In applied statistics, constructing exact confidence intervals for constrained normal means and constrained Poisson means are two long-standing challenges. Most existing inferential methods assume that the nuisance parameters in constrained models are known constants, which is often impractical. When nuisance parameters are unknown, Bayesian intervals fail to guarantee nominal coverage. To address these issues, this work developed a valid prior-free inferential model approach for normal and Poisson distributions with unknown nuisance parameters. For the constrained normal case, we constructed an inferential model interval that exactly attains the prespecified coverage probability. For the constrained Poisson case, we first proposed an inferential model interval that achieves exact frequentist coverage control; however, owing to the discreteness of the Poisson distribution, this interval is conservative. We then improved it by introducing random weighting, yielding a nonrandomized inferential model method. Simulation studies showed that the inferential model interval achieves exact coverage under the constrained normal model, but is conservative in constrained Poisson inference. In contrast, the nonrandomized inferential model interval attains coverage closest to the nominal level by shortening the interval length. Finally, two neutrino datasets from high-energy physics were used to demonstrate the advantages of the proposed new methods over Bayesian approaches.
Citation: Hezhi Lu, Qijun Wu. Prior-free probabilistic interval estimation for constrained parameters[J]. AIMS Mathematics, 2026, 11(6): 17320-17353. doi: 10.3934/math.2026709
In applied statistics, constructing exact confidence intervals for constrained normal means and constrained Poisson means are two long-standing challenges. Most existing inferential methods assume that the nuisance parameters in constrained models are known constants, which is often impractical. When nuisance parameters are unknown, Bayesian intervals fail to guarantee nominal coverage. To address these issues, this work developed a valid prior-free inferential model approach for normal and Poisson distributions with unknown nuisance parameters. For the constrained normal case, we constructed an inferential model interval that exactly attains the prespecified coverage probability. For the constrained Poisson case, we first proposed an inferential model interval that achieves exact frequentist coverage control; however, owing to the discreteness of the Poisson distribution, this interval is conservative. We then improved it by introducing random weighting, yielding a nonrandomized inferential model method. Simulation studies showed that the inferential model interval achieves exact coverage under the constrained normal model, but is conservative in constrained Poisson inference. In contrast, the nonrandomized inferential model interval attains coverage closest to the nominal level by shortening the interval length. Finally, two neutrino datasets from high-energy physics were used to demonstrate the advantages of the proposed new methods over Bayesian approaches.
| [1] |
G. J. Feldman, R. D. Cousins, Unified approach to the classical statistical analysis of small signals, Phys. Rev. D, 57 (1998), 3873–3889. https://doi.org/10.1103/PhysRevD.57.3873 doi: 10.1103/PhysRevD.57.3873
|
| [2] |
M. Mandelkern, J. Schultz, The statistical analysis of Gaussian and Poisson signals near physical boundaries, J. Math. Phys., 41 (2000), 5701–5709. https://doi.org/10.1063/1.533433 doi: 10.1063/1.533433
|
| [3] |
C. Giunti, New ordering principle for the classical statistical analysis of Poisson processes with background, Phys. Rev. D, 59 (1999), 053001. https://doi.org/10.1103/PhysRevD.59.053001 doi: 10.1103/PhysRevD.59.053001
|
| [4] |
M. Mandelkern, Setting confidence intervals for bounded parameters, Statist. Sci., 17 (2002), 149–172. https://doi.org/10.1214/ss/1030550859 doi: 10.1214/ss/1030550859
|
| [5] |
D. A. S. Fraser, N. Reid, A. C. M. Wong, Inference for bounded parameters, Phys. Rev. D, 69 (2004), 033002. https://doi.org/10.1103/PhysRevD.69.033002 doi: 10.1103/PhysRevD.69.033002
|
| [6] |
Y. Zhu, Upper limit for Poisson variable incorporating systematic uncertainties by Bayesian approach, Nucl. Instrum. Methods Phys. Res. A, 578 (2007), 322–328. https://doi.org/10.1016/j.nima.2007.05.116 doi: 10.1016/j.nima.2007.05.116
|
| [7] | B. P. Roe, M. B. Woodroofe, Setting confidence belts, Phys. Rev. D, 63 (2000), 013009. https://doi.org/10.1103/PhysRevD.63.013009 |
| [8] |
T. Zhang, M. Woodroofe, Credible and confidence sets for restricted parameter spaces, J. Stat. Plan. Infer., 115 (2003), 479–490. https://doi.org/10.1016/S0378-3758(02)00170-2 doi: 10.1016/S0378-3758(02)00170-2
|
| [9] |
D. E. Leaf, C. Liu, Inference about constrained parameters using the elastic belief method, Int. J. Approx. Reason., 53 (2012), 709–727. https://doi.org/10.1016/j.ijar.2012.02.003 doi: 10.1016/j.ijar.2012.02.003
|
| [10] |
H. Lu, H. Jin, Z. Wang, Y. Li, Confidence intervals for a Poisson parameter with background, Commun. Stat. Theory Methods, 52 (2023), 6794–6805. https://doi.org/10.1080/03610926.2022.2033268 doi: 10.1080/03610926.2022.2033268
|
| [11] |
C. Chen, S. Chen, S. Wang, D. Wang, Y. Chen, Z. Zeng, Fiducial inference framework for restricted parameter spaces: Poisson mean with background, BMC Med. Res. Methodol., 26 (2026), 48. https://doi.org/10.1186/s12874-026-02812-5 doi: 10.1186/s12874-026-02812-5
|
| [12] |
A. P. Dempster, The Dempster–Shafer calculus for statisticians, Int. J. Approx. Reason., 48 (2008), 365–377. https://doi.org/10.1016/j.ijar.2007.03.004 doi: 10.1016/j.ijar.2007.03.004
|
| [13] | G. Shafer, A mathematical theory of evidence, Princeton: Princeton University Press, 1976. https://doi.org/10.1515/9780691214696 |
| [14] |
R. Martin, C. Liu, Inferential models: A framework for prior-free posterior probabilistic inference, J. Am. Stat. Assoc., 108 (2013), 301–313. https://doi.org/10.1080/01621459.2012.747960 doi: 10.1080/01621459.2012.747960
|
| [15] |
H. Lu, H. Jin, Z. Wang, C. Chen, Y. Lu, Prior-free probabilistic interval estimation for binomial proportion, TEST, 28 (2019), 522–542. https://doi.org/10.1007/s11749-018-0588-0 doi: 10.1007/s11749-018-0588-0
|
| [16] |
Y. Fukuda, T. Hayakawa, E. Ichihara, K. Inoue, K. Ishihara, H. Ishino, et al., Evidence for oscillation of atmospheric neutrinos, Phys. Rev. Lett., 81 (1998), 1562–1567. https://doi.org/10.1103/PhysRevLett.81.1562 doi: 10.1103/PhysRevLett.81.1562
|
| [17] |
A. Nucciotti, Still too small to be measured, Nat. Phys., 18 (2022), 128–129. https://doi.org/10.1038/s41567-021-01495-7 doi: 10.1038/s41567-021-01495-7
|
| [18] |
M. Aker, D. Batzler, A. Beglarian, J. Behrens, J. Beisenkötter, M. Biassoni, et al., Direct neutrino-mass measurement based on 259 days of KATRIN data, Science, 388 (2025), 180–185. https://doi.org/10.1126/science.adq9592 doi: 10.1126/science.adq9592
|
| [19] | D. Levy, Neutrino experiments explore the unknown with ORNL expertise, equipment, In: Oak Ridge national laboratory review, 50 (2017), 20–21. Available from: https://www.ornl.gov/content/ornl-review-v50n1 |
| [20] |
K. Eitel, B. Zeitnitz, KARMEN collaboration, The search for neutrino oscillations $\overline{\nu_\mu} \to \overline{\nu_e}$ with KARMEN, Nucl. Phys. B Proc. Suppl., 77 (1999), 212–219. https://doi.org/10.1016/S0920-5632(99)00420-X doi: 10.1016/S0920-5632(99)00420-X
|
Figures(10) / Tables(4)
Hezhi Lu, Qijun Wu. Prior-free probabilistic interval estimation for constrained parameters[J]. AIMS Mathematics, 2026, 11(6): 17320-17353. doi: 10.3934/math.2026709
DownLoad: