Abstract
By extrapolating the explicit formula of the zero-bias distribution occurring in the context of Stein’s method, we construct characterization identities for a large class of absolutely continuous univariate distributions. Instead of trying to derive characterizing distributional transformations that inherit certain structures for the use in further theoretic endeavors, we focus on explicit representations given through a formula for the density- or distribution function. The results we establish with this ambition feature immediate applications in the area of goodness-of-fit testing. We draw up a blueprint for the construction of tests of fit that include procedures for many distributions for which little (if any) practicable tests are known. To illustrate this last point, we construct a test for the Burr Type XII distribution for which, to our knowledge, not a single test is known aside from the classical universal procedures.
Similar content being viewed by others
References
Allison, J. S., Santana, L. (2015). On a data-dependent choice of the tuning parameter appearing in certain goodness-of-fit tests. Journal of Statistical Computation and Simulation, 85(16), 3276–3288.
Anastasiou, A. (2018). Assessing the multivariate normal approximation of the maximum likelihood estimator from high-dimensional, heterogeneous data. Electronic Journal of Statistics, 12(2), 3794–3828.
Anastasiou, A., Gaunt, R. (2019). Multivariate normal approximation of the maximum likelihood estimator via the delta method. to appear in Brazilian Journal of Probability and StatisticsarXiv:1609.03970.
Anastasiou, A., Reinert, G. (2017). Bounds for the normal approximation of the maximum likelihood estimator. Bernoulli, 23(1), 191–218.
Anastasiou, A., Reinert, G. (2018). Bounds for the asymptotic distribution of the likelihood ratio. arXiv e-prints arXiv:1806.03666.
Barbour, A. D. (1982). Poisson convergence and random graphs. Mathematical Proceedings of the Cambridge Philosophical Society, 92(2), 349–359.
Barbour, A. D. (1990). Stein’s method for diffusion approximations. Probability Theory and Related Fields, 84(3), 297–322.
Barbour, A. D., Karoński, M., Ruciński, A. (1989). A central limit theorem for decomposable random variables with applications to random graphs. Journal of Combinatorial Theory, Series B, 47(2), 125–145.
Baringhaus, L., Henze, N. (1988). A consistent test for multivariate normality based on the empirical characteristic function. Metrika, 35(1), 339–348.
Baringhaus, L., Henze, N. (2000). Tests of fit for exponentiality based on a characterization via the mean residual life function. Statistical Papers, 41(2), 225–236.
Betsch, S., Ebner, B. (2019a). A new characterization of the Gamma distribution and associated goodness-of-fit tests. Metrika, 82(7), 779–806.
Betsch, S., Ebner, B. (2019b). Testing normality via a distributional fixed point property in the Stein characterization. TEST, https://doi.org/10.1007/s11749-019-00630-0.
Braverman, A., Dai, J. G. (2017). Stein’s method for steady-state diffusion approximations of \({M} / \mathit{Ph} / n + {M}\) systems. The Annals of Applied Probability, 27(1), 550–581.
Braverman, A., Dai, J. G., Feng, J. (2016). Stein’s method for steady-state diffusion approximations: An introduction through the Erlang-A and Erlang-C models. Stochastic Systems, 6(2), 301–366.
Cabaña, A., Quiroz, A. (2005). Using the empirical moment generating function in testing for the Weibull and the type I extreme value distributions. TEST, 14(2), 417–432.
Carrillo, C., Cidrás, J., Díaz-Dorado, E., Obando-Montaño, A. F. (2014). An approach to determine the Weibull parameters for wind energy analysis: The case of Galicia (Spain). Energies, 7(4), 2676–2700.
Chatterjee, S., Shao, Q.-M. (2011). Nonnormal approximation by Stein’s method of exchangeable pairs with application to the Curie–Weiss model. The Annals of Applied Probability, 21(2), 464–483.
Chen, L. H. Y., Goldstein, L., Shao, Q.-M. (2011). Normal approximation by Stein’s method. Berlin: Springer.
Chwialkowski, K., Strathmann, H., Gretton, A. (2016). A kernel test of goodness of fit. Proceedings of the 33rd international conference on machine learning, ICML’16 (Vol. 48, pp. 2606–2615).
Döbler, C. (2015). Stein’s method of exchangeable pairs for the Beta distribution and generalizations. Electronic Journal of Probability, 20(109), 1–34.
Döbler, C. (2017). Distributional transformations without orthogonality relations. Journal of Theoretical Probability, 30(1), 85–116.
Epps, T. W., Pulley, L. B. (1983). A test for normality based on the empirical characteristic function. Biometrika, 70(3), 723–726.
Fang, X. (2014). Discretized normal approximation by Stein’s method. Bernoulli, 20(3), 1404–1431.
Gaunt, R., Pickett, A., Reinert, G. (2017). Chi-square approximation by Stein’s method with application to Pearson’s statistic. Annals of Applied Probability, 27(2), 720–756.
Goldstein, L., Reinert, G. (1997). Stein’s method and the zero bias transformation with application to simple random sampling. The Annals of Applied Probability, 7(4), 935–952.
Goldstein, L., Reinert, G. (2005). Distributional transformations, orthogonal polynomials, and Stein characterizations. Journal of Theoretical Probability, 18(1), 237–260.
Götze, F. (1991). On the rate of convergence in the multivariate CLT. The Annals of Probability, 19(2), 724–739.
Henze, N., Jiménez-Gamero, M. D. (2019). A new class of tests for multinormality with iid and garch data based on the empirical moment generating function. TEST, 28(2), 499–521.
Henze, N., Klar, B. (2002). Goodness-of-fit tests for the inverse Gaussian distribution based on the empirical Laplace transform. Annals of the Institute of Statistical Mathematics, 54(2), 425–444.
Henze, N., Meintanis, S. G., Ebner, B. (2012). Goodness-of-fit tests for the Gamma distribution based on the empirical Laplace transform. Communications in Statistics-Theory and Methods, 41(9), 1543–1556.
Hudson, H. M. (1978). A natural identity for exponential families with applications in multiparameter estimation. The Annals of Statistics, 6(3), 473–484.
Jalali, A., Watkins, A. J. (2009). On maximum likelihood estimation for the two parameter Burr XII distribution. Communications in Statistics—Theory and Methods, 38(11), 1916–1926.
Jiménez-Gamero, M. D., Alba-Fernández, V., Muñoz-García, J., Chalco-Cano, Y. (2009). Goodness-of-fit tests based on empirical characteristic functions. Computational Statistics & Data Analysis, 53(12), 3957–3971.
Kim, S.-T. (2000). A use of the Stein-Chen method in time series analysis. Journal of Applied Probability, 37(4), 1129–1136.
Kleiber, C., Kotz, S. (2003). Statistical size distributions in economics and actuarial sciences. Wiley series in probability and statistics. Hoboken: Wiley.
Ley, C. and Swan, Y. (2011). A unified approach to Stein characterizations. arXiv e-prints arXiv:1105.4925v3.
Ley, C., Swan, Y. (2013a). Local Pinsker inequalities via Stein’s discrete density approach. IEEE Transactions on Information Theory, 59(9), 5584–5591.
Ley, C., Swan, Y. (2013b). Stein’s density approach and information inequalities. Electronic Communications in Probability, 18, 1–14.
Ley, C., Swan, Y. (2016). Parametric Stein operators and variance bounds. Brazilian Journal of Probability and Statistics, 30(2), 171–195.
Ley, C., Reinert, G., Swan, Y. (2017). Stein’s method for comparison of univariate distributions. Probability Surveys, 14, 1–52.
Linnik, Y. V. (1962). Linear forms and statistical criteria I, II. Selected Translations in Mathematical Statistics and Probability, 3,1–40: 41–90. Originally published 1953 in the Ukrainian Mathematical Journal, Vol. 5, pp. 207–243, 247–290 (in Russian).
Liu, Q., Lee, J. D., Jordan, M. (2016). A kernelized Stein discrepancy for goodness-of-fit tests. Proceedings of the 33rd International Conference on Machine Learning, ICML’16, (Vol. 46, pp. 276–284).
Nikitin, Y. Y. (2017). Tests based on characterizations, and their efficiencies: A survey. Acta et Commentationes Universitatis Tartuensis de Mathematica, 21(1), 3–24.
O’Reilly, F. J., Stephens, M. A. (1982). Characterizations and goodness of fit tests. Journal of the Royal Statistical Society: Series B (Methodological), 44(3), 353–360.
Peköz, E. A., Röllin, A. (2011). New rates for exponential approximation and the theorems of Rényi and Yaglom. The Annals of Probability, 39(2), 587–608.
Pinelis, I. (2017). Optimal-order uniform and nonuniform bounds on the rate of convergence to normality for maximum likelihood estimators. Electronic Journal of Statistics, 11(1), 1160–1179.
Prakasa Rao, B. L. S. (1979). Characterizations of distributions through some identities. Journal of Applied Probability, 16(4), 903–909.
Proakis, J. G., Salehi, M. (2008). Digital communications, 5th ed. New York: McGraw-Hill.
R Core Team (2019). R: A language and environment for statistical computing. Vienna: R Foundation for Statistical Computing.
Reinert, G., Röllin, A. (2010). Random subgraph counts and U-statistics: Multivariate normal approximation via exchangeable pairs and embedding. Journal of Applied Probability, 47(2), 378–393.
Rogers, G. L. (2008). Multiple path analysis of reflectance from turbid media. Journal of the Optical Society of America A, 25(11), 2879–2883.
Ross, N. (2011). Fundamentals of Stein’s method. Probability Surveys, 8, 210–293.
Shah, A., Gokhale, D. V. (1993). On maximum product of spacings (mps) estimation for Burr XII distributions. Communications in Statistics—Simulation and Computation, 22(3), 615–641.
Singh, S. K., Maddala, G. S. (1976). A function for size distribution of incomes. Econometrica, 44(5), 963–970.
Singh, V. P. (1987). On application of the Weibull distribution in hydrology. Water Resources Management, 1(1), 33–43.
Stein, C. (1986). Approximate computation of expectations, Vol. 7. Hayward: Institute of Mathematical Statistics.
Stein, C., Diaconis, P., Holmes, S., Reinert, G. (2004). Use of exchangeable pairs in the analysis of simulations. In P. Diaconis & S. Holmes (Eds.), Stein’s method. Lecture notes-monograph series, Vol. 46, pp. 1–25. Beachwood, OH: Institute of Mathematical Statistics.
Tenreiro, C. (2019). On the automatic selection of the tuning parameter appearing in certain families of goodness-of-fit tests. Journal of Statistical Computation and Simulation, 89(10), 1780–1797.
Wingo, D. R. (1983). Maximum likelihood methods for fitting the Burr type XII distribution to life test data. Biometrical Journal, 25(1), 77–84.
Ying, L. (2017). Stein’s method for mean-field approximations in light and heavy traffic regimes. SIGMETRICS 2017 abstracts—Proceedings of the 2017 ACM SIGMETRICS/International conference on measurement and modeling of computer systems. Association for Computing Machinery, Inc.
Zghoul, A. A. (2010). A goodness of fit test for normality based on the empirical moment generating function. Communications in Statistics—Simulation and Computation, 39(6), 1292–1304.
Acknowledgements
The authors would like to thank an associate editor as well as three anonymous reviewers for their comments and suggestions that led to a major improvement of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Below is the link to the electronic supplementary material.
About this article
Cite this article
Betsch, S., Ebner, B. Fixed point characterizations of continuous univariate probability distributions and their applications. Ann Inst Stat Math 73, 31–59 (2021). https://doi.org/10.1007/s10463-019-00735-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-019-00735-1