Konstruktion av konfidensintervall för MA(1)-parametern baserad på olika estimatorer.

The MA(1) process is a time-series process with a single parameter $\theta$. In estimating $\theta$, a major problem is the existence of pile-up probability, which is the tendency of having an estimator of $|\hat{\theta}| = 1$. We present some methods to estimate $\theta$ and discuss their advantages and disadvantages in estimation. We also take a deeper look into the residual square estimate and show that the estimate is expected to obtain $\hat{\theta} = \theta$ in most cases. We also illustrate that the residual square estimate is reliable even for values of $\theta$ near $\pm 1$ at the expense of computational power. Moreover, we also construct the confidence intervals by using residual bootstrap and for the case of maximum likelihood estimator, with the use of Fisher information. It was concluded that the residual square estimate gives a faster convergence to the confidence level as compared to other methods with the tradeoff of computational time. We also present an example on how the methods can be used in actual data as an illustration.