## Transform martingale estimating functions.(English)Zbl 1126.62074

Summary: An estimation method is proposed for a wide variety of discrete time stochastic processes that have an intractable likelihood function but are otherwise conveniently specified by an integral transform such as the characteristic function, the Laplace transform or the probability generating function. This method involves the construction of classes of transform-based martingale estimating functions that fit into the general framework of quasi-likelihood.
In the parametric setting of a discrete time stochastic process, we obtain transform quasi-score functions by projecting the unavailable score function onto the special linear spaces formed by these classes. The specification of the process by any of the main integral transforms makes possible an arbitrarily close approximation of the score function in an infinite-dimensional Hilbert space by optimally combining transform martingale quasi-score functions. It also allows an extension of the domain of application of quasi-likelihood methodology to processes with infinite conditional second moment.

### MSC:

 62M09 Non-Markovian processes: estimation 60E10 Characteristic functions; other transforms 62G20 Asymptotic properties of nonparametric inference 62M99 Inference from stochastic processes 60G42 Martingales with discrete parameter 62M05 Markov processes: estimation; hidden Markov models 46N30 Applications of functional analysis in probability theory and statistics
Full Text:

### References:

 [1] Abraham, B. and Balakrishna, N. (1999). Inverse Gaussian autoregressive models. J. Time Ser. Anal. 20 605–618. · Zbl 0939.62088 [2] Alzaid, A. A. and Al-Osh, M. A. (1990). An integer-valued $$p$$th-order autoregressive structure (INAR($$p$$)) process. J. Appl. Probab. 27 314–324. JSTOR: · Zbl 0704.62081 [3] Billard, L. and Mohamed, F. Y. (1991). Estimation of the parameters of an EAR($$p$$) process. J. Time Ser. Anal. 12 179–192. · Zbl 0743.62079 [4] Brant, R. (1984). Approximate likelihood and probability calculations based on transforms. Ann. Statist. 12 989–1005. · Zbl 0542.62001 [5] Brockwell, P. J. and Liu, J. (1992). Estimating the noise parameters from observations of a linear process with stable innovations. J. Statist. Plann. Inference 33 175–186. · Zbl 0781.62137 [6] Burrill, C. W. (1972). Measure , Integration and Probability . McGraw-Hill, New York. · Zbl 0248.28001 [7] Crowder, M. (1987). On linear and quadratic estimating functions. Biometrika 74 591–597. JSTOR: · Zbl 0635.62077 [8] DuMouchel, W. H. (1975). Stable distributions in statistical inference. II. Information from stably distributed samples. J. Amer. Statist. Assoc. 70 386–393. JSTOR: · Zbl 0321.62017 [9] Epps, T. W. and Pulley, L. B. (1985). Parameter estimates and tests of fit for infinite mixture distributions. Comm. Statist. Theory Methods 14 3125–3145. [10] Feigin, P. D., Tweedie, R. L. and Belyea, C. (1983). Weighted area techniques for explicit parameter estimation in multistage models. Austral. J. Statist. 25 1–16. · Zbl 0514.62113 [11] Feuerverger, A. (1990). An efficiency result for the empirical characteristic function in stationary time-series models. Canad. J. Statist. 18 155–161. JSTOR: · Zbl 0703.62096 [12] Feuerverger, A. and McDunnough, P. (1981). On some Fourier methods for inference. J. Amer. Statist. Assoc. 76 379–387. JSTOR: · Zbl 0463.62030 [13] Feuerverger, A. and McDunnough, P. (1981). On the efficiency of empirical characteristic function procedures. J. Roy. Statist. Soc. Ser. B 43 20–27. JSTOR: · Zbl 0454.62034 [14] Feuerverger, A. and McDunnough, P. (1984). On statistical transform methods and their efficiency. Canad. J. Statist. 12 303–317. JSTOR: · Zbl 0566.62021 [15] Godambe, V. P. and Heyde, C. C. (1987). Quasi-likelihood and optimal estimation. Internat. Statist. Rev. 55 231–274. JSTOR: · Zbl 0671.62007 [16] Godambe, V. P. and Thompson, M. E. (1989). An extension of quasi-likelihood estimation (with discussion). J. Statist. Plann. Infererence 22 137–172. · Zbl 0681.62036 [17] Grunwald, G. K., Hyndman, R. J., Tedesco, L. and Tweedie, R. L. (2000). Non-Gaussian conditional linear AR(1) models. Aust. N. Z. J. Stat. 42 479–495. · Zbl 1018.62065 [18] Heyde, C. C. (1987). On combining quasilikelihood estimating functions. Stochastic Process. Appl. 25 281–287. · Zbl 0636.62086 [19] Heyde, C. C. (1997). Quasi-Likelihood and Its Application : A General Approach to Optimal Parameter Estimation . Springer, New York. · Zbl 0879.62076 [20] Hoeting, J. A., Tweedie, R. L. and Olver, C. S. (2003). Transform estimation of parameters for stage-frequency data. J. Amer. Statist. Assoc. 98 503–514. · Zbl 1040.62101 [21] Hutton, J. E. and Nelson, P. I. (1986). Quasilikelihood estimation for semimartingales. Stochastic Process. Appl. 22 245–257. · Zbl 0616.62113 [22] Kiefer, N. M. (1978). Comment on “Estimating mixtures of normal distributions and switching regressions,” by R. E. Quandt and J. B. Ramsay. J. Amer. Statist. Assoc. 73 744–745. · Zbl 0401.62024 [23] Klimko, L. A. and Nelson, P. I. (1978). On conditional least squares estimation for stochastic processes. Ann. Statist. 6 629–642. · Zbl 0383.62055 [24] Leitnaker, M. G. (1989). Estimation of delay times in stochastic compartmental models. Biometrics 45 1239–1247. · Zbl 0718.62214 [25] McCulloch, J. H. (1996). Financial applications of stable distributions. In Statistical Methods in Finance (G. S. Maddala and C. R. Rao, eds.) 393–425. North-Holland, Amsterdam. [26] McLeish, D. L. (1984). Estimation for aggregate models: The aggregate Markov chain (with discussion). Canad. J. Statist. 12 265–285. JSTOR: · Zbl 0574.62084 [27] Merkouris, T. (1992). A transform method for optimal estimation in stochastic processes: Basic aspects. In Proc. Symposium in Honour of Professor V. P. Godambe (J. Chen, ed.). Univ. Waterloo, Waterloo, Canada. [28] Merkouris, T. (1992). A transform method for optimal estimation in stochastic processes. Ph.D. dissertation, Dept. Statistics, Univ. Waterloo. [29] Murphy, S. and Li, B. (1995). Projected partial likelihood and its application to longitudinal data. Biometrika 82 399–406. JSTOR: · Zbl 0823.62091 [30] Nikias, C. L. and Shao, M. (1995). Signal Processing with Alpha-Stable Distributions and Applications . Wiley, New York. [31] Rao, C. R. (1973). Linear Statistical Inference and Its Applications , 2nd ed. Wiley, New York. · Zbl 0256.62002 [32] Schuh, H.-J. and Tweedie, R. L. (1979). Parameter estimation using transform estimation in time-evolving models. Math. Biosci. 45 37–67. · Zbl 0418.62074 [33] Sim, C. H. (1990). First-order autoregressive models for gamma and exponential processes. J. Appl. Probab. 27 325–332. JSTOR: · Zbl 0718.60034 [34] Small, C. G. and McLeish, D. L. (1994). Hilbert Space Methods in Probability and Statistical Inference . Wiley, New York. · Zbl 0838.62002 [35] Sørensen, M. (1990). On quasi-likelihood for semimartingales. Stochastic Process. Appl. 35 331–346. · Zbl 0714.62075 [36] Wedderburn, R. W. M. (1974). Quasi-likelihood functions, generalized linear models and the Gauss–Newton method. Biometrika 61 439–447. JSTOR: · Zbl 0292.62050 [37] Yao, Q. and Morgan, B. J. T. (1999). Empirical transform estimation for indexed stochastic models. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 127–141. JSTOR: · Zbl 0913.62023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.