n = 100 n = 25 n = 50 n = 100

−. −. −. . . . −. −. −.

−. −. −. . . . −. −. −.

−. −. −. . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

β = 0.5096, γ = 1.8465, λ = 0.7846 β = 0.5476, γ = 0.5324, λ = 0.5634 β = 3.6543, γ = 2.4537, λ = 1.6533

6. Evaluation tests and real data examples In statistical literature, a plethora of distributions exist for life testing experiments. Some of them are suitable in increasing/decreasing failure rate scenario, others are appropriate for bathtub and upside down bathtub shapes and some are for both features. In this context, we studied those distributions which are very strong in their respective area, namely transmuted modified inverse Rayleigh (TMIR) defined by Khan and King (2015), transmuted inverse Weibull (TIW) introduced by Khan, King, and Hudson (2014), new modified Weibull (MW) defined by Lai, Xie, and Murthy (2003), generalized linear failure rate (GLFR) defined by Sarhan and Kundu (2009) and single parameter sine transformed distributions proposed by Kumar, Singh, and Singh (2015a). The pdfs of those competing models are expressed in Table 5. 6.1. Evaluation tests In order to demonstrate the proposed methodology, we consider four different real-world data sets, representing various failure rate pattern like increasing, decreasing, and bathtub shape, compared via the Akaike information criterion (AIC), the corrected Akaike information criterion (AICc), the Hannan-Quinn information criterion (HQIC) and the consistent Akaike information criterion (CAIC) which are used to select the best model among several models. It is also worth mentioning that we have not only confine our interest in the information criterion and statistics but also adopted the graphical displays, so that the reader can gain a perspective of the various meanings and associated interpretations. Moreover, in many applications there is a qualitative information about the failure rate shape, which can help in selecting a specified model. A hazard may be considered as a dangerous event that can lead to an emergency or disaster. A hazard analysis can be performed statistically based on the hazard rate (HR), also known as chance function, failure rate, intensity function, or risk Table . The compared distributions to the CS1E distribution. Name TMIR TIW GLFR MW ST

Pdf g(x) = (α +

2θ x

Authors −( αx + θ2 )

)( x1 )2 e

−( αx + θ2 x

x

Domain

Khan and King ()

x > 0, α, θ > 0, |λ| ≤ 1

Khan, King, and Hudson ()

x > 0, α, θ > 0, |λ| ≤ 1

)

{1 + λ − 2λe } θ α g(x) = αθ α x −α−1 e−( x ) {1 + λ θ α − 2λe−( x ) } g(x) = θ (α + λx) 2 2 (1 − e−(αx+0.5λx ) )θ−1 e−(αx+0.5λx ) α eλx α−1 λx−θx g(x) = θ (α + λx)x e g(x) = π2 βe−βx sin( π2 e−βx )

Sarhan and Kundu ()

x > 0, α, θ , λ > 0

Lai, Xie, and Murthy () Kumar, Singh, and Singh (a)

x > 0, α, θ , λ > 0 x > 0, θ > 0

®


9

Figure . Theoretical aspects of TTT plots.

rate, among other names. A nice property of the HR is that it allows us to better characterize the behavior of statistical distributions, and to differentiate models with very similar cdfs. For example, the HR may have several different shapes, such as increasing (IHR), constant, decreasing (DHR), bathtub (BT) and inverse bathtub (IBT). In this regard, a device called the total time on test (TTT) plot is useful see Almalki (2014). Usually the TTT plot is drawn by i yr:n + (n − i)yi:n i against ni , where i = 1, . . . , n and yr:n , r = 1, . . . , n plotting T ( n ) = r=1 n y r:n r=1 are the order statistics of the sample. A TTT curve may be concave (convex) is related to the IHR (DHR) class. A concave (convex) and then convex (concave) TTT curve is related to a BT (IBT) HR. Finally, a TTT curve expressed by a straight line corresponds to the exponential distribution. The TTT plots for the considered data sets are displayed in Figure 3. Therefore, these plots indicate the appropriateness of the CS1E to fit these data, since its TTT plots can present increasing, decreasing, bathtub and upside-down bathtub hazard functions (see Figure 4).

Figure . TTT plots of the selected data sets.

10

C. CHESNEAU ET AL.

Table . Data sets. Data Set I

II III IV

Values ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., ., .. ., ., , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,  , , ,  , , . , , , , , , , , , , , , , , , , , , , , . , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,  , , , , , , , , , , , , , , , , , , , , , , , .

Table . Summary results for data set I. Distribution CS1E GLFRD MW TMIR TIW ST

αˆ

θˆ

ˆ λ

ˆ −( )

KS

A∗0

p − value

. . . . . . . . . . . . . . . . . − . . . . . − . . . — . — . .

. . . . . .

W0∗

AIC

AICc

HQIC

CAIC

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2. Real data examples with statistical analyses In order to check the competency we consider four data sets. Their descriptions and our statistical analyses are given below. Data I: Generally it is observed that the brake pads of vehicles have a nominal lifetime, which is the number of miles or kilometers driven before the pads are reduced to a specified minimum thickness. To study the lifetime distribution, a manufacturer selected a random sample of 98 vehicles sold over the preceding 12 months to a specific group of dealers. Only cars that still had the initial pads were selected. For each car the brake pad lifetime (x) could have then been observed by following the cars prospectively. Table 6 contains the life times of 98 vehicles given by Lawless (2003). Statistical analysis: The TTT plot of this data is given in Figure 4(a). It indicates an increasing failure rate pattern of the failure rate. Moreover, analysis of break pad data shows that the proposed model (CS1E ) is the only suitable model from every aspects of data selection including the increasing hazard function. As we can see in Table 7, the proposed model have not only the minimum values of the test statistics and higher p − value but also have least loss of information by showing least AIC, AICc, HQIC and CAIC. The CDF plots given by Figure 5 also confirm this suitability behavior. Table . Summary results for data set II. Distribution CS1E GLFRD MW TMIR TIW ST

αˆ

θˆ

ˆ λ

. . . . . —

. . . . . .

. . . −. . —

ˆ −( )

KS

. . . . . . . . . . . .

p − value . . . . . .

A∗0

W0∗

. . . . . . . . . . . .

AIC

AICc

. . . . . . . . . . . .

HQIC

CAIC

. . . . . .

. . . . . .

®


11

Figure . CDF plots of distributions for Data I.

Data II: This data set contains the times to failure of 48 devices by Almalki (2014) and reference therein. It can be found in Table 6. Statistical analysis: The data are known to have a bathtub-shaped failure rate as portrayed in TTT plot (see Figure 4(b)). In the analysis of the data the CS1E model again shows a promising behavior with high p − value and lower information criterion values which is portrayed in Table 8. Moreover the CDF plots given by Figure 6 also show that data is adequately modeled by the proposed distribution. Data III: 21 advanced lung cancer patients, taken from a study discussed by Lawless (2003), who were randomly assigned the chemotherapy treatments termed as ”standard”. Survival times t, measured from the start of treatment for each patient, are recorded in Table 6. Statistical analysis: Clearly the TTT plot of the current data set (see Figure 4(c)) reveals decreasing hazard function pattern. Although the proposed distribution again shows minimum Log-Likelihood and higher p − value yet Sine transformed shows AIC and HQIC value a little bit smaller than the proposed distribution which seems to be lager number of parameters plenty for the proposed distribution. But the proposed distribution again achieved the adequate model fitting behavior by showing minimum AICc and CAIC which is portrayed in Table 9. Moreover the CDF plots in Figure 7 also show the appropriateness of the proposed model. Table . Summary results for data set III. Distribution CS1E GLFRD MW TMIR TIW ST

αˆ

θˆ

ˆ λ

. . . . . .×10−6 . . . . . . . . −. — . —

ˆ −( )

KS

. . . . . . . . . . . .

p − value . . . . . .

A∗0

W0∗

AIC

AICc

HQIC

CAIC

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

C. CHESNEAU ET AL.

Figure . CDF plots of distributions for Data II.

Data IV: The data in Table 6 show the number of cycles to failure for a group of 60 electrical appliances in a life test. The failure times have been ordered for convenience. This data set is reported by Lawless (2003). Statistical analysis: The TTT plot in Figure 4(d) indicates that it follows the bathtub shape failure rate pattern. The analysis of the data is shown in Table 10 which clearly shows that proposed model is the most appropriate model for such data set, which is consolidated by CDF plots too (see Figure 8).

Figure . CDF plots of distributions for Data III.

®


13

Figure . CDF plots of distributions for Data IV. Table . Summary results for data set IV.

CS1E GLFRD MW TMIR TIW ST

θˆ

αˆ

Distribution

ˆ λ

ˆ −( )

KS p − value

. . . . . . . .×10−8 . . . . . . . . . . . . . . −. . . — . — . .

. . . . . .

A∗0

W0∗

AIC

AICc

HQIC

CAIC

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Appendix of Proposition 1. Let g(x) be the pdf given by (3). Then we have E(d(X )) =

Proof +∞ d(x)g(x)dx. We can write −∞ π 3 π 2πα 1 g(x) = F (x) + θ sin F (x) , α cos f (x) 2 (2α + θ ) 2 2 (1 − H(x))2 where H(x) = gives

θ 2α+θ

(1 − sin(πF (x))). Since H(x) ∈ (0, 1), the geometric series expansion

2πα g(x) = (2α + θ )2

+∞ π 3 α cos f (x) F (x) + θ sin F (x) (k + 1)[H(x)]k . 2 2 k=0

π

On the other hand, the binomial series expansion gives k θk k [H(x)] = (−1) (sin (πF (x))) . k (2α + θ ) =0 k

By putting these equalities together, we end the proof of Proposition 1.

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Acknowledgements We thank the reviewers for providing comments that helped us to improve the presentation of our work.

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