More efficient approaches to the exponentiated half-logistic distribution based on record values

The exponentiated half-logistic distribution has various shapes depending on its shape parameter. Therefore, this paper proposes more efficient approach methods for estimating shape parameters in the presence of a nuisance parameter, that is, a scale parameter, from Bayesian and non-Bayesian perspectives if record values have an exponentiated half-logistic distribution. In the frequentist approach, estimation methods based on pivotal quantities are proposed which require no complex computation unlike the maximum likelihood method. In the Bayesian approach, a robust estimation method is developed by constructing a hierarchical structure of the parameter of interest. In addition, two approaches address how the nuisance parameter can be dealt with and verifies that the proposed methods are more efficient than existing methods through Monte Carlo simulations and analyses based on real data.

for skewed distributions with small sample sizes and censored samples. Wang et al. (2015) constructed confidence and predictive intervals by using some pivotal quantities for a family of proportional reversed hazard distributions based on lower record values.  provided an estimation equation more efficient than the maximum likelihood equation for the half-logistic distribution (HLD) based on progressively Type-II censored samples. As another alternative, the Bayesian approach can be effective if sufficient prior information can be obtained. Madi and Raqab (2007) assumed that unknown parameters of the two-parameter exponentiated exponential distribution (EED) have independently distributed gamma priors and predicted the subsequent record values based on observed record values. Asgharzadeh et al. (2016) developed estimation methods for obtaining the maximum likelihood estimators (MLEs) and the Bayes estimators of the unknown parameters in the logistic distribution based on the upper record values, and suggested the use of the Bayesian method if there is reliable prior information. However, because their approach is based on a subjective prior, it can lead to incorrect estimation results if there is no sufficient prior information. In this case, two alternatives can be considered: estimation based on noninformative or objective priors and that, based on a hierarchical prior obtained by mixing hyperparameters of a natural conjugate prior. In this regard, Jeffreys (1961) and Bernardo (1979) introduced the Jeffreys prior and a reference prior, respectively. Xu and Tang (2010) derived a reference prior for unknown parameters of the Birnbaum-Saunders distribution. Fu et al. (2012) developed an objective Bayesian analysis method to estimate unknown parameters of the Pareto distribution based on progressively Type-II censored samples. Kang et al. (2014) developed noninformative priors for the generalized half-normal distribution when scale and shape parameters are of interest, respectively.  provided a hierarchical model of a two-parameter distribution with a bathtub shape based on progressively Type-II censoring, which leads to robust Bayes estimators of unknown parameters of this distribution. Seo and Kim (2015) developed Bayesian procedures to approximate a posterior distribution of a parameter of interest in models with nuisance parameters and examined the sensitivity of the posterior distribution of interest in terms of an information measure for the Kullback-Leibler divergence.
This paper proposes more efficient methods for estimating unknown parameters of the exponentiated HLD (EHLD) based on lower record values by employing these alternatives from Bayesian and non-Bayesian perspectives. In the frequentist approach, the proposed estimation methods require no complex computation unlike the maximum likelihood method. In the Bayesian approach, a robust estimation method is developed by constructing a hierarchical structure of the parameter of interest. In addition, two approaches address how the nuisance parameter can be dealt with. The efficiency is proved through Monte Carlo simulations and analyses based on real data in "Application" section. The CDF and PDF of the EHLD are respectively given by and where θ is the reciprocal of a scale parameter and is the shape parameter. Recently, this distribution has received considerable attention as a generalized distribution of the HLD.  derived the entropy of the another generalized version of the HLD for Type-II censored samples observed from the distribution.  provided Bayesian estimation and prediction methods for the EHLD based on record values.  proved that the EHLD can be an alternative to the gamma distribution or the EED with two parameters (scale and shape). In addition, they showed that the PDF of the EHLD is a decreasing function for ≤ 1 and a right-skewed unimodal function for > 1. These findings indicate that the EHLD based on record values is a skewed distribution with small samples for > 1. In this case, the maximum likelihood method may not appropriate because it is useful for large samples. Therefore, this paper provides estimation methods more efficient than the maximum likelihood method in terms of the mean squared error (MSE) and bias as well as the computational cost in estimating a parameter of interest in a presence of the nuisance parameter and proposes a robust Bayesian estimation method using the hierarchical structure of a prior distribution. Here the paper focuses on estimating the shape parameter because the EHLD has various shapes depending on its shape parameter and the scale parameter is a nuisance parameter.
The rest of this paper is organized as follows: "Frequentist estimation" section proposes estimation methods based on pivotal quantities that require no complex computation and are more efficient than the maximum likelihood method in terms of the MSE and bias if lower record values arise from the EHLD. "Bayesian Estimation" derives a reference prior for unknown parameters, and then proposes a robust Bayesian estimation method by constructing a hierarchical structure of the parameter of interest of the EHLD based on lower record values. "Application" compares numerical results for the MSE and bias and analyzes real data, and "Conclusion" concludes the paper.

Frequentist estimation
This section estimates the parameter of interest in the presence of a nuisance parameter θ. MLEs and corresponding approximate confidence intervals (CIs) are derived, and then estimation methods are proposed based on pivotal quantities that not only are more convenient to compute but also can provide better results in terms of the MSE and bias.

Maximum likelihood estimation
Let X L(1) , . . . , X L(k) be the first k lower record values from the EHLD. Then by the definition of Arnold et al. (1998), the likelihood function based on these lower record values is given by . Kang SpringerPlus (2016) 5:1433 The log-likelihood function is given by From (2), the likelihood equations for and θ are given respectively by and Then the MLE of can be easily obtained for known θ, by solving the likelihood equation (3) for as follows: where If both and θ are unknown, then MLEs θ and ˆ can be obtained simultaneously by solving Eqs.
(3) and (4) through the Newton-Raphson method. An asymptotic variancecovariance matrix of MLEs can be obtained by inverting the Fisher information matrix for ( , θ) given by The negative second derivatives of the log-likelihood function in (5) are given by (3) Then, with the marginal density function of X L(i) , and the series expansions (see Jeffrey and Zwillinger 2007) the Fisher information matrix (5) can be obtained as where Therefore, under the suitable regularity conditions (Lehmann and Casella 1998), by the asymptotic normality of the MLE, the approximate 100(1 − α)% CIs based on the MLEs can be constructed as where z α/2 denotes the upper α/2 point of the standard normal distribution and the variances Var ˆ and Var θ are the diagonal elements of the asymptotic variance-covariance matrix I −1 ˆ ,θ . However these CIs can yield negative lower bounds although both and θ are supported by (0, ∞). The next subsection provides estimation methods based on pivotal quantities that can address the disadvantage of approximate CIs (7) and are much easier to calculate than the maximum likelihood method.
Estimation based on pivotal quantities Wang et al. (2015) provided some lemmas to construct exact confidence intervals for the family of proportional reversed hazard distributions based on lower record values.
Based on their results, this subsection develops estimation methods based on pivotal quantities. Let Then, by Lemma 1 in Wang et al. (2015), iid RVs with the standard exponential distribution can be obtained as From the spacing, a pivotal quantity for a gamma distribution with parameters (k, 1) is given by Then, because h 1 (θ) has a gamma distribution with parameters (k, ), an unbiased estimator of is given by which has an inverse gamma distribution with parameters (k, (k − 1)) and its MSE is given by Note that since the MLE ˆ (θ) has an inverse gamma distribution with parameters (k, k) from the fact that h 1 (θ) has a gamma distribution with parameters (k, ), its bias and MSE are given respectively by and In addition, since 2T k has a χ 2 distribution with 2k degrees of freedom, for 0 < α < 1, we have where χ 2 α, k is the upper α percentile of a χ 2 distribution with k degrees of freedom. Therefore, an exact 100(1 − α)% CI for is given by Note that the unbiased estimator (9) and the exact 100(1 − α)% CI (10) depend on the nuisance parameter θ. Now the method for addressing the nuisance parameter θ is discussed.
For the pivotal quantity (8), the pivotal quantities can be obtained such that they are independent RVs having a gamma distribution with parameters (j, 1). Then, since 2T j are independent RVs having a χ 2 distribution with 2j degree of freedom, the transforms are iid RVs with a uniform distribution on (0, 1). Therefore, the pivotal quantity can be obtained such that it has a χ 2 distribution with 2k − 2 degrees of freedom. Here an estimator of θ, θ p , can be found by solving the equation W (θ)/(2k − 4) = 1 for θ because W (θ)/(2k − 4) converges in probability to 1 as k → ∞. Note that the estimator θ p is unique because W (θ) is a strictly increasing function of θ (see "Appendix"). Then, by substituting the estimator θ p for θ in the unbiased estimator (9), the estimator (9) is written as which is no longer an unbiased estimator. In addition, an exact 100(1 − α)% CI for can be obtained by using the pivotal quantity W (θ) when θ is unknown. The following theorem provides an exact 100(1 − α)% CI for when θ is unknown.
The percentiles of the generalized pivotal quantity W (θ * ) are obtained based on the following algorithm: Step 1. Generate W from a χ 2 distribution with 2(k − 1) degree of freedom and solve the equation W (θ) = W for θ to obtain θ * .
Step 2. Generate 2T k from the χ 2 distribution with 2k degree of freedom.
As mentioned before, the Bayesian approach is a good alternative for small sample sizes. The next section provides the Bayesian estimation methods.  assumed independently distributed gamma priors to draw inferences for the EHLD based on lower record values. If there is sufficient information on the prior, then these subjective priors are appropriate. However, it is not easy to obtain such information in practice. Therefore, this section derives a reference prior for ( , θ) when is the parameter of interest. In addition, a robust estimation method is developed by constructing a hierarchical model of the subjective marginal prior π( ). The procedure is as follows: a subjective marginal prior for is supposed to estimate the parameter of interest , and then derive a conditional reference prior for θ for by considering the scenario proposed in Sun and Berger (1998). Then the joint prior for ( , θ) is derived. Based on this prior, a robust estimation method is developed.

Noninformative prior
The Jeffrey and reference priors can be derived based on the Fisher information matrix (6). By definition of Jeffreys (1961), the Jeffrey prior for ( , θ) is given by To derive the reference prior for ( , θ) when is the parameter of interest, the algorithm of Beger and Bernardo (1989) is applied as follows. Let and by choosing a sequence of compact sets where 0 is any fixed point. Note that because the Jeffreys prior (13) and the reference prior (14) have constraints on the parameter to obtain positive square roots, these noninformative priors are ineffective in objective Bayesian analysis. Therefore, the next subsection develops a robust estimation method by constructing a hierarchical structure of the parameter of interest .

Robust estimation
Suppose that the marginal prior is a gamma prior with the PDF which is a conjugate prior when the nuisance parameter θ is known. In addition, based on Theorem 2 of Sun and Berger (1998), a conditional reference prior π(θ | ) is given by Then the joint prior density function for ( , θ) is given by From the likelihood function (1) and the joint prior density function (16), the joint posterior density function for ( , θ) is given by The marginal posterior density function for θ is obtained by integrating the joint posterior density function (17) as Here the nuisance parameter θ can be simply estimated by maximizing the marginal posterior density function (18), which is denoted by θ MAP . Then because the marginal posterior density function for is given by where π( |α, β, θ , x) is the conditional posterior density function for , with θ substituted by θ MAP in π( |α, β, θ , x), the marginal posterior density function (19) can be approximated as (17) which is the PDF of the gamma distribution with parameters k + α, β + h 1 θ MAP .
Note that if α → 0 and β → 0, then the posterior mean ˆ B (θ) is the same as the MLE ˆ (θ) and the posterior mode ˆ MAP (θ) is the same as the unbiased estimator ˆ p (θ) when the nuisance parameter θ is known. However since the marginal distributions for and θ depend on hyperparameters α and β, if there is no sufficient information on the prior, then values of the hyperparameters cannot be determined. Now, a method for addressing these hyperparameters for robust estimation results is developed.
Let π(α, β) be a joint prior for the hyperparameters α and β. Then Based on Han (1997), values of α and β should be determined such that the marginal prior π( |α, β) is a decreasing function of . For 0 < α ≤ 1 and β > 0, the condition is satisfied. However, β is restricted to the support (0, c) because prior distributions with very thin tails may sensitive. In addition, only the case of α = 1 is considered for simplicity. Then the marginal prior π( |α, β) can be written as For robust inferences, assign a uniform distribution on (0, c) as a prior distribution of β.
Then the hierarchical prior is given by Therefore, the joint posterior density function for ( , θ) under the hierarchical prior (21) is given by The marginal posterior density function for θ is obtained by integrating the joint posterior density function (22) as where Here the nuisance parameter θ can be estimated by maximizing the marginal posterior density function (23), and it is denoted by θ HMAP . The marginal posterior density function for is provided in the following theorem.

Theorem 2 The marginal posterior density function for the parameter of interest, , is approximated as
Proof The marginal posterior density function for is given by where π( |θ , x) is the conditional posterior density function for for θ. With θ substituted by θ HMAP in π( |θ , x), the marginal posterior density function (25) can be approximated as π |θ HMAP , x , and this completes the proof. (24), the corresponding posterior mean is where Therefore if c → ∞, then the posterior mean ˆ HB (θ ) is the same as the unbiased estimator ˆ p (θ) when the nuisance parameter θ is known.

Application
This section assesses the proposed methods and verifies them using real data.

A simulation study
For a simulation study, the proposed estimators are compared in terms of their MSEs and biases over 10,000 replications. Among the estimators φ i (i = 1, 2) of the unknown parameter φ, if the MSE of φ 1 is smaller the that of φ 2 for any value of φ, then φ 1 is preferable and can be said to be more efficient than the other. For the bias, the same argument is made. The Bayes estimator ˆ B θ MAP is computed based on a vague prior with the hyperparameters α = β = 0.01. To examine the robustness of the hierarchical prior (21), the Bayes estimator ˆ HB θ HMAP is computed for different c = 5, 100, 500. Lower record values are generated from the standard EHLD with = 2(2)8. In addition, coverage probabilities (CPs) of CIs based on the MLE ˆ θ and the generalized pivotal quantity W (θ * ) are reported at the 0.95 confidence level based on 10,000 simulations. These values are given in Table 1. Because the parameter of interest is , no results for the nuisance parameter θ are reported here. Table 1 shows that ˆ HB θ HMAP is generally more efficient than other estimators in terms of the MSE and bias. In addition, it is quite robust to the choice c. CPs of CIs based on the MLE exceed corresponding nominal levels, whereas those of CIs based on the generalized pivotal quantity are well matched to their corresponding nominal levels.

Real data
This subsection analyzes real data considered in , which represent the amount of annual rainfall (in inches) recorded at the Los Angeles Civic Center from 1877 to 2012. From the data, 10 lower record values are observed as  showed that the EHLD provides a good fit to both whole data and observed lower record values. The record values are used to obtain estimates discussed in previous sections. For Bayesian inferences, hyperparameters α = β = 0.01 are set because no information on the prior is given. For the proposed hierarchical model, consider different c = 5, 100, 500. Estimates of the nuisance parameter θ are reported in Table 2, and estimates of the parameter of interest , and corresponding 95 % intervals are reported in Table 3. Figure 1 plots the likelihood function of unknown parameters ( , θ) and Fig. 2 plots the marginal posterior density functions for derived in "Frequentist estimation" section. Tables 2 and 3 show that the Bayes estimates under the hierarchical prior (21) have the same values for different values of c. The credible intervals have shorter lengths than CIs. In addition, it is observed that the MLEs ˆ and θ are exist and unique from Fig. 1. Figure 2 shows that the marginal posterior density function has the same shape for different values of c. These results indicate that the Bayesian approach based on the hierarchical prior (21) produces robust results and is superior to non-Bayesian approach in terms of the interval length. 21. 26, 11.35, 10.40, 9.21, 6.73, 5.59, 5.58, 4.85, 4.42, 3.21.

Conclusion
This paper proposes more efficient methods for estimating shape and scale parameters of the EHLD based on record values from Bayesian and non-Bayesian perspectives. The results verify that the method based on the pivotal quantity is superior to the maximum likelihood method in terms of the MSE, bias, and CP of the CI based on Monte Carlo simulations and that it is more computationally convenient. In addition, it is noted that the Bayes estimator of under the hierarchical prior (21) is a generalized version of the unbiased estimator of when the nuisance parameter θ is known. Through Monte Carlo simulations and real data analysis, it was verified that not only the Bayesian estimation under the hierarchical prior (21) are more efficient than the estimation based on the pivotal quantity in terms of the MSE and bias, but estimation results under the hierarchical prior (21) are robust to c. Therefore, the proposed robust Bayesian estimation method should be used if there is no sufficient prior information.