Exact statistical inference for quantities of loggamma distribution

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Abstract

Transformed gamma distributions are widely used in various fields such as survival analysis, reliability engineering, and environmental studies. Statistical inference for the quantities of the transformed gamma distributions is of great importance in these applications. However, traditional inference methods often rely on large sample approximations, which is not accurate for small sample sizes. To address this issue, we introduced an exact statistical inference framework for quantities of the transformed gamma distributions and applied the framework to loggamma distribution. The performance of the proposed framework was extensively evaluated via comprehensive simulation studies. The results showed that the proposed framework outperforms the parametric bootstrapping method in terms of coverage probability and type 1 error rate. Two real data applications are provided for illustrative purposes.

Keywords

generalized inference / exact inference / parametric inference / gamma distribution / loggamma distribution

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References

[1] C. Kleiber, S. Kotz, Statistical Size Distributions in Economics and Actuarial Sciences , John Wiley & Sons, 2003. https://doi.org/10.1002/0471457175

[2] S. A. Klugman, H. H. Panjer, G. E. Willmot, Loss Models: From Data to Decisions , John Wiley & Sons, 2012.

[3] R. L. Prentice, A log gamma model and its maximum likelihood estimation, Biometrika, 61 (1974), 539-544. https://doi.org/10.2307/2334737

[4] P. Veazie, O. Intrator, B. Kinosian, C. S. Phibbs, Better performance for right-skewed data using an alternative gamma model, BMC Med. Res. Methodol., 23 (2023), 298. https://doi.org/10.1186/s12874-023-02113-1

[5] M. Amini, S. M. T. K. MirMostafaee, J. Ahmadi, Log-gamma-generated families of distributions, Statistics, 48 (2014), 913-932. https://doi.org/10.1080/02331888.2012.748775

[6] B. Liu, M. M. A. Ananda, Analyzing insurance data with an exponentiated composite inverse gamma-Pareto model, Commun. Stat. Theory Methods, 52 (2023), 7618-7631. https://doi.org/10.1007/978-94-017-1431-015

[7] V. P. Singh, Log-Pearson type III distribution, in Entropy-based Parameter Estimation in Hydrology, Springer, (1998), 252-274. https://doi.org/10.1007/978-94-017-1431-0

[8] A. G. Glen, On the inverse gamma as a survival distribution, in Computational Probability Applications, Springer, (2017), 15-30. https://doi.org/10.1007/978-3-319-43317-22

[9] C. T. Lin, B. S. Duran, T. O. Lewis, Inverted gamma as a life distribution, Microelectron. Reliab., 29 (1989), 619-626. https://doi.org/10.1016/0026-2714(89)90352-1

[10] S. Weerahandi, J. Gamage, A general method of inference for two-parameter continuous distributions, Commun. Stat. Theory Methods, 45 (2016), 2612-2625. https://doi.org/10.1080/03610926.2014.887109

[11] M. Ananda, Generalized statistical inference for quantiles of two-parameter gamma distribution, J. Math. Stat., 20 (2024), 1-12. https://doi.org/10.3844/jmssp.2024.1.12

[12] S. M. Jordan, K. Krishnamoorthy, Exact confidence intervals for the common mean of several normal populations, Biometrics, 52 (1996), 77-86. https://doi.org/10.2307/2533146

[13] P. L. H. Yu, Y. Sun, B. K. Sinha, On exact confidence intervals for the common mean of several normal populations, J. Stat. Plann. Inference, 81 (1999), 263-277. https://doi.org/10.1016/S0378-3758(99)00052-X

[14] S. Weerahandi, C. R. Yu, Exact distributions of statistics for making inferences on mixed models under the default covariance structure, J. Stat. Distrib. Appl., 7 (2020), 4. https://doi.org/10.1186/s40488-020-00105-w

[15] S. Weerahandi, Generalized confidence intervals, J. Am. Stat. Assoc., 88 (1993), 899-905. https://doi.org/10.2307/2290779

[16] K. W. Tsui, S. Weerahandi, Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters, J. Am. Stat. Assoc., 84 (1989), 602-607. https://doi.org/10.2307/2289949

[17] Y. Yan, D. Cheng, J. E. Feng, H. Li, J. Yue, Survey on applications of algebraic state space theory of logical systems to finite state machines, Sci. China Inf. Sci., 66 (2023), 111201. https://doi.org/10.1007/s11432-022-3538-4

[18] G. K. K. King, O. Papaspiliopoulos, M. Ruggiero, Exact inference for a class of hidden Markov models on general state spaces, Electron. J. Stat., 15 (2021), 2832-2875. https://doi.org/10.1214/21-EJS1841

[19] R. A. Fisher, On the mathematical foundations of theoretical statistics, Philos. Trans. A Math. Phys. Eng. Sci., 222 (1922), 309-368. https://doi.org/10.1098/rsta.1922.0009

[20] J. Neyman, Su un teorema concernente le cosiddette statistiche sufficienti, G. Ist. Ital. Attuari, 6 (1935), 320-334.

[21] E. L. Lehmann, H. Scheffé, Completeness, similar regions, and unbiased estimation-Part I, in Selected Works of E. L. Lehmann, Springer, (2011), 233-268. https://doi.org/10.1007/978-1-4614-1412-423

[22] D. K. Bhaumik, K. Kapur, R. D. Gibbons, Testing parameters of a gamma distribution for small samples, Technometrics, 51 (2009), 326-334. https://doi.org/10.1198/tech.2009.07038

[23] R. E. Glaser, The ratio of the geometric mean to the arithmetic mean for a random sample from a gamma distribution, J. Am. Stat. Assoc., 71 (1976), 480-487. https://doi.org/10.2307/2285338

[24] J. Lieblein, M. Zelen, Statistical investigation of the fatigue life of deep-groove ball bearings, J. Res. Natl. Bur. Stand., 57 (1956), 273-316. http://doi.org/10.6028/jres.057.033

[25] F. Ashkar, T. B. M. J. Ouarda, Approximate confidence intervals for quantiles of gamma and generalized gamma distributions, J. Hydrol. Eng., 3 (1998), 43-51. https://doi.org/10.1061/(ASCE)1084-0699(1998)3:1(43)