Abstract
The quasi-likelihood analysis is generalized to the partial quasi-likelihood analysis. Limit theorems for the quasi-likelihood estimators, especially the quasi-Bayesian estimator, are derived in the situation where existence of a slow mixing component prohibits the Rosenthal type inequality from applying to the derivation of the polynomial type large deviation inequality for the statistical random field. We give two illustrative examples.
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Notes
The QLA is not in the sense of Robert Wedderburn. Since exact likelihood function can rarely be assumed in inference for discretely sampled continuous time processes, quasi-likelihood functions are quite often used there. Further, the word “QLA” also implies a new framework of inferential theory for stochastic processes within which the polynomial type large deviation is easily available today and plays an essential role in the theory.
Continuity is not necessary to assume as a matter of fact. Without it, the convergence of \({\mathbb Z}_T\) ensures the limit distribution is supported by \(\hat{C}\); we may assume \({\mathbb Z}\) is a continuous process after modification.
More precisely, the processes \(\tilde{b}=(b_t)_{t\in [0,T]}\), \(a=(a_t)_{t\in [0,T]}\), \(\tilde{a}=(\tilde{a}_t)_{t\in [0,T]}\) and \(\tilde{w}=(\tilde{w}_t)_{t\in [0,T]}\) are measurable mappings defined on \((\Omega ,\mathcal{F})\), and for each \(\omega '\in \Omega '\), the processes \(\tilde{b}(\omega ',\cdot )=(b_t(\omega ',\cdot ))_{t\in [0,T]}\), \(a(\omega ',\cdot )=(a_t(\omega ',\cdot ))_{t\in [0,T]}\), \(\tilde{a}(\omega ',\cdot )=(\tilde{a}_t(\omega ',\cdot ))_{t\in [0,T]}\) and \(\tilde{w}(\omega ',\cdot )=(\tilde{w}_t(\omega ',\cdot ))_{t\in [0,T]}\) satisfy the required conditions. \(\tilde{w}(\omega ',\cdot )\) is independent of \(w(\omega ',\cdot )\), i.e., \(\tilde{w}\) and w are \(\mathcal{C}\)-conditionally independent, though this independency is not indispensable.
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Acknowledgements
This work was in part supported by CREST JPMJCR14D7 Japan Science and Technology Agency; Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No. 17H01702 (Scientific Research) and by a Cooperative Research Program of the Institute of Statistical Mathematics.
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Yoshida, N. Partial quasi-likelihood analysis. Jpn J Stat Data Sci 1, 157–189 (2018). https://doi.org/10.1007/s42081-018-0006-6
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DOI: https://doi.org/10.1007/s42081-018-0006-6