Bartlett identities and large deviations in likelihood theory. (English) Zbl 0951.62014

Summary: The connection between large and small deviation results for the signed square root statistic \(R\) is studied, both for likelihoods and for likelihood-like criterion functions. We show that if \(p-1\) Barlett identities are satisfied to first order, but the \(p\)th identity is violated to this order, then \(\text{cum}_q (R)=O (n^{-q/2})\) for \(3\leq q<p\), whereas \(\text{cum}_p (R)=\kappa_p n^{-(p-2)/2} +O(n^{-p/2})\). We also show that the large deviation behavior of \(R\) is determined by the values of \(p\) and \(\kappa_p\). The latter result is also valid for more general statistics. Affine (additive and/or multiplicative) correction to \(R\) and \(R^2\) are special cases corresponding to \(p=3\) and 4.
The cumulant behavior of \(R\) gives a way of characterizing the extent to which \(R\)-statistics derived from criterion functions other than log likelihoods can be expected to behave like ones derived from true log likelihoods, by looking at the number of Bartlett identities that are satisfied. Empirical and nonparametric survival analysis type likelihoods are analyzed from this perspective via the device of “dual criterion functions”.


62E20 Asymptotic distribution theory in statistics
62F10 Point estimation
60G42 Martingales with discrete parameter
60G44 Martingales with continuous parameter
62J99 Linear inference, regression
62M99 Inference from stochastic processes
62M09 Non-Markovian processes: estimation
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[1] Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1993). Statistical Models based on Counting Processes. Springer, New York. · Zbl 0769.62061
[2] Barndorff-Nielsen, O. E. (1983). On a formula for the distribution of the maximum likelihood estimator. Biometrika 70 343-365. JSTOR: · Zbl 0532.62006
[3] Barndorff-Nielsen, O. E. (1986). Inference on full or partial parameters based on the standardized signed log likelihood ratio. Biometrika 73 307-322. JSTOR: · Zbl 0605.62020
[4] Barndorff-Nielsen, O. E. and Cox, D. R. (1979). Edgeworth and saddle-point approximations with statistical applications (with discussion). J. Roy. Statist. Soc. B 41 279-312. JSTOR: · Zbl 0424.62010
[5] Barndorff-Nielsen, O. E. and Cox, D. R. (1984). Bartlett adjustments to the likelihood ratio statistic and the distribution of the maximum likelihood estimator. J. Roy. Statist. Soc. B 46 483-495. JSTOR: · Zbl 0581.62016
[6] Barndorff-Nielsen, O. E. and Wood, A. T. A. (1998). On large deviations and choice of ancillary for p and r. Bernoulli 4 35-63. Bartlett, M. S. (1953a). Approximate confidence intervals. Biometrika 40 12-19. Bartlett, M. S. (1953b). Approximate confidence intervals. II. More than one unknown parameter. Biometrika 40 306-317. · Zbl 0894.62026
[7] Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion. Ann. Statist. 6 434-451. · Zbl 0396.62010
[8] Chaganty, N. R. and Sethuraman, J. (1985). Large deviation local limit theorems for arbitrary sequences of random variables. Ann. Probab. 13 97-114. · Zbl 0559.60030
[9] Corcoran, S. A., Davison, A. C. and Spady, R. H. (1995). Reliable inference from empirical likelihoods.
[10] Cox, D. R. (1975). Partial likelihood. Biometrika 62 269-276. JSTOR: · Zbl 0312.62002
[11] Cox, D. R. and Reid, N. (1987). Parameter orthogonality and approximate conditional inference. J. Roy. Statist. Soc. B 49 1-18. JSTOR: · Zbl 0616.62006
[12] Daniels, H. E. (1954). Saddlepoint approximations in statistics. Ann. Math. Statist. 25 631-650. · Zbl 0058.35404
[13] Davison, A. C., Hinkley, D. V. and Worton, B. J. (1992). Bootstrap likelihoods. Biometrika 79 113-130. JSTOR: · Zbl 0753.62026
[14] DiCiccio, T. J., Hall, P. and Romano, J. P. (1991). Empirical likelihood is Bartlett-correctable. Ann. Statist. 19 1053-1061. · Zbl 0725.62042
[15] DiCiccio, T. J. and Romano, J. P. (1989). On adjustments based on the signed root of the empirical likelihood ratio statistic. Biometrika 76 447-456. JSTOR: · Zbl 0676.62043
[16] Godambe, V. P. and Heyde, C. C. (1987). Quasi-likelihood and optimal estimation. Int. Statist. Rev. 55 231-244. JSTOR: · Zbl 0671.62007
[17] Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York. · Zbl 0744.62026
[18] Jacod, J. (1975). Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales. Z. Wahrsch. Verw. Gebiete 31 235-253. · Zbl 0302.60032
[19] Jensen, J. L. (1992). The modified signed likelihood statistic and saddlepoint approximations. Biometrika 79 693-703. JSTOR: · Zbl 0764.62021
[20] Jensen, J. L. (1995). Saddlepoint Approximations in Statistics. Oxford University Press. · Zbl 1274.62008
[21] Jensen, J. L. (1997). A simple derivation of r for curved exponential families. Scand. J. Statist. 24 33-46. · Zbl 0923.62026
[22] Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc. 53 457-481. JSTOR: · Zbl 0089.14801
[23] Lawley, D. N. (1956). A general method for approximating the distribution of likelihood ratio criteria. Biometrika 43 295-303. JSTOR: · Zbl 0073.13602
[24] Lazar, N. and Mykland, P. A. (1999). Empirical likelihood in the presence of nuisance parameters. Biometrika 86 203-211. JSTOR: · Zbl 0917.62029
[25] McCullagh, P. (1987). Tensor Methods in Statistics. Chapman and Hall, London. · Zbl 0732.62003
[26] McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models, 2nd ed. Chapman and Hall, London. · Zbl 0744.62098
[27] McCullagh, P. and Tibshirani, R. (1990). A simple method for the adjustment of profile likelihoods. J. Roy. Statist. Soc. B 52 325-344. JSTOR: · Zbl 0716.62039
[28] McLeish, D. L. and Small, C. G. (1992). A projected likelihood function for semiparametric models. Biometrika 79 93-102. JSTOR: · Zbl 0753.62034
[29] Mykland, P. A. (1995). Dual likelihood. Ann. Statist. 23 396-421. · Zbl 0877.62004
[30] Owen, A. B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75 237-249. JSTOR: · Zbl 0641.62032
[31] Robinson, J., H öglund, T., Holst, L. and Quine, M. P. (1990). On approximating probabilities for small and large deviations on Rd. Ann. Probab. 18 727-753. · Zbl 0704.60018
[32] Skovgaard, I. (1990). On the density of minimum contrast estimators. Ann. Statist. 18 779-789. · Zbl 0709.62029
[33] Skovgaard, I. (1991). Analytic Statistical Models. IMS, Hayward, CA. · Zbl 0755.62003
[34] Skovgaard, I. M. (1996). An explicit large-deviation approximation to one-parameter tests. Bernoulli 2 145-165. · Zbl 1066.62508
[35] Thomas, D. R. and Grunkemeier, G. L. (1975). Confidence interval estimation of survival probabilities for censored data. J. Amer. Statist. Assoc. 70 865-871. JSTOR: · Zbl 0331.62028
[36] Wallace, D. L. (1958). Asymptotic approximations to distributions. Ann. Math. Statist. 29 635- 654. · Zbl 0086.34004
[37] Wedderburn, R. W. M. (1974). Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika 61 439-447. JSTOR: · Zbl 0292.62050
[38] Wong, W. H. (1986). Theory of partial likelihood. Ann. Statist. 14 88-123. · Zbl 0603.62032
[39] Wong, W. H. and Severini, T. A. (1991). On maximum likelihood estimation in infinite dimensional parameter spaces. Ann. Statist. 19 603-632. · Zbl 0732.62026
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