Imputation of Missing Observations for Heavy Tailed Cyclostationary Time Series

Abstract

The aim of our research is to provide algorithms of data imputation for a cyclostationary time series with heavy tails. We assume that time series ${Y_t}$ of interest is K-dependent but also heavy tails of the form: ${Y_t} = {X_t} \cdot {c_t}$, where $c_t$ is the periodic function and $X_t$ is a heavy tailed stationary process.We use the multivariate t- distribution with the covariance matrix $\Sigma$ of order $2\left( K-1\right) \times 2\left( K-1\right)$. Moreover, we assume that the number of degrees of freedom $\nu$ is fixed and $2<\nu \le 6$.We use the periodic sequence $\left\{ c_t\right\}$ with the period $H$ as the periodic amplitude imposed over the stationary background time series.We propose four imputation algorithms based on the properties of the multivariate t-distribution. Using simulations, we compare the performance of those algorithms.

Appendix

The following Proposition provided by Arellano-Valle and Bolfarine [1] and worked by Garay et al. [3] is used in this paper for implementation of the simulation study and represents the marginal-conditional decomposition of a Student-$t$ random vector.

Proposition

Let $\mathbf {Y}\sim t_p({\varvec{\mu }}, {\varvec{\Sigma }},\nu )$ and $\mathbf {Y}$ be partitioned as $\mathbf {Y}^{\top }=(\mathbf {Y}^{\top }_1,\mathbf {Y}^{\top }_2)^{\top }$, with $dim(\mathbf {Y}_1) = p_1$, $dim(\mathbf {Y}_2) = p_2$, $p_1 + p_2 = p$, and where ${\varvec{\Sigma }}=\left( \begin{array}{cc} {\varvec{\Sigma }}_{11} &{} {\varvec{\Sigma }}_{12} \\ {\varvec{\Sigma }}_{21} &{} {\varvec{\Sigma }}_{22} \end{array} \right)$ and ${\varvec{\mu }}=({\varvec{\mu }}^{\top }_1, {\varvec{\mu }}^{\top }_2)^{\top }$, are the corresponding partitions of ${\varvec{\Sigma }}$ and ${\varvec{\mu }}$. Then, we have

• (i) $\mathbf {Y}_1\sim t_{p_1}({\varvec{\mu }}_1,{\varvec{\Sigma }}_{11},\nu )$; and

• (ii) the conditional cdf of $\mathbf {Y}_2|\mathbf {Y}_1={y}_1$ is given by

  $$P(\mathbf {Y}_2\le {y}_2|\mathbf {Y}_1={y}_1)=T_{p_2}\left( {y}_2|{\varvec{\mu }}_{2.1},\widetilde{{\varvec{\Sigma }}}_{22.1},\nu +p_1 \right) , $$

  where $\widetilde{{\varvec{\Sigma }}}_{22.1}=\left( \displaystyle \frac{\nu +\delta _1}{\nu +p_1}\right) {\varvec{\Sigma }}_{22.1},$ $\delta _1=({y}_1-{\varvec{\mu }}_1)^{\top }{\varvec{\Sigma }}_{11}^{-1}({y}_1-{\varvec{\mu }}_1),$ ${\varvec{\Sigma }}_{22.1}={\varvec{\Sigma }}_{22}-{\varvec{\Sigma }}_{21}{\varvec{\Sigma }}^{-1}_{11}{\varvec{\Sigma }}_{12}$, ${\varvec{\mu }}_{2.1}={\varvec{\mu }}_2+{\varvec{\Sigma }}_{21}{\varvec{\Sigma }}^{-1}_{11}({y}_1-{\varvec{\mu }}_1)$, and $T_{r}\left( .| \ldots \right)$ represents a cdf of the Student-$t$ random vector of order $r$.