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In this work, the authors proposed a four parameter potentiated lifetime model named as Transmuted Exponentiated Moment Pareto (TEMP) distribution and discussed numerous characteristic measures of proposed model. Parameters are estimated by the method of maximum likelihood and performance of these estimates is also assessed by simulations study. Four suitable lifetime datasets are modeled by the TEMP distribution and the results support that the proposed model provides much better results as compared to its sub-models.

An Italian Economist and civil engineer, Pareto (1848-1923) introduced the Power law. This law is also known as Pareto Power law and shortly turned into Pareto distribution. Unequal distribution of wealth in society was major cause to establish the Power law. 80% wealth of the population is distributed in 20% population. Thus it is also known as 80 - 20 rule and is stated as N = γx^{−k} where N is the number of individuals with income higher than x for k > 0. Under social constraints of taxation and other conditions this law is proved to be inevitable and universal. Many empirical phenomena are explained by Pareto distribution. Flexibility of Pareto distribution attracted the researchers to develop models by mixing Pareto distribution with other distributions.

Alzaatreh et al. [

Moment probability distribution or weighted distribution is introduced by Fisher [

Exponentiated CDF of a probability distribution is expressed as Exponentiated Distribution (ED). Gompertz [

Shaw and Buckley [

where F(x) and f(x) are the CDF and PDF of the corresponding QRTM.

Merovci and Puka [

The authors divided the structure of the article into several sections as follows: Section 2 describes the CDF, PDF and special cases of proposed distribution. In Section 3 and 4, various reliability measures, moments and order statistics are discussed. Quantile function, different descriptive statistics and Rényi entropy are discussed in Section 5. Simulations study is conducted to observe the behavior of MLE estimates in Section 6 while parameters of TEMP distribution are derived by the method of MLE along with two life time data sets are modeled in Section 7. Final conclusion is reported in Section 8.

We introduce a four parameter distribution named as Transmuted Exponentiated Moment Pareto distribution (TEMP distribution) with CDF as

and PDF

where α and k are positive shape parameters and | λ | < 1 is transmuted parameter of TEMP distribution.

Cumulative distribution function plot of TEMP distribution at different combinations of parameters α and λ for fixed k are given in

Some Special Cases

1) For λ = 0, α = 1, and k − 1 = β, the resulting distribution reduces to Pareto distribution.

2) For λ = 0, α = 1, the resulting distribution is Moment Pareto distribution discussed by Dara (8).

3) For k − 1 = β, α = 1, the distribution reduces to Transmuted Pareto distribution and was developed by Merovci and Puka [

TEMP distribution is developed on the basis that it provides more flexible results on highly right skewed datasets. Flexibility of TEMP distribution is assessed by comparing TEMP distribution with Pareto distribution and its related sub model (Transmuted Pareto distribution).

Survival or reliability function is used to measure the risk of occurrence of some event at a specific time. It is denoted by S(x).

Survival function S(x) of TEMP distribution is given as

Survival function of TEMP distribution (

Hazard function was introduced by Barlow et al. [

For TEMP distribution, hazard function H(x) is given by

The hazard function of TEMP distribution (

Summing up the hazard function from 0 to time (t) is considered as cumulative hazard function. It is denoted by H(t). Only continuous distributions are discussed under it. It is used to measure the overall number of failures that are added up to time t.

Cumulative hazard function is defined as

H ( x ) = − ln ( S (x))

for TEMP distribution it is described as

H ( x ) = − ln ( 1 − ( 1 + λ ) [ 1 − ( γ x ) k − 1 ] α + λ [ 1 − ( γ x ) k − 1 ] 2 α ) . (3.3)

The cumulative hazard function of TEMP distribution (

strictly increasing behavior for various combinations of parameters α and λ for fixed k.

From Equation (2.1) and Equation (3.1), reverse hazard rate function of TEMP distribution is

h r ( x ) = f ( x ) 1 − S (x)

From Equation (2.2) and Equation (3.1), mills ratio of TEMP distribution is

M ( x ) = S ( x ) f (x)

Symmetric graph of the function w.r.t the origin is said to be odd function.

For TEMP distribution it is defined as

By definition elasticity is defined as

e ( x ) = x f ( x ) F (x)

from Equation (2.1) and Equation (2.2), elasticity of TEMP distribution is written as

Moments are used to describe the mean, variance, skewness and kurtosis of the probability distribution and it is denoted by m_{1}, m_{2}, m_{3} and m_{4} respectively. Different categories of moments including Fractional, factorial, negative, incomplete, L, probability weighted and TL moments are having application in engineering, medicine, natural as well as social sciences.

The r-th moment about origin of TEMP distribution say μ ′ r is given by

μ ′ r = ∫ γ ∞ x r f ( x ) d x

μ ′ r = ∫ γ ∞ x r α ( k − 1 ) γ k − 1 x k [ 1 − ( γ x ) k − 1 ] α − 1 [ 1 + λ − 2 λ { 1 − ( γ x ) k − 1 } α ] d x .

Let

z = 1 − ( γ x ) k − 1 ⇒ x = γ z 1 1 − k ⇒ d x = − γ 1 − k z k 1 − k d z

limit x → γ ⇒ z → 1 and x → ∞ ⇒ z → 0.

Then

μ ′ r = α ( 1 + λ ) γ r ∫ 0 1 z r 1 − k ( 1 − z ) α − 1 d z + 2 α λ γ r ∫ 0 1 z r 1 − k ( 1 − z ) 2 α − 1 d z .

Simplification reduces μ ′ r

where A r = r 1 − k , B ( a , b ) = Betafunction and C r = γ r .

Fractional positive moments about the origin of r.v. X following TEMP distribution are given by

μ ′ m n = ∫ γ ∞ x m n f ( x ) d x

where A m n = m n ( 1 − k ) , B ( a , b ) = Betafunction and C m n = γ m n .

Fractional negative moments about the origin of r.v. X following TEMP distribution are given by

μ ′ ( − m n ) = ∫ γ ∞ x ( − m n ) f ( x ) d x

where A ( − m n ) = − m n ( 1 − k ) , B ( a , b ) = Betafunction and C ( − m n ) = γ ( − m n ) .

rth negative moments about the origin of r.v. X following TEMP distribution are given by

μ ′ − r = ∫ γ ∞ x − r f ( x ) d x

where A ( − r ) = − r ( 1 − k ) , B ( a , b ) = Betafunction and C ( − r ) = γ ( − r ) .

Factorial moments of TEMP distribution using Equation (2.2) is given by

E [ X ] n = ∑ r = γ n φ r μ ′ r

where A r = r 1 − k , B ( a , b ) = Betafunction , C r = γ r , [ X ] i = X ( X + 1 ) ( X + 2 ) ⋯ ( X + i − 1 ) and φ r is the Stirling number of first kind.

Moment generating function (mgf) of r.v. X following TEMP distribution using Equation (4.1) is defined as

using expansion e t x = ∑ r = 1 ∞ ( t x ) r r ! , Equation (4.6.1) is written as

M x ( t ) = ∑ r = 1 ∞ ( t ) r r ! ∫ γ ∞ x r f ( x ) d x

using Equation (4.1), mgf of TEMP distribution is

where A r = r 1 − k , B ( a , b ) = Betafunction , C r = γ r .

The central moments of probability distribution are defined by recurrence relation

μ r = ∑ i = 0 r ( r i ) ( − 1 ) i ( μ ′ 1 ) i μ ′ r − i .

For TEMP distribution

where A r = r 1 − k , B ( a , b ) = Betafunction and C r = γ r .

The cumulants of a probability distribution are defined by the recurrence relation

K r = μ ′ r − ∑ i = 1 r − 1 ( r − 1 i − 1 ) K i μ ′ r − i

for TEMP distribution

where A r = r 1 − k , B ( a , b ) = Betafunction and C r = γ r .

Symmetry of a probability distribution is defined by skewness and it is denoted by β 1

β 1 = μ 3 2 μ 2 3

The measure β 1 of TEMP distribution is followed by

where A r = r 1 − k , B ( a , b ) = Betafunction and C r = γ r .

Kurtosis is used to check the spread / peaked of a probability distribution. Kurtosis of a probability distribution is determined by β 2

β 2 = μ 4 μ 2 2

Kurtosis of TEMP distribution is given by

where A r = r 1 − k , B ( a , b ) = Betafunction and C r = γ r .

In theory of statistics, the Mellin transformation is famous as a distribution of the product as well as quotient for independent r.v.’s. By definition the Mellin transformation is

M x ( m ) = E ( x m − 1 ) = ∫ γ ∞ x m − 1 f ( x ) d x

for TEMP distribution, from Equation (4.1)

where A m − 1 = m − 1 1 − k , B ( a , b ) = Betafunction and C m − 1 = γ m − 1 .

For TEMP distribution, lower incomplete moments are defined as

M r ( l ) = E X ≤ l ( x r ) = ∫ γ l x r f ( x ) d x

From Equation (4.1),

where A r = r 1 − k , B ( γ l ) k − 1 ( a , b ) = Betafunction and C r = γ r .

For TEMP distribution, upper incomplete moments are defined as

M r ( u ) = E X > u ( x r ) = ∫ u ∞ x r f ( x ) d x

M r ( u ) = ∫ γ ∞ x r f ( x ) d x − ∫ γ u x r f ( x ) d x

from Equation (4.1), replace Beta function by B ( γ u ) k − 1 , we get

where A r = r 1 − k , B ( γ u ) k − 1 ( a , b ) = Betafunction and C r = γ r .

Let residual life m n ( w ) = E [ ( X − w ) n / X > w ] = 1 S ( w ) ∫ w ∞ ( x − w ) s f ( x ) d x of X for TEMP distribution has n-th moment.

m n ( w ) = 1 S ( w ) ∑ s = 0 n ( n s ) ( − w ) n − s ∫ w ∞ x s f ( x ) d x

m n ( w ) = α 1 − F ( w ) ∑ s = 0 n ( n s ) ( – w ) n − s C r { [ ( 1 + λ ) B ( 1 + A r , α ) − 2 λ B ( 1 + A r , 2 α ) ] − [ ( 1 + λ ) B ( γ u ) k − 1 ( 1 + A r , α ) − 2 λ B ( γ u ) k − 1 ( 1 + A r , 2 α ) ] } . (4.12)

For life expectancy or mean residual life (MRL) function say m 1 ( w ) of TEMP distribution put n = 1 in Equation (4.12), we get

where A r = r 1 − k , B ( γ u ) k − 1 ( a , b ) = Betafunction and C r = γ r .

Let reverse residual life R n ( w ) = E [ ( w − X ) n / X ≤ w ] = 1 F ( w ) ∫ γ ∞ ( w − x ) n f ( x ) d x of X for TEMP distribution has n-th moment.

R n ( w ) = 1 F ( w ) ∑ t = 0 n ( n t ) ( − 1 ) t w n − t ∫ γ ∞ x t f ( x ) d x

For mean waiting time or mean inactivity time of TEMP distribution put n = 1 in Equation (4.13), we get

where A r = r 1 − k , B ( γ w ) k − 1 ( a , b ) = Betafunction and C r = γ r .

Reliability of a system is tested by order statistic. The random sample provides important information like smallest value to largest value. To maintain the highest temperature of a medicine or lowest temperature of areas are the examples studied by order statistic to overcome the crisis or disasters in case of emergency.

Let X 1 , X 2 , X 3 , ⋯ , X m _{ }be a random sample follows to TEMP distribution and _{(}_{1)} and X_{(k) }represent the smallest and k-th smallest value follows to { X ( 1 ) , X ( 2 ) , X ( 3 ) , ⋯ , X ( m ) } respectively. The r.v.s X ( 1 ) , X ( 2 ) , X ( 3 ) , ⋯ , X ( m ) are called order statistic.

Order statistic for pdf of X_{(}_{i}_{)} is defined as

f x ( i ) ( x ) = m ! ( i − 1 ) ! ( m − i ) ! [ F ( x ) ] i − 1 [ 1 − F ( x ) ] m − i f (x)

for TEMP distribution, order statistic for pdf of X_{(i)} is

f x ( i ) ( x ) = m ! ( i − 1 ) ! ( m − i ) ! [ ( 1 + λ ) [ 1 − ( γ x ) k − 1 ] α − λ [ 1 − ( γ x ) k − 1 ] 2 α ] i − 1 ⋅ [ 1 − ( 1 + λ ) [ 1 − ( γ x ) k − 1 ] α + λ [ 1 − ( γ x ) k − 1 ] 2 α ] m − i ⋅ α ( k − 1 ) γ k − 1 x k [ 1 − ( γ x ) k − 1 ] α − 1 [ 1 + λ − 2 λ { 1 − ( γ x ) k − 1 } α ]

order statistic of TEMP distribution in reduced form

for TEMP distribution, largest order or m-th order statistic pdf X_{(m)} is given by

and first order or smallest order statistic pdf X_{(1)} for TEMP distribution, is given by

From Equation (4.15), r-th moment of order statistic for TEMP distribution in simplified and reduced form is given by

where

A = m ! α ( k − 1 ) γ k − 1 ( i − 1 ) ! ( m − i ) ! , C = m ! α γ r ( i − 1 ) ! ( m − i ) ! ,

E = ( − 1 ) j + l + p ( i − 1 j ) ( m − i l ) ( l p ) ( 1 + λ ) l − p − j λ j + p .

Statistical significance is assessed by the quantile function of the observations for known distribution. It is defined by inverting the CDF under consideration. When information about the data set is quantitatively reviewed or analyzed by the summary statistics, it is called descriptive statistics.

The q^{th} quantile function of TEMP distribution is

Median of a distribution is x q for q = 0.5. For TEMP distribution we put q = 0.5 in Equation (5.1), we get

To generate random numbers, we suppose that CDF of TEMP distribution follows uniform distribution u = U (0, 1).

Random numbers of TEMP distribution is calculated by

Coefficient of variation is defined as the quotient of standard deviation (SD) to mean.

CV = SD Mean

Coefficient of variation of TEMP distribution is

where A r = r 1 − k , B ( a , b ) = Betafunction and C r = γ r .

From Equation (3.4.1) set r = − 1 , we get harmonic mean of TEMP distribution

H M = α γ [ ( 1 + λ ) B ( k k − 1 , α ) − 2 λ B ( k k − 1 , 2 α ) ] . (5.5)

Degree of disorder or randomness in a system or our lack of information about it is defined as Entropy. In information theory, the Rényi entropy generalized Hartley entropy, Shannon entropy, Collision and min entropy. Entropies quantify the diversity, uncertainty or randomness of a system.

Rényi [

I δ ( X ) = 1 δ − 1 log ∫ 0 ∞ f δ ( x ) d x for δ > 0 and δ ≠ 1.

From Equation (2.2), the reduced form of Rényi entropy of TEMP distribution is given by

I δ ( X ) = 1 δ − 1 log [ D ∑ i = 0 ∞ [ ( δ i ) ( − 1 ) i A i B ( C + 1 , E ) ] ] . (5.7)

where A = 2 λ 1 + λ , B ( a , b ) = Betafunction , C = i α + δ ( α − 1 ) , D = γ ( 1 − δ ) k − 1 [ α ( 1 + λ ) ( k − 1 ) ] δ and E = 1 − k ( δ + 1 ) k − 1 .

The PDF of “n” mixture of TEMP distribution is followed by f ( x ) = ∑ i = 1 n p i f ( x ) , where ∑ i = 1 n p i = 1 and f i ( x ) for TEMP distribution from Equation (2.2) is

defined as

f i ( x ) = α i ( k i − 1 ) γ k i − 1 x k i [ 1 − ( γ i x ) k i − 1 ] α i − 1 [ 1 + λ i − 2 λ i { 1 − ( γ i x ) k i − 1 } α i ] .

For n = 2, mixture form of TEMP distribution is given by

f ( x ) = p 1 α 1 ( k 1 − 1 ) γ k 1 − 1 x k 1 [ 1 − ( γ x ) k 1 − 1 ] α 1 − 1 [ 1 + λ 1 − 2 λ 1 { 1 − ( γ x ) k 1 − 1 } α 1 ] + p 2 α 2 ( k 2 − 1 ) γ k 2 − 1 x k 2 [ 1 − ( γ x ) k 2 − 1 ] α 2 − 1 [ 1 + λ 2 − 2 λ 2 { 1 − ( γ x ) k 2 − 1 } α 2 ] .

For n = 3, mixture form of TEMP distribution is given by

f ( x ) = p 1 α 1 ( k 1 − 1 ) γ k 1 − 1 x k 1 [ 1 − ( γ x ) k 1 − 1 ] α 1 − 1 [ 1 + λ 1 − 2 λ 1 { 1 − ( γ x ) k 1 − 1 } α 1 ] + p 2 α 2 ( k 2 − 1 ) γ k 2 − 1 x k 2 [ 1 − ( γ x ) k 2 − 1 ] α 2 − 1 [ 1 + λ 2 − 2 λ 2 { 1 − ( γ x ) k 2 − 1 } α 2 ] + p 3 α 3 ( k 3 − 1 ) γ k 3 − 1 x k 3 [ 1 − ( γ x ) k 3 − 1 ] α 3 − 1 [ 1 + λ 3 − 2 λ 3 { 1 − ( γ x ) k 3 − 1 } α 3 ] . (5.8)

From Equation (4.1), r-th moment of mixture form of TEMP distribution is written as E ( X r ) = ∑ i = 1 n p i μ ′ r

E ( X r ) = ∑ i = 1 n p i α C r [ ( 1 + λ ) B ( 1 + A r , α ) − 2 λ B ( 1 + A r , 2 α ) ] (5.9)

where A r = r 1 − k , B ( a , b ) = Betafunction and C r = γ r .

In order to assess the behavior of estimates derived by the method of MLE from TEMP distribution, a small scaled experiment is carried out based on simulations study. Performance of MLE is evaluated on the basis of mean square errors (MSEs). For this we generate size n = 100, 200, 300, 400 and 500 samples from Equation (5.3) and results are achieved by 1000 simulations. Statistical software R is used to develop the empirical results.

Parameters of Transmuted Exponentiated Moment Pareto distribution are calculated using the method of MLE by incorporating R package (statistical software).

Log likelihood function of TEMP distribution under Equation (2.2) is stated as

Parameters | n = 25 | n = 100 | n = 200 | n = 300 | n = 400 | n = 500 |
---|---|---|---|---|---|---|

k ^ | 1.6222 (0.1702) | 1.55498 (0.0923) | 1.5443 (0.0556) | 1.5256 (0.0702) | 1.4995 (0.0444) | 1.4861 (0.0425) |

α ^ | 0.4794 (0.1492) | 0.4491 (0.1464) | 0.4709 (0.1027) | 0.6100 (0.0968) | 0.5676 (0.0809) | 0.5800 (0.0775) |

λ ^ | −0.7515 (0.2779) | −0.3283 (0.5442) | −0.5418 (0.2915) | −0.0662 (0.3622) | −0.2680 (0.2680) | −0.1773 (0.2671) |

Parameters | n = 25 | n = 100 | n = 200 | n = 300 | n = 400 | n = 500 |
---|---|---|---|---|---|---|

k ^ | 2.2491 (0.3142) | 2.7255 (0.2854) | 2.7522 (0.1678) | 2.4955 (0.2629) | 2.4734 (0.2173) | 2.3427 (0.3332) |

α ^ | 1.0669 (0.5538) | 1.5308 (0.3542) | 1.3268 (0.3525) | 1.6509 (0.1295) | 1.5801 (0.1264) | 1.5969 (0.0955) |

λ ^ | −0.3426 (0.8251) | −0.1197 (0.5371) | −0.3519 (0.4729) | 0.3599 (0.3378) | 0.2073 (0.3389) | 0.3857 (0.4775) |

Descriptive measures | n = 25 | n = 100 | n = 200 | n = 300 | n = 400 | n = 500 |
---|---|---|---|---|---|---|

μ ′ 1 | 0.1118 | 0.1033 | 0.1007 | 0.1003 | 0.1004 | 0.1002 |

μ ′ 2 | 0.0128 | 0.0107 | 0.0101 | 0.0101 | 0.0101 | 0.0100 |

μ ′ 3 | 0.0015 | 0.0011 | 0.0010 | 0.0010 | 0.0010 | 0.0010 |

μ ′ 4 | 0.0002 | 0.0001 | 0.0001 | 0.0001 | 0.0001 | 0.0001 |

Skewness | 2.3481 | 4.57777 | 9.9338 | 15.0006 | 12.662 | 22.2935 |

Kurtosis | 8.4551 | 24.8207 | 110.2899 | 240.6803 | 190.6012 | 498.0018 |

CV% | 49.5936 | 31.0745 | 16.7523 | 9.6611 | 12.7474 | 4.4775 |

AIC | 23.5142 | 2.6587 | 175.2641 | 223.5484 | 463.4593 | 624.4094 |

-Log-likelihood | 8.7571 | 1.7606 | 84.6321 | 108.7742 | 228.7297 | 309.2047 |

Descriptive measures | n = 25 | n = 100 | n = 200 | n = 300 | n = 400 | n = 500 |
---|---|---|---|---|---|---|

μ ′ 1 | 0.1956 | 0.1165 | 0.1057 | 0.1021 | 0.1035 | 0.1002 |

μ ′ 2 | 0.0924 | 0.0138 | 0.0112 | 0.0104 | 0.0107 | 0.0100 |

μ ′ 3 | − | 0.0016 | 0.0011 | 0.0011 | 0.0011 | 0.0010 |

μ ′ 4 | − | 0.0002 | 0.0001 | 0.0001 | 0.0001 | 0.0001 |

CV% | 118.1114 | 116.4815 | 96.3946 | 69.7225 | 67.2829 | 14.2311 |

Skewness | 1.1732 | 2.4445 | 3.9651 | 6.5704 | 5.3503 | 21.55994 |

Kurtosis | 3.3210 | 9.4564 | 23.3555 | 59.1233 | 37.2802 | 476.9756 |

AIC | −13.0174 | −154.2575 | −311.995 | −500.0787 | −577.9877 | −701.4997 |

-Log-likelihood | 9.5087 | 80.1287 | 158.995 | 253.0393 | 291.9938 | 353.7499 |

L L = n [ ( k − 1 ) ln γ + ln α + ln ( k − 1 ) ] − k ∑ i = 1 n ln x i + ( α − 1 ) ∑ i = 1 n ln [ 1 − ( γ x ) k − 1 ] + ∑ i = 1 n ln [ ( 1 + λ ) − 2 λ { 1 − ( γ x ) k − 1 } α ] . (7.1.1)

Partial derivatives of Equation (7.1.1) w.r.t the parameters k, α and λ are calculated and equating to zero we get.

∂ ∂ k ( L L ) = n ln γ + n k − 1 − ∑ i = 1 n ln x i − ∑ i = 1 n [ ( α − 1 ) ( γ x ) k − 1 ln ( γ x ) 1 − ( γ x ) k − 1 ] + ∑ i = 1 n [ 2 α λ [ 1 − ( γ x ) k − 1 ] α − 1 [ ( γ x ) k − 1 ln ( γ x ) ] ( 1 + λ ) − 2 λ [ 1 − ( γ x ) k − 1 ] α ] = 0 (7.1.2)

∂ ∂ α ( L L ) = n α + ∑ i = 1 n ln [ 1 − ( γ x ) k − 1 ] − 2 λ ∑ i = 1 n [ [ 1 − ( γ x ) ( k − 1 ) ] α − 1 ln [ 1 − ( γ x ) k − 1 ] ( 1 + λ ) − 2 λ [ 1 − ( γ x ) k − 1 ] α ] = 0 (7.1.3)

∂ ∂ λ ( L L ) = ∑ i = 1 n [ 1 − 2 [ 1 − ( γ x ) k − 1 ] α − 1 ( 1 + λ ) − 2 λ [ 1 − ( γ x ) k − 1 ] α ] = 0. (7.1.4)

Since γ is the initial point of PDF, as a minimum possible value of sample is the estimate of γ. Solution of simultaneous Equations (7.1.2)-(7.1.4) gives us MLE estimates of TEMP distribution. We solve these non linear equations by using R package.

Fisher Information matrix K ( φ ) of order 3 × 3 is required for hypothesis test and interval estimation. K ( φ ) is described as

K ( φ ) = [ ∂ 2 L ∂ k 2 ∂ 2 L ∂ k ∂ α ∂ 2 L ∂ α 2 ∂ 2 L ∂ λ ∂ α ∂ 2 L ∂ λ ∂ k ∂ 2 L ∂ λ 2 ] . (7.1.5)

To show that Transmuted Exponentiated Moment Pareto (TEMP) distribution is better than its sub-models Transmuted Pareto (TP) and Pareto (P) distributions, authors consider four data sets. In R, package Adequacy Model and method BFGS is used to derive the estimates.

Choulakian and Stephens [^{3}/s) of the Wheaton River in Canada. This data set is also discussed by Merovci and Puka [

Remission times (in months) of bladder cancer 128 patients sample is discussed by Lee and Wang [

Barlow et al. [

Ghitany et al. [

Models | Coefficients (Standard Error) | Information Criterion | |||||||
---|---|---|---|---|---|---|---|---|---|

k | α | λ | -LL | AIC | BIC | W | A | K-S | |

TEMP | 1.47 (0.05) | 1.88 (0.33) | −0.94 (0.06) | 280.67 | 567.35 | 574.19 | 0.73 | 4.52 | 0.19 |

TP | − | 0.35 (0.03) | −0.95 (0.05) | 286.20 | 576.40 | 580.95 | 0.72 | 4.49 | 0.23 |

PD | − | 0.24 (0.03) | − | 303.07 | 608.13 | 610.41 | 0.92 | 5.69 | 0.33 |

Models | Coefficients (Standard Error) | Information Criterion | |||||||
---|---|---|---|---|---|---|---|---|---|

k | α | λ | -LL | AIC | BIC | W | A | K-S | |

TEMP | 1.51 (0.04) | 2.26 (0.32) | −0.95 (0.05) | 452.02 | 910.04 | 918.60 | 1.59 | 8.63 | 0.21 |

TP | − | 0.35 (0.02) | −0.97 (0.03) | 466.99 | 937.99 | 943.70 | 1.53 | 8.32 | 0.29 |

PD | − | 0.24 (0.02) | − | 499.61 | 1001.22 | 1004.07 | 1.81 | 9.99 | 0.36 |

Models | Coefficients (Standard Error) | Information Criterion | |||||||
---|---|---|---|---|---|---|---|---|---|

k | α | λ | -LL | AIC | BIC | W | A | K-S | |

TEMP | 1.42 (0.04) | 1.43 (0.21) | −0.90 (0.07) | 151.07 | 308.14 | 315.98 | 1.76 | 9.67 | 0.22 |

TP | − | 0.36 (0.03) | −0.93 (0.05) | 153.88 | 311.76 | 316.99 | 1.77 | 9.67 | 0.25 |

PD | − | 0.25 (0.03) | − | 174.40 | 350.80 | 353.42 | 2.07 | 11.35 |

Models | Coefficients (Standard Error) | Information Criterion | |||||||
---|---|---|---|---|---|---|---|---|---|

k | α | λ | -LL | AIC | BIC | W | A | K-S | |

TEMP | 1.75 (0.07) | 1.42 (0.21) | −0.92 (0.05) | 358.01 | 722.03 | 729.85 | 1.37 | 8.28 | 0.22 |

TP | − | 0.63 (0.05) | −0.93 (0.05) | 360.86 | 725.73 | 730.94 | 1.37 | 8.25 | 0.26 |

PD | − | 0.45 (0.05) | − | 382.95 | 722.03 | 729.85 | 1.37 | 8.28 | 0.35 |

receives service in a bank on 100 observations (

In this article, authors have developed a new four parameter model named Transmuted Exponentiated Moment Pareto (TEMP) distribution. Numerous mathematical properties of TEMP distribution are discussed. TEMP distribution is modeled by four suitable lifetime data sets. Authors calculate the values of -LL and information criterion (AIC, BIC, A, W, K-S) on data set 1 to 4. TEMP

distribution is compared with its sub-models. Based on the minimum value of -LL and information criterion it is concluded that TEMP distribution is most favorable fit distribution as compared to its sub-models Transmuted Pareto (TP) and Pareto distribution.

In future numerous properties of Bayesian analysis of TEMP distribution will be studied.

The authors declare no conflicts of interest regarding the publication of this paper.

Arshad, M.Z., Iqbal, M.Z. and Ahmad, M. (2018) Transmuted Exponentiated Moment Pareto Distribution. Open Journal of Statistics, 8, 939-961. https://doi.org/10.4236/ojs.2018.86063