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<p>This study examines two gamma populations that share a common shape parameter but have different scale parameters. The primary objective is to derive confidence intervals and test hypotheses related to the model parameters. To achieve this, asymptotic confidence intervals for the model parameters are derived using Fisher’s information matrix. Additionally, parametric bootstrap intervals—specifically, the bootstrap-t (Boot-t) and bootstrap-p (Boot-p) intervals are generated through a bootstrap sampling procedure. Highest posterior density (HPD) intervals are also obtained using the Markov Chain Monte Carlo method, along with selected prior probabilities for the parameters. Since no closed-form expressions are available, all intervals are computed numerically. A numerical comparison of the proposed intervals is conducted based on their average length and coverage probability. For hypothesis testing, several methods are proposed, including the Likelihood Ratio Test (LRT), Parametric Bootstrap Likelihood Ratio Test (PBLRT), Standardized Likelihood Ratio Test (SLRT), Computational Approach Test (CAT), and Modified Computational Approach Test (MCAT). The sizes and powers of all proposed tests are evaluated using a Monte Carlo simulation procedure. Finally, two real-life data analyses are performed to demonstrate the practical application of the proposed model.</p>

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Interval Estimation and Hypothesis Testing on the Common Shape Parameter of Two Gamma Populations with Different and Unknown Scales

  • 期刊论文

  • 发表日期：

    2025年8月31日

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  • Kalinga Kanya Parida,
  • Manas Ranjan Tripathy

摘要

This study examines two gamma populations that share a common shape parameter but have different scale parameters. The primary objective is to derive confidence intervals and test hypotheses related to the model parameters. To achieve this, asymptotic confidence intervals for the model parameters are derived using Fisher’s information matrix. Additionally, parametric bootstrap intervals—specifically, the bootstrap-t (Boot-t) and bootstrap-p (Boot-p) intervals are generated through a bootstrap sampling procedure. Highest posterior density (HPD) intervals are also obtained using the Markov Chain Monte Carlo method, along with selected prior probabilities for the parameters. Since no closed-form expressions are available, all intervals are computed numerically. A numerical comparison of the proposed intervals is conducted based on their average length and coverage probability. For hypothesis testing, several methods are proposed, including the Likelihood Ratio Test (LRT), Parametric Bootstrap Likelihood Ratio Test (PBLRT), Standardized Likelihood Ratio Test (SLRT), Computational Approach Test (CAT), and Modified Computational Approach Test (MCAT). The sizes and powers of all proposed tests are evaluated using a Monte Carlo simulation procedure. Finally, two real-life data analyses are performed to demonstrate the practical application of the proposed model.