Marshall-Olkin generalized Erlang-truncated exponential · PDF fileOkorie et al., Cogent...

Okorie et al., Cogent Mathematics (2017), 4: 1285093http://dx.doi.org/10.1080/23311835.2017.1285093

STATISTICS | RESEARCH ARTICLE

Marshall-Olkin generalized Erlang-truncated exponential distribution: Properties and applicationsIdika E. Okorie1*, Anthony C. Akpanta2 and Johnson Ohakwe3

Abstract: This article introduces the Marshall–Olkin generalized Erlang-truncated exponential (MOGETE) distribution as a generalization of the Erlang-truncated exponential (ETE) distribution. The hazard rate of the new distribution could be increasing, decreasing or constant. Explicit-closed form mathematical expressions of some of the statistical and reliability properties of the new distribution were given and the method of maximum likelihood estimation was used to estimate the model parameters. The usefulness and flexibility of the new distribution was illustrated with two real and uncensored lifetime data-sets. The MOGETE distribution with a smaller goodness of fit statistics always emerged as a better candidate for the data-sets than the ETE, Exp Fréchet and Exp Burr XII distributions.

Subjects: Science; Mathematics & Statistics; Physical Sciences

Keywords: Erlang-truncated exponential distribution; Marshall-Olkin; reliability; failure rate; AIC

1. IntroductionThe exponential distribution is about the simplest distribution in terms of expression and analytical tractability and widely used in reliability engineering. There is no doubt that the wide applicability of the exponential distribution even in inappropriate scenarios is motivated by its simplicity. However,

*Corresponding author: Idika E. Okorie, School of Mathematics, University of Manchester, Manchester M13 9PL, UK E-mail: [email protected]

Reviewing editor:Hiroshi Shiraishi, Keio University, Japan

Additional information is available at the end of the article

ABOUT THE AUTHORSIdika E. Okorie obtained his BSc in Statistics (2009) from the Abia State University, Nigeria and MSc in Statistics (Financial Statistics) (2014) from the University of Manchester, UK. He is presently studying for a PhD in Statistics at the University of Manchester, UK. His research interest includes Statistical modelling; and Distribution theory with applications.

Anthony C. Akpanta obtained his PhD in Statistics (2008) from the Abia State University, Nigeria. He is currently working as an associate professor of Statistics at the Department of Statistics, Abia State University, Nigeria. His area of research includes Time series analysis and forecasting; Sampling methods and distribution theory.

Johnson Ohakwe obtained his PhD in Statistics (2009), MSc Statistics (2005) and BSc Statistics (1998) from the Abia State University, Nigeria. He is currently a Senior lecturer at the Department of Mathematics, Computing and Physical sciences, Federal University Otuoke, Nigeria. His research interest include Time series and Forecasting and Distribution theory.

PUBLIC INTEREST STATEMENTThe simple structure of the proposed MOGETE distribution makes it easy to work with analytically and in practical situations, it provides a better fit to data-sets than some of the already existing distributions like the Erlang-truncated exponential distribution, Exponentiated Fréchet distribution and the Exponentiated Burr XII distribution. In addition to the example of possible applications of the new distribution as shown here, MOGETE distribution is suitable for modelling infant mortality rate and failure rate of some devices/equipments due to ageing.

Received: 08 November 2016Accepted: 15 January 2017First Published: 21 January 2017

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the exponential distribution has a major problem of constant failure/hazard rate property which makes it inappropriate for modelling data-sets from various complex life phenomena that may ex-hibit increasing, decreasing or bathtub hazard rate characteristics. El-Alosey (2007) extended the standard one parameter exponential distribution to a two parameter Erlang-truncated exponential (ETE) distribution. The pdf f (x) of the ETE distribution is given by

with cdf F(x)

and hazard rate function (hrf ) h(x)

where � is the shape parameter while � is the scale parameter. It is important to note that the ETE distribution has a constant hazard rate function.

The inability of the existing standard distributions to adequately model a variety of complex real data-sets; particularly, lifetime ones has stirred huge concern amongst distribution users and re-searchers alike and has summoned enormous research attention over the last two decades. Interestingly, tremendous research breakthroughs have been recorded by many researchers in their quest to the solution of the lack of fits limitation of the standard probability distributions. Among others is Marshall and Olkin (1997) who introduced the family of distributions that is known as the Marshall–Olkin extended/generalized distributions. The Marshall–Olkin’s technique of adding an ex-tra parameter to the original distribution has remarkably been known for its ability of producing more flexible and robust distributions that can represent a wide-ranging coverage of data-sets that emanates from a variety of complex phenomena. The Marshall–Olkin family of distributions can be obtained as follows,

It follows that F(x) = 1 − F(x) and

where G(x) and g(x) are the complementary cumulative density function (survival/reliability func-tion) and density function corresponding to the baseline distribution (original distribution).

A lot of standard probability distributions have been generalized by various researchers using the Marshall–Olkin procedure. For example, Ristić and Kundu (2015) introduced the Marshall–Olkin gen-eralized exponential distribution generalizing the exponentiated exponential distribution. Ghitany, Al-Hussaini, and Al-Jarallah (2005) introduced the Marshall–Olkin extended Weibull distribution as a generalization of the standard Weibull distribution. Ghitany (2005) introduced the Marshall–Olkin extended Pareto distribution as a generalization of the standard Pareto distribution. Ristić, Jose, and Ancy (2007) introduced the Marshall–Olkin extended gamma distribution as a generalization of the standard gamma distribution. Ghitany, Al-Awadhi, and Alkhalfan (2007) introduced the Marshall–Olkin extended Lomax distribution as a generalization of the standard Lomax distribution. Jose and Krishna (2011) introduced the Marshall–Olkin extended continuous uniform distribution as a gener-alization of the standard continuous uniform distribution. Al-Saiari, Baharith, and Mousa (2014) in-troduced the Marshall–Olkin extended Burr type XII distribution as a generalization of the standard Burr type XII distribution. Alizadeh et al. (2015) introduced the Marshall–Olkin extended

(1)f (x) = 𝛽(1 − e−𝜆)e−𝛽(1−e−𝜆)x; 0 ≤ x < ∞, 𝛽, 𝜆 > 0,

(2)F(x) = 1 − e−𝛽(1−e−𝜆)x; 0 ≤ x < ∞, 𝛽, 𝜆 > 0,

(3)h(x) = 𝛽(1 − e−𝜆); 𝛽, 𝜆 > 0,

(4)F(x) =𝛼G(x)

1 − (1 − 𝛼)G(x); −∞ < x < ∞; 0 < 𝛼 < ∞

(5)f (x) =𝛼g(x)

(1 − (1 − 𝛼)G(x)

)2 ; −∞ < x < ∞; 0 < 𝛼 < ∞

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Kumaraswamy distribution as a generalization of the standard Kumaraswamy distribution. Gui (2013) introduced the Marshall–Olkin extended log-logistic distribution as a generalization of the standard log-logistic distribution. Pogány, Saboor, and Provost (2015) introduced the Marshall–Olkin extended exponential Weibull distribution generalizing the exponential Weibull distribution. Jose (2011) gave a comprehensive review of the Marshall–Olkin family of distributions and their applica-tions to reliability, time series and stress-strength analysis. For more extensive reviews of the Marshall–Olkin generalized family of distributions see, Nadarajah (2008) and Barreto-Souza, Lemonte, and Cordeiro (2013). Sandhya and Prasanth (2014) introduced the Marshall–Olkin extend-ed discrete uniform distribution as a generalizion of the standard discrete uniform distribution; etc.

Motivated by the idea of additional parameter for extra flexibility to the distribution, we introduce the three-parameter Marshall–Olkin generalized Erlang-truncated exponential (MOGETE) distribu-tion as a generalization of the standard two parameter ETE distribution. The importance of the new distribution is the ability of describing real data-sets with unimodal density as well as decreasing or increasing hazard rate function better than some already existing distributions as we show later. Hence, the MOGETE distribution has a superior fitting ability than the ETE distribution.

The remaining part of this article is organized as follows: Section 2 introduces the MOGETE distri-bution; Section 3 presents some reliability characteristics of the distribution such as the reliability function, hazard rate function and the mean residual life time; Section 4 presents some statistical properties of the new distribution such as the kth crude moment, mean, variance, coefficient of vari-ation, skewness, kurtosis, moment generating function, pth quantile function, Rényi entropy meas-ure of the new distribution and the distribution of order statistics of the distribution; Section 5 proposes the estimation of the distribution parameters through the method of maximum likelihood estimation; Section 6 presents the application of the new distribution to two real data sets; Section 7 presents the discussion of results and lastly and Section 8 is the conclusion of the study.

2. The MOGETE distributionThe cdf of the MOGETE distribution is given by

with the corresponding pdf as

where � and � are the shape parameters and � is the scale parameter.

Theorem 2.1 The MOGETE distribution with pdf in Equation (7) is identifiable i.e. ∃f (x;�1, �1, �1) = f (x; �2, �2, �2) as we show now.

Proof By setting

we have that

∀ x ≥ 0, the above equality is true ∀ �1 = �2, �1 = �2, and �1 = �2. ✷

(6)F(x) = 1 − F(x) =1 − e−𝛽(1−e

−𝜆)x

1 − (1 − 𝛼)e−𝛽(1−e−𝜆)x; 0 ≤ x < ∞, 𝛼 > 0, 𝛽, 𝜆 > 0

(7)f (x) = 𝛼𝛽(1 − e−𝜆)e−𝛽(1−e−𝜆)x

{1 − (1 − 𝛼)e−𝛽(1−e

−𝜆)x}−2

; 0 ≤ x < ∞, 𝛼 > 0, 𝛽, 𝜆 > 0

f (x; �1, �1, �1) = f (x; �2, �2, �2)

log(�1 − �2)3 + log(�1 − �2) + �1 − �2 − 3[�1(1 − e

−�1 ) − �2(1 − e−�2 )]x = 0

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Theorem 2.2 If a random variable say X is distributed according to the MOGETE distribution then the shape of its pdf as x → ∞ is decreasing when � ≤ 2 and unimodal when 𝛼 > 2.

Proof The first derivative of the pdf f ′ in Equation (7) is given by

Setting f � = 0 gives the critical point x0 at which the pdf is maximized. x0 is the root of the equation which is given by

this implies that as x → ∞ and 𝛼 > 2 there exists some x < x0 such that f (x) > 0 and some x > x0 such that f (x) < 0, hence; f(x) has a single mode at x0. Now, it makes sense to conclude that the pdf have decreasing shape as the only alternative shape when � ≤ 2; since, f �(x0) ≠ 0 and both conditions (𝛼 ≤ 2 and 𝛼 > 2) cannot be jointly satisfied in each case (monotonic decreasing and unimodality). ✷

The asymptotic behaviour of the pdf of the MOGETE distribution is f (0) = �(1 − e−�)∕� and f (∞) = 0.

3. Reliability analysis with the MOGETE distributionIn this section, we present some reliability characteristics of the MOGETE distribution that is neces-sary for reliability analysis, they are: the reliability (survival) function F(x) (R(x)), the hazard rate function h(x) and the mean residual life time M(t).

3.1. Reliability functionThe reliability function R(x) is an important tool in reliability analysis for characterizing life phenom-enon. R(x) is mathematically expressed as R(x) = 1 − F(x). Under certain predefined conditions the reliability function generally gives the estimated probability that a system will operate without fail-ure until a specified time x. The reliability function of the MOGETE distribution is given by

For various parameter values R(x) is generally a decreasing function of x and the asymptotic behav-iour of the reliability function of the MOGETE distribution is R(0) = 1; and R(∞) = 0.

3.2. Hazard rate functionThe hazard rate function (hrf) gives the probability of failure for a system that has survived up-to time x. It is mathematically expressed as h(x) = f (x)∕R(x). The hazard rate function of the MOGETE distribution is given by

Theorem 3.1 The shape of the hrf of the MOGETE distribution is constant (a special case of the ETE distribution when � = 1), decreasing when 𝛼 < 1 and increasing when 𝛼 > 1.

Proof The first derivative of the hrf h′ in Equation (9) is given by

f � = −𝛼𝛽

2(1 − e−𝜆)2e−𝛽(1−e−𝜆)x

(1 − (1 − 𝛼)e−𝛽(1−e−𝜆)x)2

−2𝛼(1 − 𝛼)𝛽2(1 − e−𝜆)2e−2𝛽(1−e

−𝜆)x

(1 − (1 − 𝛼)e−𝛽(1−e−𝜆)x)3

< 0.

x0 =log(𝛼 − 1)

𝛽(1 − e−𝜆); ∀ 𝛼 > 2,

(8)R(x) =𝛼e−𝛽(1−e

−𝜆)x

1 − (1 − 𝛼)e−𝛽(1−e−𝜆)x; 0 ≤ x < ∞, 𝛼 > 0, 𝛽, 𝜆 > 0

(9)h(x) =𝛽(1 − e−𝜆)

1 − (1 − 𝛼)e−𝛽(1−e−𝜆)x; 0 ≤ x < ∞, 𝛼 > 0, 𝛽, 𝜆 > 0.

h� = −(1 − �)�2(1 − e−�)2e−�(1−e

−�)x

(1 − (1 − �)e−�(1−e−�)x)2

.

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It is easy to see that h� = 0 has no unique root; h� < 0, ∀ 𝛼 < 1 (i.e. the hrf is decreasing) and h� > 0, ∀𝛼 > 1 (i.e. the hrf is increasing) and when � = 1, h� = 0 (i.e. the hrf is constant). ✷

The asymptotic behaviour of the hrf of the MOGETE distribution is h(0) = �(1 − e−�)∕� and h(∞) = �(1 − e−�).

3.3. The mean residual life time

Theorem 3.2 The remaining lifetime of a system that has survived up-to time t is random, as a result the failure time cannot be determined. The expected value of the random failure times is referred to as the mean residual lifetime denoted by M(t). M(t) only exists for F(t) > 0 and it is mathematical expressed as

The mean residual lifetime of the MOGETE distribution is given by

Proof

Substituting y = �(1 − e−�)x into Equation (11), we have

✷

4. Some statistical properties of the MOGETE distributionApplication of any distribution can only be possible if its basic distributional properties are available. In this section, we present explicit derivations of some important distributional properties of the MOGETE distribution.

4.1. The pth quantile function of the MOGETE distributionThe pth quantile function of the MOGETE distribution is given by

Random variables can be simulated from the MOGETE distribution through the method of the in-version of cdf by simply substituting p in Equation (12) with a U(0, 1) variates. Also, the median of the

M(t) = E(X − t|X > t) =1

R(t)

∞

∫t

R(x)dx

(10)M(t) =

�

R(t)�(1 − e−�)

∞∑j=0

(−1)2j(1 − �)je−[�(1−e

−�)t](j+1)

j + 1.

(11)M(t) =1

R(t)

∞

∫t

�e−�(1−e−�)x[1 − (1 − �)e−�(1−e

−�)x]−1dx

M(t) =�

R(t)�(1 − e−�)

∞

∫�(1−e−�)t

e−y[1 − (1 − �)e−y

]−1dy

=�

R(t)�(1 − e−�)

∞

∫�(1−e−�)t

e−y∞∑j=0

(−1)j

(j + 1 − 1

j

)(−1)j(1 − �)je−yjdy

=�

R(t)�(1 − e−�)

∞∑j=0

(−1)2j(1 − �)j

∞

∫�(1−e−�)t

e−y(j+1)dy

=�

R(t)�(1 − e−�)

∞∑j=0

(−1)2j(1 − �)je−[�(1−e

−�)t](j+1)

j + 1.

(12)�(p) = −1

�(1 − e−�)log

[1 − p

1 − p(1 − �)

]

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MOGETE distribution could be obtained from Equation (12) by setting p = 1∕2. The Bowley skewness denoted by B due to Bowley (1901–1920) and Moors kurtosis denoted by M due to Moors (1986) depends on the quantile function.

The Bowley skewness is given by

and, the Moors kurtosis is given by

Figure 1 illustrates the variability of both the Bowley skewness and Moors kurtosis on �(shape param-eter) for the MOGETE distribution. The Bowley skewness and Moors kurtosis shrinks with increasing value of �.

4.2. The kth crude moment of the MOGETE distribution

Theorem 4.1 If the kth crude moment of any random variable X exists then other essential charac-teristics of the distribution could be derived from it, such as the mean, variance, coefficient of varia-tion, skewness and kurtosis statistics. The kth crude moment of any continuous random variable X is generally given by E(Xk) = ∫∞

−∞xkf (x)dx. Hence, it follows that the kth crude moment of the MOGETE

distribution is given by

Proof

B =�(3∕4) + �(1∕4) − 2�(2∕4)

�(3∕4) − �(1∕4),

M =�(3∕8) − �(1∕8) + �(7∕8) − �(5∕8)

�(6∕8) − �(2∕8).

E(Xk) =�Γ(k + 1)

[�(1 − e−�)

]k∞∑j=0

(−1)2j(1 − �)j

(j + 1)k.

(13)

E(Xk) =

∞

∫0

xk��(1 − e−�)e−�(1−e−�)x[1 − (1 − �)e−�(1−e

−�)x]−2dx

= ��(1 − e−�)

∞

∫0

xke−�(1−e−�)x[1 − (1 − �)e−�(1−e

−�)x]−2dx

Figure 1. Plots of the Bowley skewness (left panel) and Moors kurtosis (right panel) of the MOGETE distribution.

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Substituting y = �(1 − e−�)x into Equation (13) gives

Substituting z = y(j + 1) into Equation (14) gives

✷

The mean is the first-order crude moment of the distribution and could be obtained by evaluating Equation (15) at k = 1 as

While evaluating Equation (15) at k = 2 gives the second-order crude moment of the MOGETE distri-bution as

The variance V(X) could be obtained by substituting E(X) and E(X2) into the following expression V(X) = E(X2) − {E(X)}2. Hence, the variance of the MOGETE distribution is given by

Setting E(Xk) = ��

k the coefficient of variation (CV), skewness (�1) and kurtosis (�2) statistics of the MOGETE distribution could be obtained as follows

(14)

=�[

�(1 − e−�)]k

∞

∫0

yke−y[1 − (1 − �)e−y

]−2dy

=�[

�(1 − e−�)]k

∞

∫0

yke−y∞∑j=0

(−1)j

(j + 2 − 1

j

)(−1)j(1 − �)je−yjdy

=�[

�(1 − e−�)]k

∞∑j=0

(−1)2j(j + 1)(1 − �)j

∞

∫0

yke−y(j+1)dy

(15)

=�[

�(1 − e−�)]k

∞∑j=0

(−1)2j(1 − �)j

(j + 1)k

∞

∫0

zke−zdz

=�Γ(k + 1)

[�(1 − e−�)

]k∞∑j=0

(−1)2j(1 − �)j

(j + 1)k.

E(X) =�

�(1 − e−�)

∞∑j=0

(−1)2j(1 − �)j

j + 1.

E(X2) =2�[

�(1 − e−�)]2

∞∑j=0

(−1)2j(1 − �)j

(j + 1)2.

V(X) =2�[

�(1 − e−�)]2

∞∑j=0

(−1)2j(1 − �)j

(j + 1)2−

[�

�(1 − e−�)

∞∑j=0

(−1)2j(1 − �)j

j + 1

]2

CV =

√√√√ ��

2

��21

− 1

�1 =��

3 − 3��

2��

1 + 2��31

(��

2 − ��21 )

3

2

�2 =��

4 − 4��

3��

1 + 6��

2��21 − 3��4

1

(��

2 − ��21 )

2

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4.3. The kth central moment of the MOGETE distribution

Theorem 4.2 The kth central moment of a continuous random variable X is given by E((X − �)k) = ∫∞

−∞(x − �)kf (x)dx. Hence, the kth central moment of the MOGETE distribution is given by

Proof


Substituting z = y(j + 1) into Equation (17) gives

✷

4.4. The moment generating function of the MOGETE distributionRecently, a lot of advancement both in theory and application has been achieved in statistics and probability through the moment generating function (mgf) of a random variable X. The usefulness of the mgf has been found to surpass the very trivial derivation of distributional order moments. For

E[(X − �)k] =�[

�(1 − e−�)]k

∞∑i=0

∞∑j=0

(−1)i+2jΓ(k + 1)

[��(1 − e−�)

]i(1 − �)j(j + 2)Γ(k − i + 1)

Γ(i + 1)Γ(k − i + 1)(j + 1)k−i+1.

(16)

E[(X − �)k] =

∞

∫0

(x − �)k��(1 − e−�)e−�(1−e−�)x[1 − (1 − �)e−�(1−e

−�)x]−2dx

= ��(1 − e−�)

∞

∫0

(x − �)ke−�(1−e−�)x[1 − (1 − �)e−�(1−e

−�)x]−2dx

(17)

E[(X − �)k] =�[

�(1 − e−�)]k

∞

∫0

e−y[y − ��(1 − e−�)

]k

⋅

[1 − (1 − �)e−y

]−2dy

=�[

�(1 − e−�)]k

∞

∫0

e−y∞∑i=0

(k

i

)(−1)i

[��(1 − e−�)

]iyk−i

⋅

∞∑j=0

(−1)j

(j + 2 − 1

j

)(−1)j(1 − �)je−yjdy

=�[

�(1 − e−�)]k

∞∑i=0

∞∑j=0

(−1)i+2j[��(1 − e−�)

]i(1 − �)j(j + 2)

(k

i

)

⋅

∞

∫0

yk−ie−y(j+1)dy

E[(X − �)k] =�[

�(1 − e−�)]k

∞∑i=0

∞∑j=0

(−1)i+2j

(k

i

)[��(1 − e−�)

]i(1 − �)j(j + 2)

(j + 1)k−i+1

⋅

∞

∫0

zk−ie−zdz

=�[

�(1 − e−�)]k

∞∑i=0

∞∑j=0

(−1)i+2j

⋅

Γ(k + 1)[��(1 − e−�)

]i(1 − �)j(j + 2)Γ(k − i + 1)

Γ(i + 1)Γ(k − i + 1)(j + 1)k−i+1

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instance; Villa and Escobar (2006) obtained mixture distributions with mgf, Meintanis (2010) used the mgf for testing skew normality, McLeish (2014) performed simulation of random variables using the mgf and the saddle point approximation, von Waldenfels (1987) gave a proof of an algebraic central limit theorem using the mgf, and Inlow (2010) also, proved the Lindeberg-Lévy’s central limit theorem with the mgf. The moment generating function is generally defined by

It follows from Equations (15) and (18) that the mgf of the MOGETE distribution is given by

4.5. Rényi entropy of the MOGETE distributionThe Rényi entropy denoted by H

�(x) is used to quantify the uncertainty of variation in a random vari-

able X. The limiting value of H�(x) as � → 1 is the Shannon entropy. Song (2001) compared tails and

shapes behaviour of some standard probability distributions using the Rényi entropy. The Rényi en-tropy measure is generally given by

Theorem 4.3 If X follows the MOGETE distribution, then its Rényi entropy measure is given by

Proof


✷

(18)MX(t) = E(etx) = E

[∞∑k=0

(tx)k

k!

]=

∞∑k=0

tk

k!E(xk).

MX(t) =

∞∑j=0

∞∑k=0

(−1)2jtk�Γ(k + 1)

k![�(1 − e−�)

]k(1 − �)j

(j + 1)k.

H�(x) = lim

n→∞(I

�(fn) − log(n)) =

1

1 − �log ∫ f �(x)dx =

1

1 − �log(I

�).

H�(x) =

1

1 − �log

([��

(1 − e−�

)]��(1 − e−�)

∞∑j=0

(−1)2jΓ(j + 2�)

Γ(j + 1)Γ(2�)

(1 − �)j

� + j

).

(19)

I�=

∞

∫0

[��

(1 − e−�

)e−�(1−e

−�)x(1 − (1 − �)e−�(1−e

−�)x)−2

]�dx

=[��

(1 − e−�

)]�∞

∫0

e−��(1−e−�)x

(1 − (1 − �)e−�(1−e

−�)x)−2�

dx

I�=

[��

(1 − e−�

)]��(1 − e−�)

∞

∫0

e−�y(1 − (1 − �)e−y

)−2�dy

=

[��

(1 − e−�

)]��(1 − e−�)

∞

∫0

e−�y∞∑j=0

(−1)j

(j + 2� − 1

j

)(−1)j(1 − �)je−yjdy

=

[��

(1 − e−�

)]��(1 − e−�)

∞∑j=0

(−1)2j

(j + 2� − 1

j

)(1 − �)j

∞

∫0

e−y(�+j)dy

=

[��

(1 − e−�

)]��(1 − e−�)

∞∑j=0

(−1)2j

(j + 2� − 1

j

)(1 − �)j

� + j

H�(x) =

1

1 − �log

([��

(1 − e−�

)]��(1 − e−�)

∞∑j=0

(−1)2jΓ(j + 2�)

Γ(j + 1)Γ(2�)

(1 − �)j

� + j

)

of 19


4.6. Order statistics of the MOGETE random variableThe distribution of the rth order statistics denoted by fX

(r)(x) of an n sized random sample

X1,X2,X3,… ,Xn is generally given by

The density of the rth order statistics of the MOGETE distribution could be obtained by substituting Equations (6) and (7) into Equation (20) as

The density of the rth smallest order statistics of the MOGETE distribution is given by

The density of the rth largest order statistics of the MOGETE distribution is given by

5. EstimationHere, we propose to estimate the parameters of the MOGETE distribution by the method of Maximum likelihood estimation.

5.1. Maximum likelihood estimation methodSuppose the random sample x1, x2, x3,… , xn of size n is drawn from a probability distribution with pdf f(x) then the maximum likelihood estimates (mle) of its parameters could be obtained as follows:

The likelihood () equation is given by

and the log-likelihood (�) equation is given by

then; taking the partial derivatives of Equation (22) w.r.t to �; � and � and equating to zero gives:

(20)fX(r)(x) =

n!

(r − 1)!(n − r)!(Fx(x))

r−1(1 − Fx(x))n−r fx(x).

fX(k)(x) =

n!��(1 − e−�

)e−�(1−e

−�)x[1 − (1 − �)e−�(1−e

−�)x]−2

(k − 1)!(n − k)!

⋅

[1 − e−�(1−e

−�)x

1 − (1 − �)e−�(1−e−�)x

]k−1[�e−�(1−e

−�)x

1 − (1 − �)e−�(1−e−�)x

]n−k

fX(1)(x) = n!��

(1 − e−�

)e−�(1−e

−�)x

⋅

[1 − (1 − �)e−�(1−e

−�)x]−2[ �e−�(1−e

−�)x

1 − (1 − �)e−�(1−e−�)x

]n−1

fX(n)(x) = n!��

(1 − e−�

)e−�(1−e

−�)x

⋅

[1 − (1 − �)e−�(1−e

−�)x]−2[ 1 − e−�(1−e

−�)x

1 − (1 − �)e−�(1−e−�)x

]n−1

(21)

=

n�i=1

��(1 − e−�)e−�(1−e−�)xi

�1 − (1 − �)e−�(1−e

−�)xi

�−2

=��(1 − e−�)

�ne−�(1−e−�)

n∑i=1

xin�i=1

�1 − (1 − �)e−�(1−e

−�)xi

�−2

(22)� = n log(��(1 − e−�)

)− �(1 − e−�)

n∑i=1

xi − 2

n∑i=1

log{1 − (1 − �)e−�(1−e

−�)xi

}

of 19


The analytical solution to the system of nonlinear equations in Equations (23), (24) and (25) does not exist thus, we require some nonlinear numerical optimization methods such as the Newton Raphson technique to solve the equations.

Let � = (��, 𝛽, ��)�. Under some standard regularity conditions, √n(� −�) is asymptotically mul-

tivariate normal distributed �3(�, �−1n (�)), where �n(�) is the expected information matrix defined

by �(�2�(�)∕��). The asymptotic behaviour of the expected information matrix can be ap-

proximated by the observed information matrix, denoted by �n(�). Generally speaking, the diagonal elements of �−1n (�) gives the variance of (�) while the off-diagonal elements is the covariances. The observed information matrix of the MOGETE distribution is expressed as

where the corresponding elements are:

(23)��

��=n

�− 2

n∑i=1

e−�(1−e−�)xi

1 − (1 − �)e−�(1−e−�)xi

= 0

(24)��

��=n

�− (1 − e−�)

n∑i=1

xi − 2

n∑i=1

(1 − �)(1 − e−�)xie−�(1−e−�)xi

1 − (1 − �)e−�(1−e−�)xi

= 0

(25)��

��=

ne−�

1 − e−�− �e−�

n∑i=1

xi − 2

n∑i=1

(1 − �)�xie−�−�(1−e−�)xi

1 − (1 − �)e−�(1−e−�)xi

= 0

�n(�) =

⎛⎜⎜⎜⎝

𝜕2�(�)

𝜕𝛼2

𝜕2�(�)

𝜕𝛼𝜕𝛽

𝜕2�(�)

𝜕𝛼𝜕𝜆

𝜕2�(�)

𝜕𝛽𝜕𝛼

𝜕2�(�)

𝜕𝛽2

𝜕2�(�)

𝜕𝛽𝜕𝜆

𝜕2�(�)

𝜕𝜆𝜕𝛼

𝜕2�(�)

𝜕𝜆𝜕𝛽

𝜕2�(�)

𝜕𝜆2

⎞⎟⎟⎟⎠.

�2�

��2= −

n

�2+ 2

n∑i=1

e−2�(1−e−�)x

i

[1 − (1 − �)e−�(1−e−�)x

i ]2,

�2�

��2= −

n

�2+ 2

n∑i=1

(1 − �)(1 − e−�)2x2ie−�(1−e−�)x

i

1 − (1 − �)e−�(1−e−�)x

i

+ 2

n∑i=1

(1 − �)2(1 − e−�)2x2ie−2�(1−e−�)x

i

(1 − (1 − �)e−�(1−e−�)x

i )2,

�2�

��2= −

n[e−�(1 − e−�) + e−2�]

(1 − e−�)2+ �e

−�

n∑i=1

xi

− 2

n∑i=1

[(1 − �)�x

ie−�−�(1−e−�)x

i (−1 − �xie−�)

1 − (1 − �)e−�(1−e−�)x

i

−(1 − �)2�2x2

ie−2(�+(1−e−�)x

i)

(1 − (1 − �)e−�(1−e−�)x

i )2

],

�2�

��= 2

n∑i=1

(1 − e−�)xie−�(1−e−�)x

i

1 − (1 − �)e−�(1−e−�)x

i

+ 2

n∑i=1

(1 − �)(1 − e−�)xie−2�(1−e−�)x

i

[1 − (1 − �)e−�(1−e−�)x

i ]2,

�2�

��= 2

n∑i=1

�xie−�−�(1−e−�)x

i

1 − (1 − �)e−�(1−e−�)x

i

+ 2

n∑i=1

(1 − �)�xie−�−2�(1−e−�)x

i

[1 − (1 − �)e−�(1−e−�)x

i ]2,

of 19


and

Given that √n(� −� ∼)�

3(�, �−1

n(�)), we can perform statistical inference for functions of �. For

instance, the approximate 100(1 − �)% two-sided confidence interval of the model parameters � could be calculated as:

where �−1�2𝓁(�)

��

(⋅), are the diagonal entries of the observed information matrix, and Z �

2

is the upper

�∕2th percentile of the standard normal distribution.

5.1.1. Simulation studyOne major problem of extended probability distributions is parameter estimation. In this section, we present a Monte Carlo simulation study to evaluate the performance of the mle method in estimat-ing the parameters of the distribution by drawing different samples (n = 50, 100, … , 300) from the MOGETE distribution with selected parameter values. Estimation of the parameters was carried out with the simulated random variables through the mle method to investigate the stability of the parameters and sample size effect on the estimates via bias, standard error (se), and mean square error (mse). Application of the following algorithm in (Statistical software) provides us with the results in Table 1.

5.1.2. Algorithm

(i) Simulate ui∼ Uniform(0, 1), for i = 1, 2, 3,… ,n(50, 100,… , 300);

(ii) Set xi= F−1(u

i), where F−1(⋅) is the quantile function in Equation (17) evaluated at U

i for some

parameter values (see Table 1) and X ~ MOGETE distribution;

(iii) Using x and the nlm function under the stats package in , calculate the mle estimates of the parameters of the MOGETE distribution;

(iv) Repeat steps (i–iii) in 5,000 (N) iterations;

(v) For each n and parameter, compute the mean (parameter estimate), standard deviation (standard error), bias and mse of the sequence of 5,000 parameter estimates.

�2�

��= −e−�

n∑i=1

xi − 2

n∑i=1

(1 − �)xie−�−�(1−e−�)xi

1 − (1 − �)e−�(1−e−�)xi

+ 2

n∑i=1

(1 − �)�x2i e−�−�(1−e−�)xi

1 − (1 − �)e−�(1−e−�)xi

+ 2

n∑i=1

(1 − �)2�(1 − e−�)x2i e−�−2�(1−e−�)xi

[1 − (1 − �)e−�(1−e−�)xi ]2

� ± Z �

2

√�−1�2𝓁(�)

��

(⋅),

of 19


Tabl

e 1.

Sim

ulat

ion

resu

ltsn

𝜶 𝜷

𝝀se

𝜶se

𝜷se

𝝀bias 𝜶

bias

𝜷bias 𝝀

mse

𝜶mse

𝜷mse

𝝀

�=5.000,�=5.000,�=5.000

506.

1859

85.

2149

95.

0050

63.

4368

10.

9572

70.

0335

31.

1859

80.

2149

90.

0050

613

.215

860.

9624

00.

0011

5

100

5.56

314

5.10

831

5.00

252

1.99

627

0.65

025

0.02

124

0.56

314

0.10

831

0.00

252

4.30

142

0.43

447

0.00

046

150

5.33

840

5.07

000

5.00

156

1.53

315

0.53

264

0.01

763

0.33

840

0.07

000

0.00

156

2.46

458

0.28

855

0.00

031

200

5.24

952

5.04

803

5.00

105

1.32

411

0.46

110

0.01

526

0.24

952

0.04

803

0.00

105

1.81

516

0.21

488

0.00

023

250

5.19

484

5.03

778

5.00

101

1.14

615

0.41

452

0.01

388

0.19

484

0.03

778

0.00

101

1.35

136

0.17

322

0.00

019

300

5.15

584

5.03

225

5.00

086

1.02

739

0.37

190

0.01

250

0.15

584

0.03

225

0.00

086

1.07

961

0.13

933

0.00

016

�=2.500,�=3.500,�=4.500

503.

0341

83.

6819

04.

5053

11.

6366

10.

7981

20.

0299

20.

5341

80.

1819

00.

0053

12.

9633

20.

6699

50.

0009

2

100

2.72

910

3.58

448

4.50

254

0.95

544

0.54

492

0.02

080

0.22

910

0.08

448

0.00

254

0.96

518

0.30

402

0.00

044

150

2.66

079

3.56

598

4.50

229

0.75

167

0.44

686

0.01

716

0.16

079

0.06

598

0.00

229

0.59

075

0.20

400

0.00

030

200

2.61

593

3.54

175

4.50

137

0.62

960

0.37

883

0.01

472

0.11

593

0.04

175

0.00

137

0.40

976

0.14

523

0.00

022

250

2.58

400

3.52

840

4.50

101

0.55

709

0.33

495

0.01

304

0.08

400

0.02

840

0.00

101

0.31

735

0.11

297

0.00

017

300

2.57

327

3.52

693

4.50

100

0.51

057

0.31

197

0.01

214

0.07

327

0.02

693

0.00

100

0.26

600

0.09

803

0.00

015

�=8.900,�=2.300,�=0.500

5010

.974

842.

2946

90.

5284

66.

2933

70.

0331

50.

1129

42.

0748

4−0.00531

0.02

846

43.9

0350

0.00

113

0.01

356

100

9.89

336

2.29

899

0.51

355

3.78

194

0.02

139

0.07

380

0.99

336

−0.00101

0.01

355

15.2

8695

0.00

046

0.00

563

150

9.51

749

2.29

901

0.50

832

2.87

953

0.01

755

0.05

901

0.61

749

−0.00099

0.00

832

8.67

133

0.00

031

0.00

355

200

9.36

451

2.29

955

0.50

658

2.46

095

0.01

492

0.05

059

0.46

451

−0.00045

0.00

658

6.27

081

0.00

022

0.00

260

250

9.28

069

2.29

999

0.50

541

2.15

863

0.01

332

0.04

502

0.38

069

−0.00001

0.00

541

4.80

369

0.00

018

0.00

206

300

9.17

078

2.30

029

0.50

372

1.85

221

0.01

184

0.03

961

0.27

078

0.00

029

0.00

372

3.50

334

0.00

014

0.00

158

�=0.500,�=0.500,�=0.500

500.

6322

70.

5308

80.

5241

50.

3567

40.

1184

10.

0918

30.

1322

70.

0308

80.

0241

50.

1447

40.

0149

70.

0090

1

100

0.56

720

0.51

692

0.51

308

0.22

019

0.08

190

0.06

312

0.06

720

0.01

692

0.01

308

0.05

299

0.00

699

0.00

415

150

0.54

571

0.51

280

0.50

988

0.16

805

0.06

649

0.05

124

0.04

571

0.01

280

0.00

988

0.03

032

0.00

458

0.00

272

200

0.52

970

0.50

844

0.50

651

0.14

004

0.05

660

0.04

363

0.02

970

0.00

844

0.00

651

0.02

049

0.00

327

0.00

195

250

0.52

688

0.50

708

0.50

546

0.12

643

0.05

079

0.03

914

0.02

688

0.00

708

0.00

546

0.01

670

0.00

263

0.00

156

300

0.52

068

0.50

604

0.50

465

0.11

498

0.04

709

0.03

630

0.02

068

0.00

604

0.00

465

0.01

365

0.00

225

0.00

134

�=20.000,�=20.000,�=1.000

5026

.423

5019

.994

021.

1177

118

.116

900.

1376

50.

4397

76.

4235

0−0.00598

0.11

771

369.

4178

70.

0189

80.

2072

1

100

22.3

5379

19.9

9680

1.04

063

9.24

222

0.02

020

0.19

696

2.35

379

−0.00320

0.04

063

90.9

4194

0.00

042

0.04

044

150

21.8

4860

19.9

9840

1.03

121

7.46

007

0.01

590

0.15

816

1.84

860

−0.00160

0.03

121

59.0

5884

0.00

026

0.02

598

200

21.1

9487

19.9

9851

1.02

011

5.89

144

0.01

853

0.13

151

1.19

487

−0.00149

0.02

011

36.1

2989

0.00

035

0.01

770

250

20.9

0419

19.9

9874

1.01

528

5.10

803

0.01

032

0.11

398

0.90

419

−0.00126

0.01

528

26.9

0432

0.00

011

0.01

322

300

20.8

5828

19.9

9939

1.01

474

4.66

581

0.02

018

0.10

550

0.85

828

−0.00061

0.01

474

22.5

0210

0.00

041

0.01

135

of 19


Whatever parameter values we chose in Table 1, the mle estimates ( � = 1∕Nn∑i=1

�i) approximates

to the actual value as n becomes large and the standard error �se

�=

�∑N

i=1(�

i−

�)2∕(N − 1)

�,

bias (bias = �i−�; i = 1,… ,n) and mse (mse = 1∕N

∑N

i=1[�

i−�]2) decreases with increasing

n, where � = (�, �, �)�. These results suggest that the mle method does well in estimating the pa-rameters of the MOGETE distribution.

6. ApplicationsThis section illustrates the applicability and flexibility of the MOGETE distribution with two real data-sets. The goodness of fit of the new lifetime distribution would be assessed by a comparison of its performance in modelling the real data-sets with that of its competing sub-model (ETE distribution) and the following three-parameter distributions:

• Exponentiated Burr XII

• Exponentiated Fréchet

based on the Akaike information criterion (AIC) statistic, Akaike (1981),

the AIC with a correction statistic (AICc), Sugiura (1978),

where �, k, and n corresponds to the estimate of the model maximized/minimized log-likelihood function, number of model parameters and sample size, respectively. The Chen and Balakrishnan (1995) W⋆ and A⋆ goodness of fit measures were also considered. See Oluyede, Foya, Warahena-Liyanage, and Huang (2016) for detail on the computational steps of the W⋆ and A⋆ statistics. The distribution with the smallest goodness-of-fit measure is the best. Table 2 gives the waiting times in minutes of 100 bank customers in a queue before service. The data-set was first published in Ghitany, Atieh, and Nadarajah (2008). Merovci and Elbatal (2013) and Bhati, Malik, and Vaman (2015) have also fitted the data to different distributions. The results we obtained from the data fitting are tabu-lated in Table 3.

f (x) = 𝛼kcxc−1(1 + xc)−k−1[1 − (1 + xc)−k]𝛼−1, 𝛼, c, k > 0, and

f (x) =𝛼𝛽

s

(x

s

)−𝛽−1

e−𝛼(x

s)−𝛽

, 𝛼, 𝛽, s > 0,

AIC = −2� + 2k

AICc = AIC +2k(k + 1)

n − k − 1

Table 2. 100 bank customers waiting times (min) before service0.8 0.8 1.3 1.5 1.8 1.9 1.9 2.1 2.6 2.7

2.9 3.1 3.2 3.3 3.5 3.6 4.0 4.1 4.2 4.2

4.3 4.3 4.4 4.4 4.6 4.7 4.7 4.8 4.9 4.9

5.0 5.3 5.5 5.7 5.7 6.1 6.2 6.2 6.2 6.3

6.7 6.9 7.1 7.1 7.1 7.1 7.4 7.6 7.7 8.0

8.2 8.6 8.6 8.6 8.8 8.8 8.9 8.9 9.5 9.6

9.7 9.8 10.7 10.9 11.0 11.0 11.1 11.2 11.2 11.5

11.9 12.4 12.5 12.9 13.0 13.1 13.3 13.6 13.7 13.9

14.1 15.4 15.4 17.3 17.3 18.1 18.2 18.4 18.9 19.0

19.9 20.6 21.3 21.4 21.9 23.0 27.0 31.6 33.1 38.5

of 19


The variance-covariance matrix of the MOGETE distribution under the fitted 100 bank customers waiting times data is given by

The second example is on the annual maximum daily precipitation in millimetre that was recorded in Basan, Korea, from 1904 to 2011. The data are presented in Table 4. The data-set has been ana-lysed by Jeong, Murshed, Am Seo, and Park (2014) and was recently reported in Mansoor et al. (2016). Results from the data fitting to the distributions are presented in Table 5.

�−1n (�) =

⎛⎜⎜⎝

1.83309877 0.2559214 0.00352501

0.25592137 85.9418670 −4.10769366

0.00352501 −4.1076937 0.19651703

⎞⎟⎟⎠

Table 3. Results from modelling the 100 bank customers waiting times dataModels Estimatess s.e −� AIC AICc W

⋆A⋆

MOGETE

�� 4.11562602 1.353920

𝛽 2.10379157 9.270484 320.7120 647.4241 647.6741 0.0008333 0.6351718

�� 0.09588135 0.443302

ETE

𝛽 0.2428209 5.284512 329.0209 662.0418 662.1655 27.8976100 4.2290630

�� 0.5394967 15.56389

Exp Fréchet

�� 4.200080 1.978491e+03

𝛽 1.162912 7.995092e-02 334.3810 674.7620 675.0120 0.3832246 2.504610

s 1.461971 5.921992e+02

Exp Burr XII

�� 39.882445 39.3966644

k 2.829186 1.2939812 327.5301 661.0601 661.3101 0.2297712 1.561101

c 0.595938 0.2128391

Table 4. Rainfall data24.8 140.9 54.1 153.5 47.9 165.5 68.5 153.1 254.7 175.3 87.6 150.6

147.9 354.7 128.5 150.4 119.2 69.7 185.1 153.4 121.7 99.3 126.9 150.1

149.1 143 125.2 97.2 179.3 125.8 101 89.8 54.6 283.9 94.3 165.4

48.3 69.2 147.1 114.2 159.4 114.9 58.5 76.6 20.7 107.1 244.5 126

122.2 219.9 153.2 145.3 101.9 135.3 103.1 74.7 174 126 144.9 226.3

96.2 149.3 122.3 164.8 188.6 273.2 61.2 84.3 130.5 96.2 155.8 194.6

92 131 137 106.8 131.6 268.2 124.5 147.8 294.6 101.6 103.1 247.5

140.2 153.3 91.8 79.4 149.2 168.6 127.7 332.8 261.6 122.9 273.4 178

177 108.5 115 241 76 127.5 190 259.5 301.5

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The variance–covariance matrix of the MOGETE distribution under the fitted Rainfall data is given by

7. Discussion of resultsThe density plots in Figure 2 (left panel) depict some monotonic decreasing function of x for � ≤ 2 and for α > 2 the distribution is unimodal, while the cdf plot (right panel) shows some monotonic increasing curves for all 𝛼 < 1. The plots in Figure 2 indicate that the reliability function (left panel) is a monotonically decreasing function of x for all � while the hazard rate function (right panel) could increasing (if 𝛼 < 1), decreasing (if 𝛼 < 1), or constant (if � = 1), these characteristics make it more reasonable for analysing complex lifetime data-sets. The results in Tables 3 and 5 show that the MOGETE distribution with smaller minimized log-likelihood value and smaller information statistics provides better fit to the data-sets than the ETE and the other competing distributions. Also, the P-P plots in Figures 4 and 5 does not raise any alarm against the suggestion of the AIC, AICc, W* and A⋆ statistics.

8. ConclusionsThis article introduces a new lifetime distribution—the (MOGETE) distribution. The new distribution generalizes the ETE distribution and has the ETE distribution as a sub-model. We have given explicit mathematical expressions for some of its basic statistical properties such as the probability density function, cumulative density function, kth raw moment,kth central moment, mean, variance, coef-ficient of variation, skewness, kurtosis, moment generating function, pth quantile function, the rth order statistics and the Rényi’s entropy measure. Also, some of its reliability characteristics like the

�−1n (�) =

⎛⎜⎜⎝

193.60658478 0.2749396 0.03408826

0.27493958 0.4427812 −0.11510233

0.03408826 −0.1151023 0.02999145

⎞⎟⎟⎠

Table 5. Results from modelling the rainfall dataModels Estimates s.e −� AIC AICc W

⋆A⋆

MOGETE

�� 35.58880229 13.9142583

𝛽 0.33061261 0.6654180 583.7588 1173.518 1173.755 0.0007939 1.2923280

�� 0.08257185 0.1731804

ETE

𝛽 0.01535297 0.01651167 627.2663 1258.533 1258.650 104.3609000 15.7744200

�� 0.59875101 0.8855356

Exp Fréchet

�� 14.21029 227.2744212

𝛽 1.64995 0.1031434 608.5969 1223.194 1223.432 0.7123182 4.1213390

s 19.99516 193.8275346

Exp Burr XII

�� 591.1682852 181.9098298

k 0.9348673 0.4196369 612.2200 1230.440 1230.678 0.5777131 3.3452710

c 1.4783094 0.6605398

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Figure 4. Probability-Probability (P-P) plots of the fitted distributions with the waiting times data.

Figure 3. Possible shapes of the survival/reliability function F(x) (left) and hazard rate function h(x) (right) of the MOGETE distribution for fixed parameter values of � and � and selected values of � parameter.

Figure 2. Possible shapes of the probability density function f (x) (left) and cumulative distribution function F(x) (right) of the MOGETE distribution for fixed parameter values of � and � and selected values of � parameter.

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reliability function, hazard rate function and the mean residual life time was given. Estimation of the model parameters was approached through the method of maximum likelihood estimation. The applicability, flexibility and robustness of the new lifetime distribution was demonstrated with the 100 bank customers waiting times data and 105 Rainfall data, and the results obtained show that the MOGETE distribution provides a more reasonable fit than the ETE, Exp Fréchet and Exp Burr XII distributions. We hope that the MOGETE distribution would receive a high rate of application, par-ticularly, because of its hazard rate characteristics.

Figure 5. Probability-Probability (P-P) plots of the fitted distributions with the Rainfall data.

FundingThis work was supported by the University of Manchester, UK.

Author detailsIdika E. Okorie1

E-mail: [email protected] ID: http://orcid.org/0000-0001-7770-3036Anthony C. Akpanta2

E-mail: [email protected] ID: http://orcid.org/0000-0003-4178-9370Johnson Ohakwe3

E-mail: [email protected] ID: http://orcid.org/0000-0001-5193-86541 School of Mathematics, University of Manchester,

Manchester M13 9PL, UK.2 Department of Statistics, Abia State University, Uturu, Abia

State, Nigeria.3 Faculty of Sciences, Department of Mathematics & Statistics,

Federal University Otuoke, P.M.B 126 Yenagoa, Bayelsa, Bayelsa State, Nigeria.

Citation informationCite this article as: Marshall-Olkin generalized Erlang-truncated exponential distribution: Properties and applications, Idika E. Okorie, Anthony C. Akpanta & Johnson Ohakwe, Cogent Mathematics (2017), 4: 1285093.

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http://orcid.org/0000-0001-7770-3036


http://orcid.org/0000-0003-4178-9370


http://orcid.org/0000-0001-5193-8654

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