Robust linear regression is one of the most popular problems in the robust statistics community. The parameters of this method are often estimated via least trimmed squares, which minimizes the sum of the k smallest squared residuals. So, the estimation method in contrast to the common least squares estimation method is very computationally expensive. The main idea of this paper is to propose a new estimation method in partial linear models based on minimizing the sum of the k smallest squared residuals which determines the set of outlier point and provides robust estimators. In this regard, first, difference based method in estimation parameters of partial linear models is introduced. Then the method of obtaining robust difference based estimators in partial linear models is introduced which is based on solving an optimization problem minimizing the sum of the k smallest squared residuals. This method can identify outliers. The simulated example and applied numerical example with real data found the proposed robust difference based estimators in the paper produce highly accurate results in compare to the common difference based estimators in partial linear models.

In statistical research, experimental designs are used to investigate the effect of control variables on output responses. These methods are based on the assumption of normal distribution of data and face fundamental challenges in dealing with outliers. The present study examines five different examples of experimental design methods to deal with this challenge: Huber, quadratic, substitution, ranking, and fuzzy regression robustness methods. By providing empirical evidence from real data on seedling growth and weld quality, it is shown that fuzzy can be used as an efficient alternative to conventional methods in the presence of outliers. It is shown that fuzzy not only outperforms the classical experimental design method in the presence of outliers, but also outperforms standard robustness methods in handling outliers.