The U Family of Distributions: Properties and Applications

2.1 Definition

The following result is at the basis of the newly defined family.

2.2 Discussion

As suggested by its name, the U family unifies a plethora of existing distributions defined with a polynomial-exponential function as main term. For instance, when a = 0 and b = +∞, it includes various extensions or generalizations of the Weibull, linear failure rate and Lindley distributions. Some of them are presented below.

When θ = 0 (so ϕ plays no role) and α = 1, the distribution defined by F(x) with R(x) = λx corresponds to the exponential distribution, with R(x) = λx^k, it corresponds to the Weibull distribution, with R(x) = λx + βx², we obtain the linear failure rate distribution (see [6]), with R(x) = λx^ke^−ϵ/x, it becomes the new extended Weibull distribution (see [19]), with R(x) = λx + βx^k, it corresponds to the MW distribution (see [24]), with R(x) = λx^k + βx^γ provided k > γ, we get the additive Weibull distribution (see [27]), with R(x) = λ(e^xk −1), it corresponds to the Chen distribution, (see [9]) and with R(x) = (1 + λx^k)^1/γ − 1, it corresponds to the generalized Weibull distribution (see [18]). When θ = 0 and α > 0, the distribution defined by F(x) with R(x) =λx corresponds to the two-parameter generalized exponential distribution (see [12]), with R(x) = λx^k, we get the exponentiated Weibull distribution (see [17]), with R(x) = λx + βx², it corresponds to the generalized linear failure rate distribution (see [23]) and with R(x) = λx^ke^γx, we obtain the GMW distribution (see [8]). Also, when θ = 1 and α = 1, the distribution defined by F(x) with R(x) = λx and ϕ = 1/(λ + 1) corresponds to the Lindley distribution (see [16]), and with R(x) = λx^k and ϕ = 1/(λ + 1), it becomes the power Lindley distribution (see [11]). As a last example, when θ = 1 and α > 0, the distribution defined by F(x) with R(x) = λx^k and ϕ = 1/(λ + 1) corresponds to the exponentiated power Lindley distribution (see [4]).

The list of the referenced distributions belonging to the U family becomes endless if various values for a and b are considered, such as the distributions defined on a bounded interval or the real line, i.e., a = −∞ and b = +∞ (including various symmetric and skewed distributions…).

Also, the U family can be expressed by considering a baseline cdf denoted by G(x), with corresponding pdf denoted by g(x). Indeed, one can remark that the function R(x) = −log(1 − G(x)), corresponding to the cumulative hrf associated to G(x), satisfies all the conditions of Proposition 2.1. This choice of function is also motivated by [2] in the context of the TX family. Hence, by substituting this expression of R(x) into (2.1), we get

The corresponding pdf is obtained as

and f(x) = 0 for x ∉ [a, b].

Also, the corresponding hrf is given by

and h(x) = 0 for x ∉ [a, b]. The qf of the U family, say Q(y), is involved in the following nonlinear equation: F[Q(y)] = y, i.e., if θϕ = 0, G[Q(y)] = y^1/α and, if θ_ϕ ≠ 0,

To the best of our knowledge, under this quite standard formulation involving a baseline distribution, the U family has no analogue in the literature. Even though some special members are well known, the general definition of the U family opens up unexplored horizons that deserve full attention. In this study, with the utopian idea to find the ultimate flexible lifetime distribution, we use the U family to propose a new promising extension of the Weibull distribution.

3.1 Definition

After preliminary investigations, a special member of the U family stands up in several desirable aspects. It constitutes a new four-parameters lifetime distribution defined by the cdf given as (2.2) with a = 0, b = +∞, θ = 1, ϕ = 1 and R(x) = λx + βx^k, i.e.,

and F(x) = 0 for x < 0. Here, it is assumed that λ + β > 0, allowing λ = 0 if β > 0 and vice-versa, and k > 0. The definition of F(x) uses the same configuration of the MW distribution, with the activation of the parameters θ, ϕ and a such as θ = 1, ϕ = 1 and a > 0. For the purposes of this paper, the corresponding distribution is naturally called the extended MW (EMW) distribution and denoted by EMW (a, λ, β, k) when the parameters need to be specified. Another point of view is to remark that F(x) is defined as (2.2) with the cdf of the MW distribution, i.e., G(x) = 1−e^{−(λx+βxk)} for x ≥ 0, and G(x) = 0 for x < 0, and θ = 1, ϕ = 1 and α > 0.

The remainder of the study presents the main motivations and interests of the EMW distribution, both from a theoretical side and from an applied perspective.

3.2. On the pdf and hrf

First of all, the pdf of the EMW distribution is derived from the differentiation of the F(x) according to x; it is thus given by

and f(x) = 0 for x < 0.

The corresponding hrf is given by

and h(x) = 0 for x < 0.

The corresponding qf satisfies the following equation:

Due to their analytical complexity, the shape properties of f(x) and h(x) are complicated to analyse in full generality. Figure 1 helps in this regard by plotting these functions for different values of the parameters α, β, λ and k. From Figure 1 (i), we see that the pdf may be almost symmetric, left skewed, right skewed and decreasing with reverse J shape. Also, from Figure 1 (ii), increasing, decreasing, increasing-decreasing, constant, upside-down bathtub and bathtub shapes for the hrf are observed. This panel of shapes is not obtained for the former MW distribution as attested by [23: Figure 2]. These plots reveal the high level of flexibility of these functions, which constitutes a great advantage for the EMW distribution; it can be used as an appropriate statistical model for a great diversity of data sets.

FIGURE 1.

Some plots of the (i) pdf curve and (ii) hrf curve of the EMW distribution.

FIGURE 2.

Plots of the average biases of the estimates for the selected parameter values.

3.3 Some properties

Here, we present some important properties of the EMW distribution.

4.1 Estimation method

For the statistical purposes of the EMW model, the estimation of unknown parameters is an essential task. In this regard, several methods exist in the literature, but the maximum likelihood estimation method remains the most commonly employed. The resulting estimators, called maximum likelihood estimators (MLEs), enjoy desirable properties. In particular, the normal approximation for the MLEs in large sample distribution theory is easily handled either analytically or numerically, allowing to construct asymptotic confidence intervals and test statistics. In this section, we investigate the MLEs of the parameters of the new distribution, provide a simulation study to validate their numerical efficiency and specify four goodness-of-fit statistics to compare the fitted pdfs to any data set. Given n observed values x₁, x₂, …, x_n from a random variable following the EMW(α, λ, β, k) distribution, the MLEs of the parameters in θ = (α, λ, β, k) are given by θ^=argmaxθ L(θ), where θ^=(α^,λ^,β^,k^) and L(θ) denotes the likelihood function defined by

In our context, a more tractable framework can be considered; the MLEs are also defined as θ^=argmaxθ ℓ(θ), where ℓ(θ) denotes the log-likelihood function defined by ℓ(θ) = log[L(θ)], i.e.,

Now, consider the first partial derivatives of 𝓁(θ) given by

By solving the equations of the system ∂𝓁(θ)/∂θ= 0₄ simultaneously, we get the MLEs. One can notice that, thanks to the first equation, the MLEs satisfy the following relation:

Some numerical iterative techniques may be used for evaluating the MLEs; the global maxima of the log-likelihood can be investigated by setting different starting values for the parameters.

Under standard regularity conditions, the asymptotic distribution behind the random estimator version of θ^−θ is the multivariate normal N₄(0₄, I(θ)⁻¹), where I(θ) = E₄[J(θ)] is the expected Fisher information matrix. The approximate multivariate normal distribution given by N4(04,J(θ^)−1), where J(θ^)−1 is the inverse of the observed Fisher information matrix evaluated at θ=θ^, can be used to set up approximate confidence intervals for the model parameters. The elements of J(θ) can be obtained from the authors upon request.

4.2 A simulation study

We now perform a simulation study to investigate the numerical performance of the MLEs of the EMW model parameters. In this regard, we generate N = 3000 samples of sizes n = 50, 150 and 300 from the EMW(α, λ, β, k) distribution with various target values for α, λ, β and k, and we determine the average estimates and mean square errors (MSEs) of the parameters. The results of the simulation are reported in Table 2. We use the R program for the random generation and specially, the optim-CG routine to obtain the MLEs.

From this table, as expected, we observe that the average estimates approach the corresponding target values as n increases. For n = 300 only, these estimates are almost equal to the corresponding target values. Also, when n increases, we see that the MSEs decrease to zero, with very small values for n = 300.

We complete the previous numerical study by graphical work. We generate N = 3000 samples of sizes n = 10 to 120 from the EMW distribution with target parameters values α = 5, β = 1.2, λ = 0.2 and k = 1.2 and plot the curves of the average biases for the estimates in Figure 2 and the curves of the MSEs for the estimates in Figure 3.

FIGURE 3.

Plots of the MSEs of the MLEs for the selected parameter values.

From these figures, it is clear that the average biases and MSEs decrease as n increases; all the curves tend quickly to the axis y = 0. Also, one can remark that n = 80 seems enough to obtain a precise estimation in our considered setting.

4.3 Statistical criteria

To check the goodness-of-fit of the fitted models, we compute four goodness–of–fit statistics, namely the Anderson-Darling (A*), Cramér-von Mises (W*) and Kolmogorov-Smirnov (K-S) statistics with their K-S p-values. These statistics can determine how closely a distribution fits the empirical distribution of the data. The distribution with a better fit than the others will be the one having the smallest statistics and largest p-values. Mathematically, these statistics are given by

where z_i = F_*(y_i), F_*(x) denotes the cdf of the considered distribution and y₁, y₂, …, y_n are the ordered observations.

We also consider the Akaike information criterion (AIC) and Bayesian information criterion (BIC) defined as

respectively, where − log L denotes the minus estimated log-likelihood function and k is the number of parameters. As usual, the best model is the one having the smallest AIC and BIC.

5.1 Data sets

The considered data sets are presented below, roughly named “Carbon dioxide”, “Airborne” and “Rainfall”.

Rainfall

The third data set contains the mean of maximum daily rainfall during the period of 1975 to 2004 at 35 stations in Malaysia. This data set was recently studied by [13].

Elementary statistical descriptions of these three data sets are provided in Table 3.

Table 3.

Descriptive statistics for the three considered data sets.

	n	means	medians	standard deviations	skewness	kurtosis
Carbon dioxide	56	1.5	1.42	0.61	0.2	2.19
Airborne	46	3.61	1.75	4.94	2.79	11.29
Rainfall	35	1.05	1.05	0.13	−0.34	2.76

From this table, we see that Carbon dioxide and Airborne are right skewed since their skewness are positive, whereas Rainfall is moderate left skewed, corresponding to a negative skewness. Since their kurtosis are lower to 3, Carbon dioxide and Rainfall are platykurtic, whereas Airborne is leptokurtic with a kurtosis equal to 11.29.

Figures 4 gives the box-plots of the three considered data sets.

FIGURE 4.

Box-plots of the three considered data sets.

From this figure, we see that three outliers are present for Airborne, explaining the highly right skewed nature of this data set. Also, from the quantile points of view, we see that Carbon dioxide and Rainfall are similar.

Let us now investigate the nature of the underlying hrf behind the data. In this regard, we present the total time test (TTT) plots of the three data sets in Figure 5. Further detail on the definition and interpretation of the TTT plot can be found in [1].

FIGURE 5.

TTT plots for the three considered data sets.

From this figure, we observe that the underlying hrfs are probably increasing for Carbon dioxide and Rainfall, and probably bathtub for Airborne. Those cases are covered by the EMW model, motivating its use to fit these data sets.

5.2 Competitors and parametric estimation

The EMW model is compared with the models corresponding to the following distributions, all having support on (0, +∞):

• The transmuted-Weibull (TW) distribution proposed by [3]; it is defined by the following pdf:

  f(x)=ησe−(x/σ)ηxση−12λe−(x/σ)η−λ+1,  σ,η>0,λ∈(−1,1).

• The MW distribution introduced by [24]; it is defined by the following pdf:

  f(x)=λ+βkxk−1e−λx+βxk,  λ,β,k>0.

• The Marshall-Olkin exponential Weibull (MOEW) distribution developed by [20]; it is defined by the following pdf:

  f(x)=αa+bcxc−1e−ax+bxc1−(1−α)e−ax+bxc2,  a,b,c,α>0.

• The GMW distribution introduced by [8]; it is defined by the following pdf:

  f(x)=aαxγ−1(γ+λx)eλx−αxγeλx1−e−αxγeλxa−1,  a,α,γ,λ>0.

• The odd log-logistic modified-Weibull (OLLMW) distribution proposed by [22]; it is defined by the following pdf:

  f(x)=αβ+γλxγ−1e−αβx−αλxγ1−e−βx−λxγα−11−e−βx−λxγα+e−αβx−αλxγ2,  β,λ,γ,α>0.

The maximum likelihood method is used for estimating the parameters of all the models. The MLEs with their standard errors (computed by inverting the observed information matrices) are given in Tables 4, 5 and 6 for the three data sets in order, respectively.