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Kitagawa, Hayato; Uchida, Masayuki

Adaptive test statistics for ergodic diffusion processes sampled at discrete times. (English) Zbl 1287.62016

J. Stat. Plann. Inference 150, 84-110 (2014).
Summary: We consider adaptive test statistics for ergodic diffusion processes based on discrete observations. Since an exact likelihood function for the discretely observed diffusion process cannot been generally obtained, the quasi-likelihood function based on the Ito-Taylor expansion is used and three kinds of test statistics, the likelihood ratio type test statistic, Wald type test statistic and Rao’s score type test statistic, for diffusion processes, are proposed. It is shown that the test statistics converge in distribution to \(\chi^2\) (the chi-squared distribution) under null hypothesis and the tests are consistent. Moreover, we prove that the test statistics converge in distribution to the noncentral \(\chi^2\) under a local alternative hypothesis.

Cited in 7 Documents

MSC:

62M02 Markov processes: hypothesis testing
62F05 Asymptotic properties of parametric tests

Keywords:

consistent tests; discrete time observations; likelihood ratio test; maximum likelihood type estimator; Rao’s score test; stochastic differential equations; Wald test

Software:

sde
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References:

[1] Bibby, B. M.; Sørensen, M., Martingale estimating functions for discretely observed diffusion processes, Bernoulli, 1, 17-39 (1995) · Zbl 0830.62075
[2] De Gregorio, A.; Iacus, S. M., Adaptive LASSO-type estimation for ergodic diffusion processes, Econometric Theory, 28, 838-860 (2012) · Zbl 1419.62170
[3] De Gregorio, A.; Iacus, S. M., On a family of test statistics for discretely observed diffusion processes, J. Multivariate Anal., 122, 292-316 (2013) · Zbl 1280.62094
[4] Florens-Zmirou, D., Approximate discrete time schemes for statistics of diffusion processes, Statistics, 20, 547-557 (1989) · Zbl 0704.62072
[5] Genon-Catalot, V.; Jacod, J., On the estimation of the diffusion coefficient for multidimensional diffusion processes, Ann. Inst. H. Poincaré Probab. Statist., 29, 119-151 (1993) · Zbl 0770.62070
[6] Gobet, E., LAN property for ergodic diffusions with discrete observations, Ann. Inst. H. Poincare Probab. Statist.´, 38, 711-737 (2002) · Zbl 1018.60076
[7] Iacus, S. M., Simulation and Inference for Stochastic Differential Equations: With R Examples (2008), Springer-Verlag: Springer-Verlag New York · Zbl 1210.62112
[8] Kessler, M., Estimation des paramètres d׳une diffusion par des contrastes corrigés, C. R. Acad. Sci. Paris Ser. I Math., 320, 359-362 (1995) · Zbl 0831.62059
[9] Kessler, M., Estimation of an ergodic diffusion from discrete observations, Scand. J. Statist., 24, 211-229 (1997) · Zbl 0879.60058
[12] Kutoyants, Yu. A., Statistical Inference for Ergodic Diffusion Processes (2004), Springer-Verlag: Springer-Verlag London · Zbl 1038.62073
[13] Lehmann, E. L., Elements of Large-Sample Theory (1999), Springer-Verlag: Springer-Verlag New York · Zbl 0914.62001
[14] Polansky, A. M., Introduction to Statistical Limit Theory (2011), Chapman & Hall: Chapman & Hall Boca Raton · Zbl 1344.60002
[15] Prakasa Rao, B. L.S., Asymptotic theory for nonlinear least squares estimator for diffusion processes, Math. Operationsforsch. Statist. Ser. Statist., 14, 195-209 (1983) · Zbl 0532.62060
[16] Prakasa Rao, B. L.S., Statistical inference from sampled data for stochastic processes, Contemp. Math., 80, 249-284 (1988) · Zbl 0687.62069
[17] Sen, P. K.; Singer, J. M., Large Sample Methods in Statistics (1993), Chapman & Hall: Chapman & Hall New York · Zbl 0867.62003
[18] Uchida, M., Contrast-based information criterion for ergodic diffusion processes from discrete observations, Ann. Inst. Statist. Math., 62, 161-187 (2010) · Zbl 1422.60135
[19] Uchida, M.; Yoshida, N., Estimation for misspecified ergodic diffusion processes from discrete observations, European Series in Applied and Industrial Mathematics: Probability and Statistics, 15, 270-290 (2011) · Zbl 1395.62258
[20] Uchida, M.; Yoshida, N., Adaptive estimation of an ergodic diffusion process based on sampled data, Stochastic Process. Appl., 122, 2885-2924 (2012) · Zbl 1243.62113
[21] Uchida, M.; Yoshida, N., Quasi likelihood analysis of volatility and nondegeneracy of statistical random field, Stochastic Process. Appl., 123, 2851-2876 (2013) · Zbl 1284.62539
[22] Yoshida, N., Estimation for diffusion processes from discrete observation, J. Multivariate Anal., 41, 220-242 (1992) · Zbl 0811.62083
[23] Yoshida, N., Polynomial type large deviation inequalities and quasi-likelihood analysis for stochastic differential equations, Ann. Inst. Statist. Math., 63, 431-479 (2011) · Zbl 1333.62224
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