In Small Area Estimation (SAE), the usual practice is to assume that the random components follow the normal distribution. A simulation study showed that under the one-fold nested error regression model assuming normal distribution may lead to large bias and significant increase of the mean squared error (MSE) of complex parameters (nonlinear function of the mean) estimation when the unit level errors' distribution is skewed. Hence, in this thesis, the assumption of normal distribution for the random components is relaxed by considering the skew-normal (SN) class of distributions. The SN
class of distributions is very interesting because it contains the normal distribution family as a special case (when the skewness parameter is equal to zero).
For the one-fold nested error regression model with the random components following SN distribution, the empirical best linear unbiased prediction (EBLUP) and the empirical best (EB) estimators of linear parameters are provided. Under the same model, EB estimators of complex parameters are developed following the approach introduced by Molina and Rao (2010). A simpler conditional alternative method is compared to the previously
mentioned method. The semi-parametric method developed by Elbers et al. (2003) is improved by correctly assigning the area effects. A parametric bootstrap approach for the MSE estimation of the EB estimator and a semi-parametric bootstrap method for the ELL method are specified. An HB method for estimating complex parameters is also presented. The HB method uses Monte Carlo (MC) simulations and sampling importance resampling (SIR) techniques no Markov chain Monte Carlo (MCMC) is required.
For the two-fold nested error regression model with the random components following SN distribution,
the empirical best linear unbiased prediction (EBLUP) and the empirical best (EB) for linear parameters for linear parameters are discussed. The best predictor of the area and sub-area random effects is provided under the general linear mixed model setup.
For repeated cross-sectional surveys, Rao and Yu (1992, 1994) used AR(1) series to model the area-by-time effects and doing so improved the estimation of the small areas linear parameters. In this thesis an extension to the Rao-Yu model which does not required stationarity assumption is proposed.