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Estimating a gradual parameter change in an AR(1)-process

100%

Hušková M., Prášková Z., Steinebach J. G.

Śląski Przegląd Statystyczny

2020

issue 18 (24)

254-262

The authors discuss the estimation of a change-point 𝑡0 at which the parameter of a (non-stationary) AR(1)-process possibly changes in a gradual way. Making use of the observations 𝑋1, … , 𝑋𝑛 , we study the least squares estimator 𝑡̂0 for 𝑡0, which is obtained by minimizing the sum of squares of the residuals with respect to the given parameters. As the first result it can be shown that, under certain regularity and moment assumptions, 𝑡̂0/𝑛 is a consistent estimator for 𝜏0, where 𝑡0 = ⌊𝑛𝜏0⌋, with 0 < 𝜏0 < 1, i.e., 𝑡̂0/𝑛 →𝑃 𝜏0 (𝑛 → ∞). Based on the rates obtained in the proof of the consistency result, a rough convergence rate statement can also be given. Some possible further investigations are briefly discussed, including the weak limiting behaviour of the (suitably normalized) estimator.

Estimation of the density and cumulative distribution functions of the exponentiated Burr XII distribution

88%

Hassan A. S., Assar S. M., Ali K. A., Nagy H. F.

Statistics in Transition new series

2021

vol. 22

issue 4

171-189

The exponentiated Burr Type XII (EBXII) distribution has wide applications in reliability and economic studies. In this article, the estimation of the probability density function and the cumulative distribution function of EBXII distribution is considered. We examine the maximum likelihood estimator, the uniformly minimum variance unbiased estimator, the least squares estimator, the weighted least squares estimator, the maximum product spacing estimator, the Cramér–von-Mises estimator, and the Anderson–Darling estimator. We derive analytical forms for the bias and mean square error. A simulation study is performed to investigate the consistency of the suggested methods of estimation. Data relating to the wind speed and service times of aircraft windshields are used with the studied methods. The simulation studies and real data applications have revealed that the maximum likelihood estimator performs more efficiently than its remaining counterparts.

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1 Statistics in Transition new series

1 Śląski Przegląd Statystyczny

1 Ali K. A.

1 Assar S. M.

1 Hassan A. S.

1 Hušková M.

1 Nagy H. F.

1 Prášková Z.

1 Steinebach J. G.

1 2021

1 2020

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Estimating a gradual parameter change in an AR(1)-process

Estimation of the density and cumulative distribution functions of the exponentiated Burr XII distribution