For the MLTIL distribution the quantile function, say x = Q_MLTIL(q), can be obtained by inverting (6) as

(12) F-1(q)=QMLTIL(q)=θ(λ-1λ(1-λ-q))1α-1.

We can easily generate X by taking q as a uniform random variable in (0,1). Based on (12), one can compute the measures of skewness (Sk) and kurtosis (Ku) as follows: Sk=Q(34)+Q(14)-2Q(12)Q(34)-Q(14) and Ku=Q(78)-Q(58)+Q(38)-Q(18)Q(68)-Q(28), respectively, where Q(⋅) refers to the quantile function.

The nth moment of MLTIL distribution can be derives as μn′=E(Xn)=∫-∞∞xnf(x)dx=∑k=0∞ϑk∫0∞xnh(x;α(k+1),θ)dx=∑k=0∞ϑkα(k+1)θnB(1-n,α(k+1)+n),n=1,2,3,… where B(a,b) is the beta function, Γ(0)=-γ and Γ(-n)=(-1)nn!ϕ(n)-(-1)nn!γ for n=1,2,3,…,γ denotes Euler’s constant and ϕ(n)=∑j=1n1j. For more details see [13]. Similarly, for the MLTIL distribution we can obtain the nth inverse moment as follows: E(1Xn)=∑k=0∞ϑkB(1+n,α(k+1)-n)θn.

In particular, μ1′=E(X)=∑k=0∞ϑkα(k+1)θΓ(0), where γ denotes Euler’s constant. μ2′=E(X2)=∑k=0∞ϑkα(k+1)(α(k+1)+1)θ2Γ(-1) and var(X)=μ2′-(μ1′)2.

The rth central moment μr of X is derived as μn=E(X-μ1′)n=∑k=0n(-1)r-k(nk )μn-k′(μ1′)k.

The Sk and Ku measures can be computed using the following expressions: Sk=μ3′−3μ2′μ+2μ3(μ2′−μ2)32,Ku=μ4′−4μ3′μ+6μ2′μ2−3μ4(μ2′−μ2)2.

The following propositions are a description of three different types of moments such as incomplete moments, moment generating function (mgf) and conditional moment.

prop 3.1. If X~MLTIL, then the incomplete moments of X is given as follows:

(13) IX(t)=∑k=0∞ϑk∫−∞txnh(x;α(k+1),θ)dx            =∑k=0∞ϑk∫0txn(αθ(k+1)x−2(1+θx)−α(k+1)−1)dx            =αθ∑k=0∞ϑk(k+1)∫0txα(k+1)+n−1(θ+x)−α(k+1)−1dx            =∑k=0∞ϑkα(k+1)tn(tθ)α(k+1)α(k+1)+nF2:1(α(k+1)+1;α(k+1)+n;α(k+1)+n+1;−tθ),

where, F2:1(u;v;c;y)=1B(u,c-v)∫ 01zv-1(1-z)c-v-1(1-zy)udz is the hypergeometric function see [14]. See also [15–17].

The first incomplete moment, I₁(t), follows from Eq. (13) with n = 1.

prop 3.2. If X~MLTIL, then the mgf of X is MX(t)=∑j=0∞tjj!E(Xj)=∑j=0∞∑k=0∞ϑk(θt)jj!α(k+1)B(1-j,α(k+1)+j).

prop 3.3. If X~MLTIL, then the characteristic generating function (cgf) of X is MX(t)=∑j=0∞(it)jj!E(Xj)=∑j=0∞∑k=0∞ϑk(θit)jj!α(k+1)B(1-j,α(k+1)+j).

prop 3.4. If X~MLTIL, then the conditional moment of X is CX(t)=E(E(Xn∣X>x)=ψn(x)1−F(x), where ψn(x)=∑k=0∞ϖk∫x∞ynh(x;α(k+1),θ)dy=∑k=0∞ϑkα(k+1)θ(-θ)n-1B-θx(1-n,-α(k+1)), where, Bz(a,b)=∫ 0zya-1(1-y)b-1dy is the incomplete beta function.

Entropy has been used in areas like in physics (sparse kernel density estimation), medicin (molecular imaging of tumors) and engineering (measure the randomness of systems). The entropy is a measure of variation of the uncertainty. The Rényi entropy (RE) of order ρ is defined as

(14) IR(ρ)=11-ρlog(∫ fρ(x)dx).

For the MLTIL distribution in (1), we can write IR(ρ)=11−ρlog{ ((λ−1)αθlog(λ))ρ∫0∞(x−2(1+θx)−(α+1)λ−(λ−1)(1+θx)−α)ρdx }, ρ≥0, ρ≠1           =11−ρlog{ ((λ−1)αλθlog(λ))ρ∑k=0∞((λ−1)λ)kΓ(ρ+k)k!B(2ρ−1,α(k+ρ)−ρ+1). }.

The ρ-entropy, Hρ-R(x), can be written as

(15) Hρ-R(ρ)=11-ρlog(1-∫ 0∞fρ(x)dx).

Shannon’s entropy (SE) is defined as E[-logf(X)] and it is a special case of (15) for ρff1. Limiting ρff1 in (15) and using L´Hospital´s rule, one obtains E[−logf(X)]=E[ −log((λ−1)αθx−2(1+θx)−(α+1)log(λ)[ λ−(λ−1)(1+θx)−α ]) ]      =−log[ (λ−1)αθlog(λ) ]−E[ log(x−2(1+θx)−(α+1)λ−(λ−1)(1+θx)−α) ]      =log[ log(λ)αθ(λ−1) ]+2E(log(x))+(α+1)E(log(1+θx))      +E(log[ λ−(λ−1)(1+θx)−α ]).

Let X(1)<X(2),…,<X(n) be the order statistics of a random sample of size n taken from the MLTIL distribution. In this case, we can write the PDF of the sth order statistic as follows

(16) fX(s)(x)=f(x)B(s,n-s+1)∑k=0n-s(-1)k(n-sk )Fk+s-1(x),

where B(., .) is the beta function. Using (1), (6) and a power series expansion, we obtain

(17) f(x)Fk+s-1(x)=αθ(λ-1)(log(λ))k+sx-2(1+θx)-(α+1)λ-(λ-1)(1+θx)-α[log(λλ-(λ-1)(1+θx)-α)]k+s-1.

From (17) and (16), we can write

(18) fX(s)(x)=αθ(λ−1)(B(s,n−s+1)∑k=0n−s(−1)k(log(λ))k+s(n−sk)x−2(1+θx)−(α+1)(λ−(λ−1)(1+θx)−α)                   ×[ log(λλ−(λ−1)(1+θx)−α) ]k+s−1.

Particularly, we can obtain the PDF of the first and last order statistics from (18), respectively, as

(19) fX(1)(x)=αθ(λ−1)n∑k=0n−1(−1)k(log(λ))k+1(n−1[3pt]k)x−2(1+θx)−(α+1)(λ−(λ−1)(1+θx)−α)   ×[ log(λλ−(λ−1)(1+θx)−α) ]k,

and

(20) fX(n)(x)=αθ(λ-1)n(log(λ))nx-2(1+θx)-(α+1)(λ-(λ-1)(1+θx)-α)[log(λλ-(λ-1)(1+θx)-α)]n-1.

The (u,v)th probability weighted moments of the random variable X is defined by

(21) ρu,v=E{XuF(X)v}=∫ -∞∞xuf(x)Fv(X)dx.

Based on Eq. (21), we can obtain the probability weighted moments of the MLTIL distribution as

(22) ρu,v=αθ(λ-1)ρ(log(λ))v+1,

where

(23) ρ=∫ 0∞xu-2(1+θx)-(α+1)λ-(λ-1)(1+θx)-α[log(λλ-(λ-1)(1+θx)-α)]vdx.

The nth moment of the residual life (RL) of the random variable X is defined as follows Rn(t)=E((X-t)n∣X>t)=11-F(t)∫ t∞(x-t)nf(x)dx,n≥1

Based on Eq. (1) and applying the binomial expansion of (x − t)ⁿ, we have Rn(t)=11−F(t)∑m=0n(−t)n−m(nm)∫t∞xnf(x)dx            =αθ(λ−1)λ∑m=0n∑k=0∞(−t)n−m(−θ)n−1(n[3pt]m)(λ−1λ)kB−θt(1−n,−(k+1)α)log(λ−(λ−1)(1+θx)−α).

The nth moment of the reversed RL can be derived using the general formula mn(t)=E((t-X)n∣X≤t)=1F(t)∫ 0t(t-x)nf(x)dx,n≥1.

Based on Eq. (1) and applying the binomial expansion of (t − x)ⁿ, we can write mn(t)=αθ(λ-1)λlog(θ)F(t)∑m=0n(-t)n-m(nm )∫0txnf(x)dx=α(λ-1)λlog(θ)F(t)∑m=0n∑k=0∞(-1)n-m(nm )(λ-1λ)kt2n-m+α(k+1)θα(k+1)(n+α(k+1))×F2:1(a(k+1)+1;n+a(k+1);n+a(k+1)+1;-tθ).

Let Y1~MLTIL(λ1,α1,θ1) and Y2~MLTIL(λ2,α2,θ2). If Y₁ represents stress and Y₂ represents strength, then the stress-strength parameter, denoted by R, for the MLTIL distribution is given by R=P(Y2>Y1)=∫0∞F1(y)dF2(y)=1−∫0∞F¯1(y)dF2(y) =1−(λ2−1)α2θ2log(λ1)log(λ2)∫0∞y−2(1+θ2y)−(α2+1)log[ λ1−(λ1−1)(1+ θ 1x)−α1 ]λ2−(λ2−1)(1+θ2x)−α2dy =1−(λ2−1)α2θ2log(λ1)log(λ2)∫0∞y−2(1+θ2y)−(α2+1)[ log(λ1)+log(1−(1−λ1)(1+ θ 1 x)−α1) ]λ2−(λ2−1)(1+θ2x)−α2dy =1−(λ2−1)α2θ2log(λ2)∫0∞y−2(1+θ2y)−(α2+1)λ2−(λ2−1)(1+θ2x)−α2dy −(λ2−1)α2θ2log(λ1)log(λ2)∫0∞y−2(1+θ2y)−(α2+1)log(1−(1−λ1)(1+ θ 1 x)−α1)λ2−(λ2−1)(1+θ2x)−α2dy.

Using series expansion in the last equation, we obtain R=1−(λ2−1)α2θ2log(λ2)∑m=0∞(λ2−1λ2)m∫0∞y−2(1+θ2y)−α2(m+1)−1dy−(λ2−1)α2θ2log(λ1)log(λ2) ×∑m=0∞∑k=0∞(−1)k−1k(λ2−1λ2)m(1−λ1λ1)k∫0∞y−2(1+θ1y)−α1k(1+θ2y)−α2(m+1)−1dy =1−(λ2−1)α2θ2log(λ2)∑m=0∞(λ2−1λ2)m1λ2θ2(m+1)−(λ2−1)α2θ2log(λ1)log(λ2)∑m=0∞∑k=0∞∑j=0∞(−1)k−1(−θ1)jk ×(λ2−1λ2)m(1−λ1λ1)k(α1k+j−1j)∫0∞y−2−j(1+θ2y)−α2(m+1)−1dy =1−(λ2−1)α2θ2log(λ2)∑m=0∞(λ2−1λ2)m1λ2θ2(m+1)−(λ2−1)α2log(λ1)log(λ2)∑m=0∞∑k=0∞∑j=0∞(−1)k−1(−θ1θ2)jk ×(λ2−1λ2)m(1−λ1λ1)k(α1k+j−1j)B(j+1,α2(m+1)−j).

Using a random sample of size m taken from the MLTIL distribution, then based on (5) the log-likelihood function can be written as

(24) E1(ϖ)=mlog((λ-1)αθlog(λ))-2∑j=1m log(xj)-(α+1)∑j=1m log(1+θxj)-∑j=1m log[λ-(λ-1)(1+θxj)-α],

where ϖ=(λ,α,θ)T. The maximum likelihood estimates (MLEs) of λ, α, and θ, denoted by λ^ML, α^ML, and θ^ML, are the solution of the following three equations ∂E1(ϖ)∂λ=mλ-1-mλlog(λ)-∑j=1m1-(1+θxj)-αλ-(λ-1)(1+θxj)-α=0, ∂E1(ϖ)∂α=mα-∑j=1m log(1+θxj)-∑j=1m(λ-1)(1+θxj)-α log(1+θxj)λ-(λ-1)(1+θxj)-α=0 and ∂E1(ϖ)∂θ=mθ-(α+1)∑j=1m1xj(1+θxj)+α∑j=1m(λ-1)xj-1(1+θxj)-α-1λ-(λ-1)(1+θxj)-α=0.

It is observed that these equations cannot be solved analytically for λ, α, and θ, therefore to obtain λ^ML, αML, and θML one can use any numerical technique for this purpose.

Kao et al. [18,19] originally proposed the percentile method of estimation and recently used by many authors to obtain the parameters of some proposed distributions, see for example [20,21]. It is very useful when the distribution has a closed form quantile function. Since the MLTIL distribution has an explicit quantile function as in (12), then we can obtain the percentile estimates (PEs) of λ, α, and θ, denoted by λ^P, α^P, and θ^P, by minimizing

(25) E2(ϖ)=∑j=1m[x(j)-θ(λ-1λ(1-λ-qj))1α-1]2,

where x_(j) is the ordered observation of x_j, j=1,…,n and q_j = j/(m + 1). Instead of minimizing (25) with respect to λ, α, and θ, we can obtain λ^P, α^P, and θ^P by solving ∂E2(ϖ)∂λ=∑j=1m[x(j)-θ(λ-1λ(1-λ-qj))1α-1]Φ1j(ϖ)=0, ∂E2(ϖ)∂α=∑j=1m[x(j)-θ(λ-1λ(1-λ-qj))1α-1]Φ2j(ϖ)=0 and ∂E2(ϖ)∂θ=∑j=1m[x(j)-θ(λ-1λ(1-λ-qj))1α-1]Φ3j(ϖ)=0, where Φ1j(ϖ)=θ(1-λλvj)1/α-1[1-λ-1λ+qj(λ-1)λqj+1vj]αλvj[(1-λλvj)1/α-1]2, Φ2j(ϖ)=θ(1-λλvj)1/α log(1-λλvj)α2[(1-λλvj)1/α-1]2 and Φ3j(ϖ)=[(1-λλvj)1/α-1]-1, where vj=1λqj-1.