Mean estimation of small areas using penalized spline mixed-model under informative sampling

Penalized spline is a suitable nonparametric approach in estimating mean model in small area. However, application of the approach in informative sampling in a published article is uncommon. We propose a semiparametric mixed-model using penalized spline under informative sampling to estimate mean of small area. The response variable is explained in terms of mean model, informative sample effect, area random effect and unit error. We approach the mean model by penalized spline and utilize a penalized spline function of the inclusion probability to account for the informative sample effect. We determine the best and unbiased estimators for coefficient model and derive the restricted maximum likelihood estimators for the variance components. A simulation study shows a decrease in the average absolute bias produced by the proposed model. A decrease in the root mean square error also occurred except in some quadratic cases. The use of linear and quadratic penalized spline to approach the function of the inclusion probability provides no significant difference distribution of root mean square error, except for few smaller samples.