The ELo distribution is defined by the cdf given by (4). We recall that, as introduced in [36], the LP distribution is defined by the following cdf: (6)FLP(x;α,λ)={1,x≥1,xα[1−λαlog⁡(x)],x∈(0,1),0,x≤0,with λ∈[0,1] and α>0. An overview of the LP distribution is as follows. The LP distribution is created by applying an original logarithmic weighted function to the cdf of the transmuted power (TP) distribution. It can be viewed as a straightforward modification of the log-Lindley distribution created in [37]. Its definition establishes a special stochastic ordering that includes power, transmuted power, and log-weighted power distributions. A large panel of decreasing and sharp (mesokurtic) increasing-decreasing shapes for the pdf and flexible bathtub shapes for the hrf are established. The LP distribution thus offers a statistical substitute for the TP distribution. This claim was also demonstrated practically; with the use of a standard estimation strategy, the LP model can offer a better fit than the TP model using real-life data sets.

The following result shows the link existing between the ELo and LP distribution.

Proposition 2.1. Let X be a random variable following the LP distribution. The random variable Y=(1/β)(X−1−1) with β>0 then follows the ELo distribution.

Proof. We proceed by using the cdfs of the involved random variables. Let us denote by FY(x) the cdf of Y. First, we notice that the domain of definition of Y is (0,+∞), implying that FY(x)=0 for x≤0. For x>0, we have (7)FY(x)=P(Y≤x)=P(1β(X−1−1)≤x)=P(X−1≤1+βx)=P(X≥(1+βx)−1)=1−FLP((1+βx)−1;α,λ)=1−(1+βx)−α[1+λαlog⁡(1+βx)].

We recognize the cdf of the ELo distribution, ending the proof of the proposition.

First, let us investigate the asymptotic behavior of the cdf of the ELo distribution presented in Eq. (4). We have (8)FELo(x;α,β,λ)=αβx(1−λ+αβλx)(1+o(1))→0asx→0and, for λ∈(0,1], (9)FELo(x;α,β,λ)=1−λαβ−αx−αlog⁡(x)(1+o(1))→1asx→+∞.

Since the asymptotic property 1−FELo(x;α,β,λ)=Cx−γ(1+o(1)) as x→+∞, where C>0 and γ>0, is not satisfied due to the presence of a logarithmic term log⁡(x), the ELo distribution has not a fat tail, contrary to the Lo distribution corresponding to the case λ=0. The rest of this part is devoted to the analysis of the pdf of the ELo distribution. First, it is specified by (10)fELo(x;α,β,λ)={αβ(1+βx)−α−1[1−λ+λαlog⁡(1+βx)],x>0,0,x≤0,

Let us examine the analytical behavior of this function, beginning with the asymptotic behavior. We have (11)limx→0+fELo(x;α,β,λ)=αβ(1−λ).

Thus, the parameter λ modulates the value of fELo(x;α,β,λ) at x=0; this value can be 0 for λ=1, which is not possible for the Lo distribution. For λ∈(0,1], we have (12)fELo(x;α,β,λ)=λα2β−αx−α−1log⁡(x)(1+o(1))→+∞asx→+∞.

As a result, the right tail of the ELo distribution decreases with a “logarihtmic-polynomial” decay, which is slightly slower than the right tail of the Lo distribution.

The following proposition studies the mode properties of the ELo distribution.

Proposition 2.2.

For λ∈[0,(1+α)/(1+2α)], fELo(x;α,β,λ) is decreasing.

In this case, the ELo distribution is unimodal.

Proof. For x∈(0,1), the first derivative of fELo(x;α,β,λ) can be expressed as (14)fELo′(x;α,β,λ)=−αβ2(1+βx)−α−2[α(1−2λ)+1−λ+λα(α+1)log⁡(1+βx)].

Therefore, for λ∈[0,(1+α)/(1+2α)], since the term in bracket is immediately positive, it is clear that fELo′(x;α,β,λ)<0, so fELo(x;α,β,λ) is decreasing. For λ∈[(1+α)/(1+2α),1], we have fELo′(x;α,β,λ)=0 if and only if x=x∗ as given in (13). This point is a maximum since fELo′(x;α,β,λ)>0 for x∈(0,x∗) and fELo′(x;α,β,λ)<0 for x∈(x∗,+∞). Thus x∗ is the mode of the ELo distribution, it is thus unimodal in this case.

It is clear that for any t>0, we have ∫0+∞etxfELo(x;α,β,λ)dx=+∞; the ELo distribution is heavy-tailed (but it is not fat-tailed, as previously shown).

Fig. 1 illustrates the mathematical result aboves by displaying several plots of fELo(x;α,β,λ).

As expected, fELo(x;α,β,λ) has decreasing or unimodal asymetrical shapes, with varying weights on the left tail and the right tail remaining more or less heavy depending on the values of the parameters. The functional possibilities of the Lo distribution are overpassed, as demonstrated in Fig. 2, where the unimodal shapes are totally abscent.

Now, consider the ratio function fELo(x;α,β,λ)/fLo(x;α,β) for a more in-depth comparison of the pdfs of the ELo and Lo distributions. We have (15)fELo(x;α,β,λ)fLo(x;α,β)=1−λ+λαlog⁡(1+βx).

Therefore, for λ∈(0,1], if x∈(0,(e1/α−1)/β), then we have fELo(x;α,β,λ)<fLo(x;α,β), and of x>(e1/α−1)/β, we have fELo(x;α,β,λ)>fLo(x;α,β). As a result, x∗=(e1/α−1)/β can be regarded as a transition point between the ELo and Lo distributions.

The hrf of the ELo distribution is defined by (16)hELo(x;α,β,λ)={αβ1−λ+λαlog⁡(1+βx)(1+βx)[1+λαlog⁡(1+βx)],x>0,0,x≤0,

The following limit holds: (17)limx→0+hELo(x;α,β,λ)=αβ(1−λ).

The remarks made on fELo(x;α,β,λ) when x→0+ are still valid here; the parameter λ has a non-negligible effect on the initial value of hELo(x;α,β,λ), can be equal to 0. We have (18)hELo(x;α,β,λ)=αβ1+βx(1+o(1))→0asx→+∞.

Thus, for all the configurations of the parameters, the hrf decays to 0 with a polynomial rate.

The shape behavior of the hrf is studied in the next proposition.

Proposition 2.3.

For λ∈[0,2/(1+4α+1)], hELo(x;α,β,λ) is decreasing.

In this last case, the shape of hELo(x;α,β,λ) is upside-down bathtub.

Proof. We have (20)hELo′(x;α,β,λ)=−αβ2P[αλlog⁡(1+βx)](1+βx)2[1+λαlog⁡(1+βx)]2.where (21)P(u)=u2+(2−λ)u−αλ2+1−λ,u>0.

Then the polynomial P(u) has two roots given by (22)u1=12(λ−2−λ4α+1),u2=12(λ−2+λ4α+1)and we have P(u)=(u−u1)(u−u2). Since λ∈[0,1], we always have u1<0. Therefore, if u2≤0, i.e., λ∈[0,2/(1+4α+1)], we have P(u)>0 for all u>0, implying that hELo′(x;α,β,λ)<0, so hELo(x;α,β,λ) is decreasing. If u2≥0, i.e., λ∈(2/(1+4α+1),1], u2 is a valid root for P(u), and thus hELo′(x∗;α,β,λ)=0 if and only if αλlog⁡(1+βx∗)=u2, so (23)x∗=1β(eu2/(αλ)−1),which can be expressed as (19). This point is a maximum since hELo′(x;α,β,λ)>0 for x∈(0,x∗) and hELo′(x;α,β,λ)<0 for x∈(x∗,+∞). Hence, the shape of hELo(x;α,β,λ) is upside-down bathtub. This ends the proof.

Fig. 3 provides various plots of hELo(x;α,β,λ) to demonstrate the above mathematical result.

The UBFR property is particularly illustrated in Fig. 3b. For modeling purposes in the analysis of financial, survival, and environmental data, this is a desired property of the heavy-tailed distribution. We recall that the UBR property is not a quality of the Lo distribution, as shown in Fig. 4.

Following the approach presented in [38], we now study certain stochastic ordering aspects of the ELo distribution.

Proposition 3.1. The following first-order stochastic (FOS) comparisons hold:

For α2≥α1, the ELo (α1,β,λ) distribution FOS dominates the ELo (α2,β,λ) distribution; it is understood that the dominance is strict for α2>α1, with equality if and only if α2=α1.

Proof. We proceed by studying the monotonicity of FELo(x;α,β,λ) according to the parameters.

For x>0, we have (24)∂∂αFELo(x;α,β,λ)=(1+βx)−αlog⁡(1+βx)[1−λ+λαlog⁡(1+βx)].

Since λ∈[0,1], we have ∂FELo(x;α,β,λ)/(∂α)≥0, so FELo(x;α,β,λ) is increasing with respect to α. Thus, for α2≥α1, we have FELo(x;α1,β,λ)≤FELo(x;α2,β,λ), meaning that the ELo(α1,β,λ) distribution FOS dominates the ELo(α2,β,λ) distribution.

For x>0, we have (25)∂∂βFELo(x;α,β,λ)=αx(1+βx)−α−1[1−λ+λαlog⁡(1+βx)].

Since λ∈[0,1], we have ∂FELo(x;α,β,λ)/(∂β)≥0, so FELo(x;α,β,λ) is increasing with respect to β. Thus, for β2≥β1, we have FELo(x;α,β1,λ)≤FELo(x;α,β2,λ), meaning that the ELo(α,β1,λ) distribution FOS dominates the ELo(α,β2,λ) distribution. An alternative proof can be made for this point by using Proposition 2.1.

For x>0, we have (26)∂∂λFELo(x;α,β,λ)=−(1+βx)−αlog⁡(1+βx)

It is clear that ∂FELo(x;α,β,λ)/(∂λ)≤0, so FELo(x;α,β,λ) is decreasing with respect to λ. Thus, for λ2≥λ1, we have FELo(x;α,β,λ2)≤FELo(x;α,β,λ1), meaning that the ELo(α,β,λ2) distribution FOS dominates the ELo(α,β,λ1) distribution. By taking λ2=λ and λ1=0, we conclude that the ELo(α,β,λ) distribution FOS dominates the Lo distribution with parameters α and β.

This ends the proof.

Thanks to Proposition 3.1, we now see how random values from the ELo distribution can be relatively located in comparison to other random values of the ELo distribution with different parameters. This could be useful in cases where the practitioner is unsure which ELo distribution to employ while dealing with data. For further details on stochastic dominance, we may refer to [39–40].

Because they can be used to explain statistical distribution properties, moment properties are important when specifying our probability distribution to work with. As a result, they help to describe the distribution. In this part, different types of moments of the ELo distribution are examined, along with their interpretation.

Hereafter, we designate by X a random variable with the ELo distribution. The following result suggests a clear and simple finite sum expression for the rth moment of X.

Proposition 3.2. For any integer r<α−2, the rth moment of X exists, and it is given by mr=E(Xr), where E stands for the expectation operator. It can be expressed by a finite sum as (27)mr=rβr∑k=0r−1(r−1k)(−1)r−1−kα−k−1[1+λαα−k−1].

Proof. Since the ELo distribution has the support (0,+∞), we have (28)mr=r∫0+∞xr−1[1−FELo(x;α,β,λ)]dx=r∫0+∞xr−1(1+βx)−α[1+λαlog⁡(1+βx)]dx.

By performing the change of variable x=(1/β)(y−1) and applying the standard binomial formula, by noticing that α−k−1≥α−r−2>0 for k=0,…,r−1, we obtain (29)E(Xr)=rβr∫1+∞(y−1)r−1y−α[1+λαlog⁡(y)]dy=rβr∑k=0r−1(r−1k)(−1)r−1−k∫1+∞yk−α[1+λαlog⁡(y)]dy=rβr∑k=0r−1(r−1k)(−1)r−1−k[1α−k−1+λα∫1+∞yk−αlog⁡(y)dy].

For the remaining integral term, upon the change of variable y=eu/(α−k−1), since ∫0+∞ue−udu=Γ(2)=1, we have (30)∫1+∞yk−αlog⁡(y)dy=1(α−k−1)2∫0+∞ue−udu=Γ(2)(α−k−1)2=1(α−k−1)2.

Therefore, by combining the above equalities, we get (31)mr=rβr∑k=0r−1(r−1k)(−1)r−1−kα−k−1[1+λαα−k−1].

The stated result is obtained, ending the proof.

It is worth noting from Proposition 3.2 that the ELo distribution does not admit moments of all orders.

In particular, according to Proposition 3.2, for α>4, the two first moments of X are given by (32)m1=1(α−1)β(1+λαα−1)and (33)m2=2β2[−1α−1(1+λαα−1)+1α−2(1+λαα−2)]=2(2α2λ+α2−3αλ−3α+2)β2(α−2)2(α−1)2.

Similarly, Proposition 3.2 allows the variance, moments skewness and kurtosis of X to be expressed in terms of α, β and λ, but they lack an especially interesting concise form.

Among the possible generalizations of the moments are the unconditional moments, which naturally appear in various survival measures and are more suitable for use in a practical setting with censored data. On this topic, we may refer to [41,42].

The next result expresses the rth unconditional moment of X at a fixed value t.

Proposition 3.3. For any integer r and fixed t>0, the rth unconditional moment of X at a fixed value t exists, and is given by mr(t)=E(Xr∣X≤t). Under the condition that r<α−2, it can be expressed as (34)mr(t)=11−(1+βt)−α[1+λαlog⁡(1+βt)]×{−tr(1+βt)−α[1+λαlog⁡(1+βt)]+rβr∑k=0r−1(r−1k)(−1)r−1−kα−k−1[1−(1+βt)k−α+1+λαγ(2,log⁡[(α−k−1)(1+βt)])α−k−1]}, where γ(a,x)=∫0xta−1e−tdt is the standard incomplete gamma function.

Proof. We have (35)mr(t)=1FELo(t;α,β,λ)∫0txrfELo(x;α,β,λ)dx.

For t>0, we have FELo(t;α,β,λ)=1−(1+βt)−α[1+λαlog⁡(1+βt)], and an integration by part yields (36)∫0txrfELo(x;α,β,λ)dx=−tr[1−FELo(t;α,β,λ)]+r∫0txr−1[1−FELo(x;α,β,λ)]dx=−tr(1+βt)−α[1+λαlog⁡(1+βt)]+r∫0txr−1(1+βx)−α[1+λαlog⁡(1+βx)]dx.

By performing the change of variable x=(1/β)(y−1) and applying the standard binomial formula, we get (37)∫0txr−1(1+βx)−α[1+λαlog⁡(1+βx)]dx=1βr∫11+βt(y−1)r−1y−α[1+λαlog⁡(y)]dy=1βr∑k=0r−1(r−1k)(−1)r−1−k∫11+βtyk−α[1+λαlog⁡(y)]dy=1βr∑k=0r−1(r−1k)(−1)r−1−k[1−(1+βt)k−α+1α−k−1+λα∫11+βtyk−αlog⁡(y)dy].

For the remaining integral term, by using the incomplete gamma function and doing the change of variable y=eu/(α−k−1), we obtain (38)∫11+βtyk−αlog⁡(y)dy=1(α−k−1)2∫0log⁡[(α−k−1)(1+βt)]ue−udu=γ(2,log⁡[(α−k−1)(1+βt)])(α−k−1)2.

Therefore, by combining the above equalities, we get (39)mr(t)=11−(1+βt)−α[1+λαlog⁡(1+βt)]×{−tr(1+βt)−α[1+λαlog⁡(1+βt)]+rβr∑k=0r−1(r−1k)(−1)r−1−kα−k−1[1−(1+βt)k−α+1+λαγ(2,log⁡[(α−k−1)(1+βt)])α−k−1]}.

The stated result is obtained, ending the proof of Proposition 3.3.

Based on the conditional moments, we can define the moments of the residual life of the ELo distribution, and the mean residual life (MRL) function in particular. We can refer to [43–45] to support the importance of this last function in various branches of probability and statistics.

Because the ELo distribution does not admit moments of all orders, its quantile properties are fascinating to investigate.

The next result is a closed-form expression for the quantile function (qf) of the ELo distribution.

Proposition 3.4. The qf of the ELo distribution is expressed as (40)QELo(u;α,β,λ)={1β[(1−u)−1/α−1]forλ=0,1β[{λu−1W[u−1λe−1/λ]}1/α−1]forλ∈(0,1],u∈(0,1), where W(x) denotes the Lambert function, i.e., satisfying the equation W(x)eW(x)=x.

Proof. The qf is readily defined as QELo(u;α,β,λ)=FELo−1(u;α,β,λ). Thus, we determine it via the following equivalences: For λ>0, we have (41)1−(1+βx)−α[1+λαlog⁡(1+βx)]=u⇔(1+βx)−α[1+λαlog⁡(1+βx)]=1−u⇔log⁡[(1+βx)α]=1λ[(1−u)(1+βx)α−1]⇔(1+βx)α=e−1/λe(1−u)(1+βx)α/λ⇔u−1λ(1+βx)αe(u−1)(1+βx)α/λ=u−1λe−1/λ⇔u−1λ(1+βx)α=W[u−1λe−1/λ]⇔x=1β[{λu−1W[u−1λe−1/λ]}1/α−1].

The desired expression is established, ending the proof of Proposition 3.4.

There are several interests in having a closed-form expression of the qf. First of all, the main quartiles of the ELo distribution can be exhibited. In particular, the median is obtained as (42)M=QELo(12;α,β,λ)={1β(21/α−1)forλ=0,1β[{−2λW[−12λe−1/λ]}1/α−1]forλ∈(0,1].

The qf and random values from the uniform distribution over [0, 1] can be used to generate random values from a random variable X following the ELo distribution.

Since the ELo distribution does not admit moments of all orders, quantile measures of skewness and kurtosis, as proposed in [46,47] respectively can be useful. They are respectively defined as (43)SGalton=QELo(6/8;α,β,λ)−2QELo(4/8;α,β,λ)+QELo(2/8;α,β,λ)QELo(6/8;α,β,λ)−QELo(2/8;α,β,λ),and (44)KMoors=QELo(7/8;α,β,λ)−QELo(5/8;α,β,λ)+QELo(3/8;α,β,λ)−QELo(1/8;α,β,λ)QELo(6/8;α,β,λ)−QELo(2/8;α,β,λ).

In our heavy-tailed distribution setting, we propose to focus on actuarial measures based on the qf. The first measure is the value at risk (VaR). In the setting of the ELo distribution, it is simply defined by (45)VaRq=QELo(q;α,β,λ),q∈(0,1).

The second mesure is the expected shortfall (ES) introduced by [48], and generally considered as a better measure than VaR. It is defined by (46)ESq=1q∫0qVaRxdx,q∈(0,1).

Fig. 5 presents the shape of these actual measures for various values of the parameters with respect to q.

According to Fig. 5, both VaRq and ESq are increasing and convex, with a more or less angular shape.

In addition to the above quantile material, advanced quantile modeling can be done using the qf. See [49] for further details.

Let x1,…,xn represent n independent observations from a random variable X following the ELo(α,β,λ) distribution. Then, based on the pdf specified by (10) and x=(x1,…,xn), the likelihood function is specified by (47)L(α,β,λ;x)=∏i=1nfELo(xi;α,β,λ)=∏i=1nαβ(1+βxi)−α−1[1−λ+λαlog⁡(1+βxi)].

Clearly, the ELo distribution does not belong to the exponential family form; the Pitman-Koopman theorem applies. The ML estimates (MLEs) of α and λ are given by (48)(α^,β^,λ^)=argmax(α,β,λ)L(α,β,λ;x).

Alternatively, they may be defined as (49)(α^,β^,λ^)=argmax(α,β,λ)ℓ(α,β,λ;x),where ℓ(α,λ;x) refers to the log-likelihood function given by (50)ℓ(α,β,λ;x)=log⁡[L(α,β,λ;x)]=nlog⁡(α)+nlog⁡(β)−(α+1)∑i=1nlog⁡(1+βxi)+∑i=1nlog⁡[1−λ+λαlog⁡(1+βxi)].

They can be obtained by solving the following non-linear equations with respect to α and λ (51)∂ℓ(α,β,λ;x)∂α=0,∂ℓ(α,β,λ;x)∂β=0,∂ℓ(α,β,λ;x)∂λ=0,where (52)∂ℓ(α,β,λ;x)∂α=nα−∑i=1nlog⁡(1+βxi)+∑i=1nλlog⁡(1+βxi)1−λ+λαlog⁡(1+βxi), (53)∂ℓ(α,β,λ;x)∂β=nβ−(α+1)∑i=1nxi1+βxi+∑i=1nλαxi(1+βxi)|1−λ+λαlog⁡(1+βxi)]and (54)∂ℓ(α,β,λ;x)∂λ=∑i=1nαlog⁡(1+βxi)−11−λ+λαlog⁡(1+βxi).

Unfortunately, because the above equations have no explicit solutions, any numerical approximation technique can be utilized to obtain them. The related standard errors (SEs) can be determined through the calculation of the estimated Fisher information matrix. To accomplish the above calculations, the AdequacyModel package in R−Statistical\ Computing\ Environment can be used.

The ML method has the advantage of ensuring interesting features for MLEs, such as asymptotic unbiasedness and normality. In particular, the asymptotic unbiasedness provides theoretical guarantees on the fact that, for n large enough, the MLEs must be close to the true unknown parameter values. However, there is no solid guarantee for small n. The asymptotic normality allows us to construct confidence intervals and statistical tests on the unknown parameters based on the normal or normal-transformaiton distribution [50]. Contains more information about these features. We can estimate all of the underlying functions of the ELo distribution using MLEs. In particular, an estimate of the cdf and pdf are given by F^ELo(x)=F(x;α^,β^,λ^) and f^ELo(x)=f(x;α^,β^,λ^), respectively. With the same substitution approach, one can estimate various measures, such as the actuarial measures as defined in (45) and (46).

The above methodology is for complete samples. Other types of samples, such as censored data samples, can be investigated with appropriate modifications to the definition of the likelihood function. On this topic, we may refer the reader to [7,8,11,51].

The rest of this section is devoted to simulated tests proving the nice behavior of the MLEs.

We carry out a Monte Carlo simulation study in order to underline the accuracy of the MLEs parameters of the ELo distribution. To this end, N=600 samples are considered at varying sample sizes, chosen as: n=20,40,…,600 for the following parameter value scenarios: Set 1: (α=1,β=1.5,λ=0.2), Set 2: (α=1.5,β=0.5,λ=0.2), and Set 3: (α=2,β=0.5,λ=0.1). The average MLEs are calculated, as well as the bias and the mean squared error (MSE) defined by (55)Bias(θ^)=∑i=1600θi^600−θ,MSE(θ^)=∑i=1600(θi^−θ)2600,respectively, where θ∈{α,β,λ}, and the index i refers to the ith sample for the given sample size. The detailed summary of the simulation is depicted in Tables 1–3 for Sets 1, 2 and 3, respectively. For a visual approach, the corresponding plots are given in Figs. 6–8, respectively, with n in absisse.