Conditional inference in parametric models
[Inférence conditionnelle dans les modèles paramétriques]
Journal de la société française de statistique, Tome 160 (2019) no. 2, pp. 48-66.

Cet article propose une nouvelle approche d’inférence statistique, fondée sur la simulation d’échantillons conditionnés par une statistique des données. L’approximation de la vraisemblance conditionnelle de longues séries d’échantillons sachant la statistique des données admet une forme explicite qui est présentée. Lorsque la statistique de conditionnement est exhaustive par rapport à un paramètre fixé, on montre que la densité approchée est également invariante par rapport à ce même paramètre. Une nouvelle procédure de Rao-Blackwell est proposée et les simulations réalisées montrent que le théorème de Lehmann Scheffé reste valide pour cette approximation. L’inférence conditionnelle sur les familles exponentielles avec paramètre de nuisance est également étudiée, menant à des tests de Monte Carlo, dont les performances sur échantillonnage conditionnel sont comparées à celles sur bootstrap paramétrique. Enfin, on s’intéresse à l’estimation du paramètre d’intérêt par la vraisemblance conditionnelle.

This paper presents a new approach to conditional inference, based on the simulation of samples conditioned by a statistics of the data. Also an explicit expression for the approximation of the conditional likelihood of long runs of the sample given the observed statistics is provided. It is shown that when the conditioning statistics is sufficient for a given parameter, the approximating density is still invariant with respect to the parameter. A new Rao-Blackwellisation procedure is proposed and simulation shows that Lehmann Scheffé Theorem is valid for this approximation. Conditional inference for exponential families with nuisance parameter is also studied, leading to Monte Carlo tests; comparison with the parametric bootstrap method is discussed. Finally the estimation of the parameter of interest through conditional likelihood is considered.

Mots clés : Inférence conditionnelle, Théorème de Rao Blackwell, Théorème de Lehmann Scheffé, Familles exponentielles, Paramètre de nuisance, Simulation
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Broniatowski, Michel; Caron, Virgile. Conditional inference in parametric models. Journal de la société française de statistique, Tome 160 (2019) no. 2, pp. 48-66. http://www.numdam.org/item/JSFS_2019__160_2_48_0/

[1] Barndorff-Nielsen, O.E. (1978). Information and exponential families in statistical theory. Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley & Sons. | MR 489333 | Zbl 0387.62011

[2] Bardnörff-Nielsen, O.E. and Cox, R.R. (1994). Inference and Asymptotics. Chapman & Hall, London. | MR 1317097 | Zbl 0826.62004

[3] Basu, D. (1977). On the elimination of nuisance parameters. J. Amer. Statist. Assoc. 72 (1977), no. 358, 355–366. | MR 451477 | Zbl 0395.62003

[4] Broniatowski M. and Caron, V. (2011). Long runs under a conditional limit distribution. Ann. Appl. Probab. 24 (2014), no. 6, 2246–2296. | MR 3262503

[5] Broniatowski M. and Caron, V. (2011). Small variance estimators for rare event probabilities. ACM Trans. Model. Comput. Simul. 23 (2013), no. 1, Art. 7, 23 pp. | MR 3034217

[6] Casella, G. and Robert, C. P. (1996). Rao-Blackwellisation of sampling schemes. Biometrika 83 , no. 1, 81–94. | MR 1399157 | Zbl 0866.62024

[7] Casella, G. and R. C. P. (1998). Post-processing accept-reject samples: recycling and rescaling. J. Comput. Graph. Statist. 7 , no. 2, 139–157 | MR 1649370

[8] Cheng, R.C.H. (1984). Generation of inverse Gaussian variates with given sample mean and dispersion. Appl. Statist. 33, 309–16 | MR 782074 | Zbl 0582.65003

[9] Dembo, A. and Zeitouni, O. (1996) Refinements of the Gibbs conditioning principle. Probab. Theory Related Fields 104 1–14. | MR 1367663 | Zbl 0838.60025

[10] Diaconis, P. and Freedman, D.A. (1988) Conditional limit theorems for exponential families and finite versions of de Finetti’s theorem. J. Theoret. Probab. 1 381–410. | MR 958245 | Zbl 0655.60029

[11] Efron, B. (1975). Defining the curvature of a statistical problem (with applications to second order efficiency) (with discussion). Ann. Statist. 3, 1189-1242. | MR 428531 | Zbl 0321.62013

[12] Efron, B. (1978). The geometry of exponential families. Ann. Statist. 6, 362-376. | MR 471152 | Zbl 0436.62027

[13] Efron, B. (1979), Bootstrap methods: another look at the jackknife. Ann. Statist. 7 , no. 1, 1–26. | MR 515681 | Zbl 0406.62024

[14] Engen, S. and Lillegard, M. (1997). Stochastic simulations conditioned on sufficient statistics. Biometrika, 84, 235–240. | MR 1450203 | Zbl 0887.62021

[15] Fraser, D. A. S. (2004). Ancillaries and conditional inference. With comments by Ronald W. Butler, Ib M. Skovgaard, Rudolf Beran and a rejoinder by the author. Statist. Sci. 19 , no. 2, 333–369 | MR 2140544 | Zbl 1100.62534

[16] Iacobucci, A., Marin, J.-M., Robert, C. (2010). On variance stabilisation in population Monte Carlo by double Rao-Blackwellisation. Comput. Statist. Data Anal. 54 , no. 3, | MR 2744426

[17] Jöckel, K.-H. (1986). Finite sample properties and asymptotic efficiency of Monte Carlo tests. Ann. of Stat., 14, 336–347. | MR 829573 | Zbl 0589.62015

[18] Lehmann, E.L. (1986). Testing Statistical Hypotheses. Springer. | MR 852406

[19] Lindqvist, B.H., Taraldsen, G., Lillegärd, M. and Engen, S. (2003). A counterexample to a claim about stochastic simulations. Biometrika 90 , no. 2, 489–490. | MR 1986665 | Zbl 1040.65001

[20] Lindqvist, B.H. and Taraldsen, G. (2005). Monte Carlo conditioning on a sufficient statistic. Biometrika 92 , no. 2, 451–464. | MR 2201370 | Zbl 1094.62013

[21] Lockhart, R. and O’Reilly, F. (2005) A note on Moore’s conjecture. Statist. Probab. Lett. 74 , no. 2, 212–220. | MR 2169379 | Zbl 1070.62032

[22] Lockhart, R A., O Reilly, F J. and Stephens, A. (2007) Use of the Gibbs sampler to obtain conditional tests, with applications. Biometrika 94 , no. 4, 992–998. | MR 2416805 | Zbl 1156.62307

[23] Lockhart, R.A. and Stephens, M. A. (1994). Estimation and tests of fit for the three–parameter Weibull distribution. J. Roy. Statist. Soc.. B. 56:491–500. | MR 1278222 | Zbl 0800.62145

[24] O’Reilly, F. and Gravia-Medrano, L. (2006). On the conditional distribution of goodness-of-fit tests. Commun. Statist. A, 35:541–9. | MR 2274070 | Zbl 1084.62001

[25] Moore, David S. (1973). A note on Srinivasan’s goodness-of-fit test. Biometrika 60 , 209–211. | MR 418331 | Zbl 0258.62026

[26] Pedersen, B.V., (1979). Approximating conditional distributions by the mixed Edgeworth-saddlepoint expansion. Biometrika, 66(3), 597–604. | Zbl 0417.62014

[27] Pace L. and Salvan A., (1992). A note on conditional cumulants in canonical exponential families. Scand. J. Statist., 19, 185–191. | MR 1173599 | Zbl 0748.62012

[28] Perron, F. (1999). Beyond accept-reject sampling. Biometrika 86 , no. 4, 803–813 | MR 1741978 | Zbl 0943.65014

[29] Reid, N. (1995). The roles of conditioning in inference. With comments by V. P. Godambe, Bruce G. Lindsay and Bing Li, Peter McCullagh, George Casella, Thomas J. DiCiccio and Martin T. Wells, A. P. Dawid and C. Goutis and Thomas Severini. With a rejoinder by the author. Statist. Sci. 10 , no. 2, 138–157, 173–189, 193–196. | MR 1368097 | Zbl 0955.62524

[30] Douc, R. and Robert, C.P.(2011). A vanilla Rao-Blackwellization of Metropolis-Hastings algorithms. Ann. Statist. 39 , no. 1, 261–277 | MR 2797846 | Zbl 1209.62023

[31] Sundberg, R. (2009). Flat and multimodal likelihoods and model lack of fit in curved exponential families. Research Report 2009:1, http://www.math.su.se/matstat. | MR 2779640 | Zbl 1226.62007