We develop uniformly most powerful unbiased (UMPU) two sample equivalence test for a difference of canonical parameters in exponential families. This development involves a non-unique reparameterization. We address this issue via a novel characterization of all possible reparameterizations of interest in terms of a matrix group. Furthermore, our procedure involves an intractable conditional distribution which we reproduce to a high degree of accuracy using saddle point approximations. The development of this saddle point-based procedure involves a non-unique reparameterization, but we show that our procedure is invariant under choice of reparameterization. Our real data example considers the mean-to-variance ratio for normally distributed data. We compare our result to six competing equivalence testing procedures for the mean-to-variance ratio. Only our UMPU method finds evidence of equivalence, which is the expected result. We also perform a Monte Carlo simulation study which shows that our UMPU method outperforms all competing methods by exhibiting an empirical significance level which is not statistically significantly different from the nominal 5% level for all simulation settings.
R. Zhao and R. L. Paige, "Optimal Equivalence Testing in Exponential Families," Statistical Papers, Springer, Jan 2022.
The definitive version is available at https://doi.org/10.1007/s00362-022-01346-4
Mathematics and Statistics
Keywords and Phrases
Equivalence tests; Exponential families; Saddlepoint approximations; Uniformly most powerful test
International Standard Serial Number (ISSN)
Article - Journal
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01 Jan 2022