Resampling \(m\)-dependant random variables with applications to forecasting (Q2742766)

The distributions of sums of \(m\)-dependent, possibly differently distributed, real-valued random variables are estimated by resampling methods. Let \(Y_n =\bigl\{ Y_{jn}\bigr \}_{j=1}^n\), \(n=1, 2, \ldots\), be a triangular array of \(m\)-dependent random variables, \(E[Y_{jn}] = \mu_{jn}.\) Let \(Y_{jn} = \mu_{jn} + U_{jn}\) with \(E[U_{jn}] =0\), \(E[U_{jn}^2] = \sigma_{jn}^2\), \(E[U_{in} U_{jn}] = \gamma_{ijn}.\) The random variables \(U_{in}\) and \(U_{jn}\) are independent if \(|i-j|>m.\) The author assumes that one observed value exists for each of the random variables \(Y_{1n}, \ldots, Y_{nn}\), but \(\bigl\{ \mu_{jn}\bigr\}_{j=1}^n\) and \(U_{jn}\) are not observable. He proposes specific resampling methods to estimate the distribution laws \({\mathcal L}(Y_{\cdot n} -\mu_{\cdot n}) = {\mathcal L}(U_{\cdot n})\) consistently.NEWLINENEWLINENEWLINEFor example, he proves that under some assumptions the following assertion holds true: NEWLINE\[NEWLINE{\mathcal L}(Y_{\cdot n} -\mu_{\cdot n}) {\buildrel wa \over \longleftrightarrow} N(0, \sigma_{\cdot n}^2)\quad \text{as} n \to \infty,NEWLINE\]NEWLINE where \(N(0, \sigma_{\cdot n}^2)\) denotes the normal distribution with mean zero and variance \(\sigma_{\cdot n}^2.\) The estimated distribution of \((Y_{\cdot n} -\mu_{\cdot n})\) is used to make inferences on \(\mu_{\cdot n}\), e.g., constructing confidence intervals for \(\mu_{\cdot n}.\) The resampling methods have potential applications to time series analysis, to distinguish between two different forecasting models. This is illustrated with a numerical example consisting of Swedish export prices for coated paper products.