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 (Figure 3) shows the decreasing behavior on several combinations of parameters α and λ for fixed k.

Hazard function was introduced by Barlow et al. [18] . It is time dependent function. It is used to measure the failure rate of some components in a particular period of time x.

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

The hazard function of TEMP distribution (Figure 4) for various combinations of parameters for fixed k indicates the increasing trend at initial phase. Longer tail to right shows the decreasing behavior 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 (Figure 5) indicates

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

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), , replace Beta function by B ( γ l ) k − 1 , we get

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 be its arranged form where X₍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 .

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 [19] entropy is defined as

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 .

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

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.