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More efficient approaches to the exponentiated halflogistic distribution based on record values
 JungIn Seo^{1} and
 SukBok Kang^{2}Email authorView ORCID ID profile
 Received: 8 March 2016
 Accepted: 11 August 2016
 Published: 30 August 2016
Abstract
The exponentiated halflogistic 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 nonBayesian perspectives if record values have an exponentiated halflogistic 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.
Keywords
 Exponentiated halflogistic distribution
 Hierarchical Bayesian model
 Record value
 Pivotal quantity
 Robust estmation
Background
Record values introduced by Chandler (1952) arise in many realworld situations involving weather, sports, economics, lifetests and stock markets, among others. Let \(\{X_1, \, X_2, \, \ldots \}\) be a sequence of independent and identically distributed (iid) random variables (RVs) with a cumulative distribution function (CDF) and a probability density function (PDF). Lower records are values in the sequence lower than all preceding ones, and the observation \(X_1\) is the first record value. Indices for which lower record values occur are given by record times \(\{L(k), \, k\ge 1\}\), where \(L(k)=\min \{jj>L(k1), \, X_{j}<X_{L(k1)}\}, \, k>1\), with \(L(1)=1\). Therefore, a sequence of lower record values is denoted by \(\{X_{L(k)}, k=1,2,\ldots \}\) from the original sequence \(\{X_1, \, X_2, \ldots \}\). However, because record occurrences are rare in practice, the maximum likelihood method can entail a substantial bias for inferences based on record values. Alternately, a method based on a pivotal quantity can be considered. Some authors studied estimation methods based on pivotal quantities when censored samples or record values are observed. Wang and Jones (2010) proposed an estimation method based on a pivotal quantity if progressively TypeII censored samples are observed from a certain family of twoparameter lifetime distributions. Yu et al. (2013) provided new estimation equations using a pivotal quantity and showed that those equations to be particularly effective 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. Seo and Kang (2015) provided an estimation equation more efficient than the maximum likelihood equation for the halflogistic distribution (HLD) based on progressively TypeII 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 twoparameter 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 BirnbaumSaunders distribution. Fu et al. (2012) developed an objective Bayesian analysis method to estimate unknown parameters of the Pareto distribution based on progressively TypeII censored samples. Kang et al. (2014) developed noninformative priors for the generalized halfnormal distribution when scale and shape parameters are of interest, respectively. Seo et al. (2014) provided a hierarchical model of a twoparameter distribution with a bathtub shape based on progressively TypeII 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.
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 \(\lambda \) in the presence of a nuisance parameter \(\theta \). 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
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.
Theorem 1
 Step 1.:

Generate W from a \(\chi ^2\) distribution with \(2(k1)\) degree of freedom and solve the equation \(W(\theta )=W\) for \(\theta \) to obtain \(\theta ^*\).
 Step 2.:

Generate \(2T_k\) from the \(\chi ^2\) distribution with 2k degree of freedom.
 Step 3.:

Compute \(W \left( \theta ^* \right) \).
 Step 4.:

Repeat \(N(\ge 10,000)\) times.
As mentioned before, the Bayesian approach is a good alternative for small sample sizes. The next section provides the Bayesian estimation methods.
Bayesian estimation
Seo and Kang (2014) 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 \((\lambda , \theta )\) when \(\lambda \) is the parameter of interest. In addition, a robust estimation method is developed by constructing a hierarchical model of the subjective marginal prior \(\pi (\lambda )\). The procedure is as follows: a subjective marginal prior for \(\lambda \) is supposed to estimate the parameter of interest \(\lambda \), and then derive a conditional reference prior for \(\theta \) for \(\lambda \) by considering the scenario proposed in Sun and Berger (1998). Then the joint prior for \((\lambda , \theta )\) is derived. Based on this prior, a robust estimation method is developed.
Noninformative prior
Note that because the Jeffreys prior (13) and the reference prior (14) have constraints on the parameter \(\lambda \) 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 \(\lambda \).
Robust estimation
Note that if \(\alpha \rightarrow 0\) and \(\beta \rightarrow 0\), then the posterior mean \(\hat{\lambda }_B(\theta )\) is the same as the MLE \(\hat{\lambda }(\theta )\) and the posterior mode \(\hat{\lambda }_{MAP}(\theta )\) is the same as the unbiased estimator \(\hat{\lambda }_p(\theta )\) when the nuisance parameter \(\theta \) is known. However since the marginal distributions for \(\lambda \) and \(\theta \) depend on hyperparameters \(\alpha \) and \(\beta \), 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.
Theorem 2
Proof
Corollary 1
Application
This section assesses the proposed methods and verifies them using real data.
A simulation study
MSEs(biases) and \(95\,\%\) CIs for \(\lambda \)
\(\lambda \)  k  MSE(bias)  c  MSE(bias)  CI based on  

\(\hat{\lambda }\)  \(\hat{\lambda }_p \left( \hat{\theta }_p \right) \)  \(\hat{\lambda }_B \left( \hat{\theta }_{MAP} \right) \)  \(\hat{\lambda }_{HB} \left( \hat{\theta }_{HMAP} \right) \)  \(\hat{\lambda }\)  \(W \left( \theta ^* \right) \)  
2  10  0.357 (0.239)  0.175 (0.005)  0.260 (0.115)  5  0.170 (−0.003)  0.977  0.954 
100  0.173 (−0.004)  
500  0.173 (−0.004)  
12  0.230 (0.180)  0.135 (0.003)  0.179 (0.086)  5  0.132 (−0.002)  0.972  0.951  
100  0.135 (−0.002)  
500  0.135 (−0.002)  
14  0.151 (0.139)  0.100 (−0.003)  0.123 (0.065)  5  0.099 (−0.002)  0.971  0.949  
100  0.100 (−0.002)  
500  0.100 (−0.002)  
16  0.121 (0.119)  0.085 (−0.003)  0.101 (0.056)  5  0.084 (−0.001)  0.970  0.950  
100  0.085 (−0.001)  
500  0.085 (−0.001)  
4  10  0.771 (0.330)  0.270 (0.020)  0.427 (0.178)  5  0.228 (−0.004)  0.972  0.954 
100  0.229 (−0.005)  
500  0.229 (−0.005)  
12  0.406 (0.230)  0.189 (0.012)  0.266 (0.126)  5  0.172 (−0.005)  0.968  0.951  
100  0.172 (−0.005)  
500  0.172 (−0.005)  
14  0.208 (0.167)  0.120 (0.001)  0.157 (0.089)  5  0.115 (−0.003)  0.968  0.950  
100  0.115 (−0.003)  
500  0.115 (−0.003)  
16  0.151 (0.136)  0.096 (0.000)  0.119 (0.072)  5  0.094 (−0.002)  0.966  0.949  
100  0.094 (−0.002)  
500  0.094 (−0.002)  
6  10  1.923 (0.454)  0.468 (0.046)  0.624 (0.227)  5  0.294 (0.004)  0.975  0.954 
100  0.294 (0.004)  
500  0.294 (0.004)  
12  0.783 (0.298)  0.289 (0.027)  0.383 (0.159)  5  0.224 (0.002)  0.968  0.951  
100  0.224 (0.002)  
500  0.224 (0.002)  
14  0.306 (0.204)  0.153 (0.008)  0.208 (0.110)  5  0.141 (−0.001)  0.967  0.950  
100  0.141 (−0.001)  
500  0.141 (−0.001)  
16  0.195 (0.159)  0.113 (0.003)  0.146 (0.086)  5  0.108 (−0.001)  0.966  0.950  
100  0.108 (−0.001)  
500  0.108 (−0.001)  
8  10  5.376 (0.604)  0.871 (0.079)  0.769 (0.264)  5  0.355 (0.009)  0.975  0.952 
100  0.355 (0.009)  
500  0.355 (0.009)  
12  1.522 (0.377)  0.452 (0.047)  0.488 (0.188)  5  0.279 (0.006)  0.967  0.951  
100  0.279 (0.006)  
500  0.279 (0.006)  
14  0.458 (0.247)  0.202 (0.018)  0.268 (0.130)  5  0.171 (−0.003)  0.964  0.950  
100  0.171 (−0.003)  
500  0.171 (−0.003)  
16  0.254 (0.186)  0.134 (0.009)  0.176 (0.099)  5  0.124 (−0.003)  0.962  0.950  
100  0.124 (−0.003)  
500  0.124 (−0.003) 
Table 1 shows that \(\hat{\lambda }_{HB} \left( \hat{\theta }_{HMAP} \right) \) 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
Proposed estimates of \(\theta \)
\(\hat{\theta }\)  \(\hat{\theta }_p\)  \(\hat{\theta }_{MAP}\)  \(\hat{\theta }_{HMAP}\)  

\(c=5\)  \(c=100\)  \(c=500\)  
0.184  0.138  0.156  0.137  0.137  0.137 
Proposed estimates of \(\lambda \)
\(\hat{\lambda }\)  \(\hat{\lambda }_p\left( \hat{\theta }_p \right) \)  \(\hat{\lambda }_B \left( \hat{\theta }_{MAP} \right) \)  \(\hat{\lambda }_{HB}\left( \hat{\theta }_{HMAP} \right) \)  

\(c=5\)  \(c=100\)  \(c=500\)  
Estimates  7.996  5.897  7.051  5.891  5.890  5.890 
Intervals  (1.433, 14.560)  (2.739, 15.661)  (3.055, 11.530)  (2.428, 9.889)  (2.428, 9.889)  (2.428, 9.889) 
Lengths  13.127  12.922  8.475  7.461  7.461  7.461 
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 \(\hat{\lambda }\) and \(\hat{\theta }\) 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 nonBayesian approach in terms of the interval length.
Conclusion
This paper proposes more efficient methods for estimating shape and scale parameters of the EHLD based on record values from Bayesian and nonBayesian 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 \(\lambda \) under the hierarchical prior (21) is a generalized version of the unbiased estimator of \(\lambda \) when the nuisance parameter \(\theta \) 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.
Declarations
Authors' contributions
Both authors contributed in process of manuscript writing. JS developed the theoretical and methodology, and wrote the article. SK worked on the literature review, collected the data sets and analyzed the data. Both authors read and approved the final manuscript.
Acknowlegements
This work was supported by the 2015 Yeungnam University Research Grant. The authors are very grateful to the editors and the reviewers for their helpful comments.
Competing interests
The authors declare that they have no competing interests
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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