### Abstract

A sequential estimator is proposed for the autoregression parameter of first-order (AR(1)), which is constructed on the basis of a generalized least square method (GLSM) using a special choice of the weight coefficients in the sum of residual squares. Under some natural requirements on the noise distribution function, this is the prescribed precision estimator in the sense that it provides the unknown parameter estimation with any fixed square average accuracy at the moment of termination of the observation. In contrast to the sequential least square estimator, our estimator has the important property of uniform asymptotic normality with respect to the parameter on the whole axis. Using this result one can show that the sequential least square estimator is asymptotically optimal in the minimax sense for the power loss function, in a wide class of sequential and nonsequential procedures.

Original language | English |
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Pages (from-to) | 678-694 |

Number of pages | 17 |

Journal | Theory of Probability and its Applications |

Volume | 41 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 1996 |

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### Keywords

- Autoregression process
- Local asymptotic normality
- Prescribed precision estimators
- Uniform asymptotic normality

### ASJC Scopus subject areas

- Statistics and Probability

### Cite this

*Theory of Probability and its Applications*,

*41*(4), 678-694. https://doi.org/10.1137/S0040585X97975691

**The prescribed precision estimators of the autoregression parameter using the generalized least square method.** / Konev, V. V.; Pergamenshchikov, S. M.

Research output: Contribution to journal › Article

*Theory of Probability and its Applications*, vol. 41, no. 4, pp. 678-694. https://doi.org/10.1137/S0040585X97975691

}

TY - JOUR

T1 - The prescribed precision estimators of the autoregression parameter using the generalized least square method

AU - Konev, V. V.

AU - Pergamenshchikov, S. M.

PY - 1996/12

Y1 - 1996/12

N2 - A sequential estimator is proposed for the autoregression parameter of first-order (AR(1)), which is constructed on the basis of a generalized least square method (GLSM) using a special choice of the weight coefficients in the sum of residual squares. Under some natural requirements on the noise distribution function, this is the prescribed precision estimator in the sense that it provides the unknown parameter estimation with any fixed square average accuracy at the moment of termination of the observation. In contrast to the sequential least square estimator, our estimator has the important property of uniform asymptotic normality with respect to the parameter on the whole axis. Using this result one can show that the sequential least square estimator is asymptotically optimal in the minimax sense for the power loss function, in a wide class of sequential and nonsequential procedures.

AB - A sequential estimator is proposed for the autoregression parameter of first-order (AR(1)), which is constructed on the basis of a generalized least square method (GLSM) using a special choice of the weight coefficients in the sum of residual squares. Under some natural requirements on the noise distribution function, this is the prescribed precision estimator in the sense that it provides the unknown parameter estimation with any fixed square average accuracy at the moment of termination of the observation. In contrast to the sequential least square estimator, our estimator has the important property of uniform asymptotic normality with respect to the parameter on the whole axis. Using this result one can show that the sequential least square estimator is asymptotically optimal in the minimax sense for the power loss function, in a wide class of sequential and nonsequential procedures.

KW - Autoregression process

KW - Local asymptotic normality

KW - Prescribed precision estimators

KW - Uniform asymptotic normality

UR - http://www.scopus.com/inward/record.url?scp=0030311448&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030311448&partnerID=8YFLogxK

U2 - 10.1137/S0040585X97975691

DO - 10.1137/S0040585X97975691

M3 - Article

VL - 41

SP - 678

EP - 694

JO - Theory of Probability and its Applications

JF - Theory of Probability and its Applications

SN - 0040-585X

IS - 4

ER -