Highly robust estimation of the autocovariance function (Q154478)

The autocovariance function \(\gamma(h)=\text{ Cov}(X_{t+h}X_t)\) of a stationary process \(X_t\) is estimated via the equation NEWLINE\[NEWLINE\gamma(h)=4^{-1}[(X_t+X_{t+h}) - Var(X_t-X_{t+h})].NEWLINE\]NEWLINE To estimate \(Var(Z)\) the authors use the highly robust estimator NEWLINE\[NEWLINEQ_n=c \{|Z_i-Z_j|: i<j\}_{[k]},NEWLINE\]NEWLINE where \(Z_j,\) \(j=1,\dots, n\) are observations of \(Z\), subscript \([k]\) denotes the \(k\)-th order statistics, NEWLINE\[NEWLINEk=\left\lfloor\left( {\binom n2}+2\right)\slash 4 \right\rfloor+1,\quad c=2,2191,NEWLINE\]NEWLINE for Gaussian \(Z\). The authors demonstrate that the obtained estimator has a breakdown point 0.25 and for MA(1) and AR(1) Gaussian models its asymptotic relative efficiency is near 80\%. Results of simulations and application to the autocovariance estimation for the data of monthly interest rates of an Austrian bank are considered.