This paper studies techniques for fitting parsimonious periodic time series models to periodic data. Large sample standard errors for the parameter estimates in a periodic autoregressive moving-average time series model under parametric constraints are derived. Likelihood ratio statistics for hypothesis testing are examined. The techniques are applied in modeling daily temperatures at Griffin, Georgia, USA.
Tail estimates are developed for power law probability distributions with exponential tempering, using a conditional maximum likelihood approach based on the upperorder statistics. Tempered power law distributions are intermediate between heavy power-law tails and Laplace or exponential tails, and are sometimes called "semiheavy" tailed distributions. The estimation method is demonstrated on simulateddata from a tempered stable distribution, and for several data sets from geophysics and finance that show a power law probability tail with some tempering.
This paper studies correlation and partial autocorrelation properties of periodic autoregressive moving-average (PARMA) time series models. An efficient algorithm to compute PARMA autocovariances is first derived. An innovations based algorithm to compute partial autocorrelations for a general periodic series is then developed. Finally, periodic moving averages and autoregressions are characterized as periodically stationary series whose autocovariances and partial autocorrelations, respectively, are zero at all lags that exceed some periodically varying threshold.
The article considers the Yule-Walker estimator of the autoregressive coefficient based on the observed time series that contains an unknown trend function and an autoregressive error term. The trend function is estimated by means of B-splines and then subtracted from the observations. The Yule-Walker estimator is obtained from the residual sequence. Asymptotic properties of this estimator are derived. The performance of the estimator is illustrated by simulation studies and real data analysis.
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