1.1 Overview of the TBI problematic

Trauma is a multisystem health condition that represents the third cause of death worldwide, surpassed only by cerebrovascular diseases and cancer [10]. It is estimated that over sixty million people have trauma each year, and nearly 16,000 people die every day after some traumatic injury. Traumatic Brain Injury (TBI) represents one of the significant causes of death and disability among the trauma epidemiology data [10, 11]. Most of the patients with TBI are young, economically active adults and more likely to have been involved in a traffic accident [12–16]. Therefore, TBI is considered a public health concern that leads to high costs of hospitalization with various economic and social burdens.

The limited available data in the TBI problematic, usually presented in a small number of patients, motivated the current study, which relied on a data set of a longitudinal observational investigation of patients after TBI due to traffic accidents admitted to a Brazilian Emergency Department. Investigating optimal statistical analyses in this population is essential for providing impactful information applied not only to patients but also to their families, caregivers, and society in general [17].

1.2 Current statistical methods and their limitations

In the literature, there are various classical methods for estimating the unknown parameters of probability distributions. Under the frequentist approach, the primary interest is to compare the maximum likelihood estimation method with other estimation procedures. Related studies about different distributions have also been presented in the literature [18–22]. In this work, we consider several of these estimation procedures, such as least squares, the weighted least squares, the maximum product of spacings, and the Anderson-Darling maximum goodness-of-fit estimators.

An intensive simulation study is conducted to compare these estimation methods. However, we observe two significant problems. The first is that, for some methods, the estimation procedures fail in finding the target estimates, i.e., report convergence problems in the maximization/minimization process. The second is related to the occurrence of a significant bias in the obtained estimates for small samples. In order to overcome this problem, we propose a penalized maximum likelihood estimator, which, combined with a very useful practical algorithm called Simulated Annealing (SANN) (for further details, see Kirkpatrick et al. [23]), guarantees the best convergence not depending on the conditions of the problem (i.e., the initial values), even when it has several local extrema. Prentice [24] argued that the approximately normal distribution for ϕ, using the maximum likelihood theory, may not be achieved even for sample sizes equal to or larger than 400. Due to the asymptotic relationship of the maximum likelihood estimator with the penalized maximum likelihood estimator, the same problem may be observed. Therefore, we consider a bootstrap approach to building accurate confidence intervals (see DiCiccio and Efron [25]) for small and moderate samples. Finally, by combining all these approaches, one can perform inference for the flexible GG distribution with good precision even for small sample sizes.

The paper is organized as follows. Section 2 presents some properties of the GG distribution, including its cumulative distribution, survival and hazard functions, and its moments. Additionally, the SANN algorithm is also discussed in detail, and implementation procedures are presented. Section 3 discusses the eight estimation methods considered in this paper. Section 4 shows a simulation study, using synthetic data, designed to identify the most efficient estimation procedure. In Section 5, we apply our proposed methodology to three new real data sets provided by a medical school, which contain the TBI patients’ lifetime risk among different hospitalization stages. Finally, some final comments are given in Section 6.

2.1 Simulated annealing algorithm

The SANN algorithm was developed via the generalization of the Metropolis algorithm (Metropolis et al. [27]) to simulate the changes in the energy of molten metal when lowering its temperature slowly. The purpose of the cooling process is to reach a globally minimum energy state, i.e., to obtain a solid that is in its ground state. However, as pointed out by Salter and Pearl [28], if the temperature is lowered too fast, the resulting solid can become trapped in a metastable state that is not its ground state. Some authors, such as Kirkpatrick et al. [23], noticed the analogy between the cooling process of a substance to its minimum energy state and the minimization of a function by using a stochastic search strategy. In this case, the metastable state represents a local minimum, the ground state represents the global minimum, and the cooling rate corresponds to some parameters that control the possible solutions by the search algorithm.

Let be the function to be minimized (objective function), let be the initial solution (initial points), and let k₀ be the initial value of the control parameter (initial temperature). The SANN algorithm can be described in a formal and general way as follows.

1. Set i = 0.
2. From the current solution, x_i, generate a potential solution, x_j, according to a specific generation scheme.
3. If Δg = g(x_j) − g(x_i) ≤ 0, then set x_i+1 = x_j with probability p = 1. Otherwise, set x_i+1 = x_j with probability p = exp {− Δg / k_i}.
4. Update the value of the control parameter, k_i, and set i = i + 1. Then, go to step 2.

Steps 2-4 are repeated, e.g., until the value of the control parameter, k_i, is sufficiently small, or the same solution is repeatedly generated in many successive iterations.

Next, some useful remarks about the above algorithm are given.

1. For some choices of k₀, including k₀ = log (g (x₀)), see, e.g., Aarts and Korst [29].
2. In step 2, the potential solutions, x_j’s, are randomly chosen within a range. For instance, through x_j = x_j + rV, where r is a uniformly distributed random number in the interval (−1, 1), i.e. r ~ U(−1, 1), and V is a vector (of length d) of step sizes. After s iterations, the SANN adjusts its search bounds for each variable so that 50% of all moves will be accepted, either enlarging these bounds to select a new ground to move to or shrinking them to a minimum.
3. In step 3, for the cases where Δg > 0, we generate u ~ U(0, 1) and move to x_j only if u < p. Thus, accepting worse solutions may prevent the process from becoming stuck at local minima.
4. In step 4, after m iterations, temperature (control parameter) k drops as k′ = r_k × k, where 0 ≤ r_k ≤ 1 is the rate of temperature reduction given the initial annealing/cooling schedule. Usually, r_k = 0.95.
5. As pointed out by Salter and Pearl [28], by the Markov chain theory, the SANN algorithm can be shown to converge to a stationary distribution for which the set of optimal solutions has probability 1, under certain conditions of both the sequence of control parameters and the generation scheme (see, e.g., Aarts and Korst [29], Haario and Saksman [30]).

Although presented above as a minimization problem, the SANN algorithm can be easily modified/adapted to cases where the interest resides on maximizing the function g(⋅).

Several variants of the SANN algorithm have been proposed in the recent literature. Among them, we can mention the relevant works of Torres-Jimenez and Rodriguez-Tello [31], Torres-Jimenez et al. [32], and Izquierdo-Marquez et al. [33].

3.1 Common estimators

The method of moments (MM) is one of the oldest procedures used for estimating parameters in statistical models. It is still widely used mainly because of its simplicity. For instance, for the two-parameter Gamma distribution, MM estimators have closed-form expressions.

Huang and Hwang [7] derived the MM estimators for the GG distribution, pointing out the need for solving two nonlinear equations, to find such estimators. Let t₁, …, t_n be a random sample of size n from T ∼ GG(ϕ, μ, α). The moments estimators are obtained by solving (7) where and are the sample mean and standard deviation, respectively; while the estimate of ϕ can be obtained by (8) where and are obtained by solving the nonlinear equations in (7). The MM estimators of all GG model parameters do not have closed-form expressions, which is undesirable. Another disadvantage of this approach is that the authors did not discuss the asymptotic properties of the MM estimators. Therefore, no interval estimates for ϕ, μ and α can be constructed without further research.

Another common procedure is to consider the ordinary least squares (OLS) estimators. The , and estimates can be obtained by minimizing, with respect to ϕ, μ and α, the following equation: (9) where t₍₁₎ ≤ t₍₂₎ ≤ ⋯ ≤ t_(n) are the order statistics of a random sample of size n. Equivalently, these estimates can be obtained by solving the nonlinear equations: (10) where (11)

Note that the solution of Δ₁(t_(i)|ϕ, μ, α) involves a non-trivial partial derivative of the lower incomplete gamma function. However, this can be easily achieved numerically with high precision.

The weighted least squares (WLS) estimators, , and , can be obtained by minimizing (12) with respect to ϕ, μ and α. These estimates can also be obtained by solving the nonlinear equations: (13) for j = 1, 2, 3.

3.2 Maximum likelihood estimators

The maximum likelihood (ML) estimation method is widely used for the GG distribution due to the invariance and asymptotic properties of the obtained estimators. Let t = (t₁,…, t_n)′ be a random sample of size n from a GG(ϕ, μ, α) population. Then, the likelihood function of (1) is given by (14) The log-likelihood function of (14) is given by (15)

By solving the expressions: , and , the following nonlinear equations can be obtained, respectively: (16) (17) (18) where is the digamma function. The solutions of the above Eq (17) yield the MLEs. After some algebraic manipulations, we have (19) (20) and the MLE of α is obtained by solving the nonlinear equation: (21)

Although only one nonlinear equation has to be solved, there are usually different local maxima, which lead to different estimates than expected. On the other hand, under mild conditions, the MLEs are asymptotically normally distributed with a joint trivariate normal distribution given by (22) where I(ϕ, μ, α) is the Fisher information matrix (see Hager and Bain [5] for a detailed discussion) given by (23) and is the trigamma function.

3.3 Penalized maximum likelihood estimators

Firth [34] proved that the bias of MLEs can be reduced by considering a penalization in the likelihood function. Moreover, the author showed that in exponential families with canonical parameterization, the first-order term is removed by using the Jeffreys prior [35] as a penalization term. The Jeffreys prior for the GG distribution is computed by |I(ϕ, μ, α)|^1/2, where |⋅| stands for the determinant of the Fisher information matrix (23), which results in (24)

As previously stated, the first-order term related to the bias is removed in the case of distributions that belong to the exponential family. On the other hand, the GG distribution is not a member of the exponential family. However, this penalization also allows us to improve the estimates, even not ensuring that the improvement is of the first order. Note that when ϕ = 1, the GG distribution reduces to the Weibull distribution, for which the Jeffreys prior is given by π_J (μ, α) ∝ (μα)⁻¹. The extra α⁻¹ helps us to decrease the bias of α; therefore, since π_J (ϕ, μ, α) is not a function of α, we consider the following penalization: (25)

The penalized likelihood function of ϕ, μ and α, using the Jeffreys prior (25), is given by (26)

The log-likelihood function of (26) is given by (27)

By solving the expressions: , and , the following nonlinear equations can be obtained, respectively: (28) (29) (30) Note that one of the parameters can be isolated, in order to obtain two nonlinear equations. The three possible expressions are given by (31) (32) (33) From Eqs (31)–(33), we observe that, considering (32), the penalized maximum likelihood (PML) estimators are achieved with less computational effort. Therefore, we will consider the nonlinear Eqs (28) and (30), where is obtained from (32).

Although Firth [34] proved that the PML estimators obtained from the penalized likelihood or log-likelihood function in the exponential family of distributions are always finite and, in addition, always exist, the same cannot be done for the GG distribution. For this model, the MLEs can have no solution or several solutions (see Wingo [36]). This problem is observed computationally, since it is very complex to prove analytically. It is also complicated to demonstrate analytically the results for the PML estimators. Note that our main goals here are to propose a method to circumvent these computation difficulties, and achieve improved estimates for the parameters.

The Fisher information matrix I_P (ϕ, μ, α) is given by (34) where (35)

It can be easily noted that I_P(ϕ) → I(ϕ) as n → ∞. Additionally, (36) Therefore, the PML estimators of ϕ, μ and α converge to the MLEs. Hence, under the same mild conditions of the MLEs, the PML estimators are asymptotically normally distributed with a joint trivariate normal distribution given by (37)

It is important to point out that it is not simple to check the regularity conditions necessary to ensure asymptotically normal distribution (see Lehman [37], Theorem 5.1, page 463). In fact, Prentice [24] showed that the approximate normal distribution for ϕ, using the ML theory, could not be achieved even for sample sizes equal to or larger than 400. This result can also be extended to the PML theory, since the MLEs and PML estimators are asymptotically equivalent. In order to overcome this problem, for small sample sizes, we considered the bootstrap approach presented by DiCiccio and Efron [25] to construct improved confidence intervals based on the PML estimates.

3.4 Maximum product of spacings estimators

As an alternative to the ML estimation method, the maximum product of spacings (MPS) is a robust method for estimating the unknown parameters of continuous univariate distributions. Cheng and Amin [38, 39] introduced this method, which was independently developed by Ranneby [40] as an approximation to the Kullback-Leibler information measure. Moreover, Cheng and Amin [39] proved some desirable properties of the MPS estimators, such as asymptotic efficiency, invariance and, more importantly, the consistency of these estimators holds under more general conditions than for MLEs.

The uniform spacings of a random sample from the GG distribution are defined as (38) for i = 1, 2, …, n + 1, where t_(i) is the i-th order statistics, F(t₍₀₎|ϕ, μ, α) = 0 and F(t_(n+1)|ϕ, μ, α) = 1. This implies that .

The MPS estimates, , and , are obtained by maximizing the geometric mean of the spacings (39) with respect to ϕ, μ and α. Or equivalently, by maximizing the logarithm of the geometric mean of sample spacings (39): (40) Thus, , and can be obtained by solving the nonlinear equations: (41)

In practice, one problem that may occur is the presence of ties due to multiple observations with the same value. In this case, if t_(i) = t_(i−1) for some i ∈ {1, 2, …, n + 1}, then D_i(ϕ, μ, α) = D_i−1(ϕ, μ, α) = 0. Thus, the MPS estimators are sensitive to closely-spaced observations, especially ties. Notice that (42) Hence, D_i(ϕ, μ, α) should be replaced by the corresponding likelihood L(ϕ, μ, α|t_(i)) = f(t_(i)|ϕ, μ, α) when t_(i) = t_(i−1).

Cheng and Amin [38] presented a useful comparison between the MLEs and MPS estimators: (43) where R(t_(i), t_(i−1)|ϕ, μ, α) is essentially of order O(|t_(i) − t_(i−1))|) and |t_(i) − t_(i−1)| → 0 in probability as n increases. For standard situations, log(D_i(ϕ, μ, α)) is basically the same as log (f(t_(i)|ϕ, μ, α)) with respect to ϕ, μ and α, except for a negligible number of terms. Therefore, the MLEs and the MPS estimators are asymptotically equal and have the same properties, i.e. (44)

3.5 Anderson-Darling estimators

Here, we present one type of minimum distance estimators (also referred to as the maximum goodness-of-fit estimators), which is based on the Anderson-Darling statistic and, due to this, is known as the Anderson-Darling (AD) estimator. The AD estimates, , and , of the GG model parameters ϕ, μ and α, are obtained by minimizing, with respect to ϕ, μ and α, the function (45) These estimates can also be obtained by solving the nonlinear equations: (46)

Turning now to a modified version of the AD statistic, the right-tail Anderson-Darling (RAD) estimates, and , of the parameters ϕ, μ and α, are obtained by minimizing the function (47) with respect to ϕ, μ and α. These estimates can also be obtained by solving the nonlinear equations: (48) where Δ_j (⋅|ϕ, μ, α), j = 1, 2, 3, are given in (11).