Tanaka, Toyoto; Hirose, Yoshihiro; Komaki, Fumiyasu Second-order matching prior family parametrized by sample size and matching probability. (English) Zbl 1452.62201 Stat. Pap. 61, No. 4, 1701-1717 (2020). Summary: We propose a family of priors that satisfies the second-order probability matching property. The posterior quantile of a probability matching prior is exactly or approximately equal to the frequentist one. Most models lack an exact matching prior. If all quantiles of a prior’s posterior converge to the frequentist ones up to \(o(n^{-1/2})\) or \(o(n^{-1})\) as the sample size \(n\) increases, the prior is called a first-order probability matching prior and a second-order probability matching prior, respectively. Although a second-order matching prior does not necessarily exist, a first-order matching prior always exists. We introduce a class of priors that depend on the sample size and matching probability. We derive the condition under which the family satisfies the second-order probability matching property even when a second-order probability matching prior does not exist. The superiority of the proposed priors is illustrated in several numerical experiments. MSC: 62F05 Asymptotic properties of parametric tests 62F15 Bayesian inference 62F25 Parametric tolerance and confidence regions Keywords:Bayesian asymptotics; frequentist validity; posterior quantile; probability matching prior PDFBibTeX XMLCite \textit{T. Tanaka} et al., Stat. Pap. 61, No. 4, 1701--1717 (2020; Zbl 1452.62201) Full Text: DOI References: [1] Amari, S., Differential geometrical methods in statistics (1985), New York: Springer, New York · Zbl 0559.62001 [2] Bhattacharya, RN; Ghosh, JK, On the validity of the formal Edgeworth expansion, Ann Stat, 6, 434-451 (1978) · Zbl 0396.62010 [3] Cox, DR; Reid, N., Parameter orthogonality and approximate conditional inference, J R Stat Soc Ser B, 49, 1-39 (1987) · Zbl 0616.62006 [4] Datta, GS; Mukerjee, R., Probability matching priors: higher order asymptotics (2004), New York: Springer, New York · Zbl 1044.62031 [5] Garvan, CW; Ghosh, M., Noninformative priors for dispersion models, Biometrika, 84, 976-982 (1997) · Zbl 0892.62010 [6] Johnson, RA, Asymptotic expansions associated with posterior distributions, Ann Math Stat, 41, 851-864 (1970) · Zbl 0204.53002 [7] Kim, DH; Kang, SG; Lee, WD, Noninformative priors for linear combinations of the normal means, Stat Pap, 47, 249-262 (2006) · Zbl 1104.62025 [8] Mukerjee, R.; Dey, DK, Frequentist validity of posterior quantiles in the presence of a nuisance parameter: higher order asymptotics, Biometrika, 80, 499-505 (1993) · Zbl 0788.62025 [9] Mukerjee, R.; Ghosh, M., Second-order probability matching priors, Biometrika, 84, 970-975 (1997) · Zbl 0895.62003 [10] Ong, SH; Mukerjee, R., Data-dependent probability matching priors of the second-order, Statistics, 44, 291-302 (2010) · Zbl 1291.62070 [11] Peers, HW, On confidence points and Bayesian probability points in the case of several parameters, J R Stat Soc Ser B, 27, 9-16 (1965) · Zbl 0144.41403 [12] Sun, D.; Ye, K., Frequentist validity of posterior quantiles for a two-parameter exponential family, Biometrika, 83, 55-65 (1996) · Zbl 1059.62530 [13] Tibshirani, RJ, Noninformative priors for one parameter of many, Biometrika, 76, 604-608 (1989) · Zbl 0678.62010 [14] Ventura, L.; Cabras, S.; Racugno, W., Prior distributions from pseudo-likelihoods in the presence of nuisance parameters, J Am Stat Assoc, 104, 768-774 (2009) · Zbl 1388.62060 [15] Welch, BL; Peers, HW, On formulae for confidence points based on integrals of weighted likelihoods, J R Stat Soc Ser B, 25, 318-329 (1963) · Zbl 0117.14205 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.