Asymptotic ancillarity and conditional inference for stochastic processes.

*(English)*Zbl 0757.62013The author aims to obtain limit theorems for the distributions of parameter estimators conditional on observed information. In greater generality, conditioning on any asymptotically ancillary statistic which includes data asymptotically equivalent to the observed information, is considered. The principal result obtained says that the conditional distributions of the randomly-normed score statistic and the randomly- normed maximum likelihood estimator, given such an asymptotically ancillary statistic, are asymptotically normal.

Such a general statement has been rigorously formulated and proved in this paper. The theorems are stated under conditions that allow to apply them to inference from general stochastic processes and are relevant in particular for nonergodic models. The results can be considered as an extension of previous results of the same author [ibid. 14, 925-933 (1986; Zbl 0633.62084)] for supercritical branching processes. The mode of conditional convergence used in the present paper is weaker.

An example illustrates the applicability and the importance of the results. The limit distributions derived can be used to improve the poor conditional performance of the unconditional confidence intervals.

Such a general statement has been rigorously formulated and proved in this paper. The theorems are stated under conditions that allow to apply them to inference from general stochastic processes and are relevant in particular for nonergodic models. The results can be considered as an extension of previous results of the same author [ibid. 14, 925-933 (1986; Zbl 0633.62084)] for supercritical branching processes. The mode of conditional convergence used in the present paper is weaker.

An example illustrates the applicability and the importance of the results. The limit distributions derived can be used to improve the poor conditional performance of the unconditional confidence intervals.

Reviewer: S.Penev (Kensington)

##### MSC:

62F12 | Asymptotic properties of parametric estimators |

62M99 | Inference from stochastic processes |

62E20 | Asymptotic distribution theory in statistics |

62M09 | Non-Markovian processes: estimation |