[Paper Review] Generalized Confidence Interval for the Common Coefficient of Variation

Javad Behboodian, Ali Akbar Jafari|arXiv (Cornell University)|May 2, 2014

Statistical Distribution Estimation and Applications8 references4 citations

TL;DR

This paper proposes a new generalized confidence interval for the common coefficient of variation (CV) across multiple normal populations using generalized p-values and generalized variables. By combining a novel method with Tian's (2005) approach, the resulting interval achieves shorter length and better coverage probability than existing methods, as validated through simulation and real medical data applications.

ABSTRACT

In this article, we consider the problem of constructing the confidence interval and testing hypothesis for the common coefficient of variation (CV) of several normal populations. A new method is suggested using the concepts of generalized p-value and generalized confidence interval. Using this new method and a method proposed by Tian (2005), we obtain a shorter confidence interval for the common CV. This combination method has good properties in terms of length and coverage probability compared to other methods. A simulation study is performed to illustrate properties.. Finally, these methods are applied to two real data sets in medicine.

Motivation & Objective

• To address the challenge of constructing accurate confidence intervals for the common coefficient of variation (CV) when multiple normal populations share the same CV.
• To overcome limitations of traditional methods that struggle with nuisance parameters and non-pivotal distributions in CV inference.
• To develop a new generalized variable-based method that improves interval precision and coverage properties compared to existing approaches.
• To combine the new method with Tian’s (2005) generalized confidence interval to achieve optimal performance in terms of length and coverage probability.
• To validate the proposed method using Monte Carlo simulations and real-world medical datasets, demonstrating practical utility.

Proposed method

• Uses the concept of generalized pivotal variables to construct confidence intervals for the common CV, ensuring the distribution is free of nuisance parameters.
• Derives a generalized variable based on the ratio of sample standard deviations and means, adjusted for the common CV under the null hypothesis of equal CVs.
• Employs Monte Carlo simulation with 5,000 replicates to estimate quantiles of the generalized variable distribution for interval construction.
• Combines the new method with Tian’s (2005) generalized p-value approach to form a hybrid method that improves interval length and coverage.
• Applies the method to real data using the generalized variable approach to compute confidence intervals and p-values for hypothesis testing.
• Uses simulation to compare coverage probability and interval length across four methods: Tian (2005), Verrill and Johnson (2007), the new method (13), and the combined method (17).

Experimental results

Research questions

• RQ1 Can a new generalized variable-based method improve the coverage probability and length of confidence intervals for the common coefficient of variation compared to existing methods?
• RQ2 How does the performance of the combined method (new method + Tian’s method) compare to individual methods in terms of interval length and coverage probability?
• RQ3 What is the empirical performance of the proposed method under small sample sizes, particularly in terms of maintaining nominal coverage levels?
• RQ4 How do the proposed methods perform on real medical datasets with heterogeneous sample sizes and varying CVs?
• RQ5 Does the new method produce more accurate point estimates of the common CV compared to existing estimators like MLE or Feltz and Miller’s estimator?

Key findings

• The combined method (17) produced the shortest confidence interval length (0.0092) in Example 1, outperforming Tian’s (0.0100) and Verrill and Johnson’s (0.0103) methods.
• In Example 2, the combined method yielded a 2.8020-length interval, shorter than Tian’s (5.4416) and Verrill and Johnson’s (0.6479), though the latter had a shorter interval, it showed under-coverage in small samples.
• The coverage probability of the Verrill and Johnson (2007) method was below the nominal 95% level when sample sizes were small, indicating poor reliability in such settings.
• The new method (13) had slightly shorter average interval length than Tian’s method but exhibited lower coverage probability in some scenarios, indicating a trade-off between precision and reliability.
• The combined method (17) maintained coverage probability close to the nominal 95% level while achieving shorter interval length than Tian’s method in both real data examples.
• In Example 1, the combined method’s 95% confidence interval was (0.0333, 0.0425), which was narrower than Tian’s (0.0347, 0.0447), demonstrating improved precision without sacrificing coverage.