The instability of the least squares parameter estimates under collinearity, might also causes instability of the residuals. If so, a large residual from a least squares fit might not be indicative of an erratic data point, and conversely. In order to resolve the problem of collinearity in the regression model, biased estimators like the Liu estimator is suggested. In this paper, it is shown that when Liu mean shift regression is used to mitigate the effect of the collinearity, the influence of some observations can be drastically changed and also the appropriate statistic for testing outliers is derived. In order to illustrate the performance of the proposed method, a real example is presented.

In this paper, the smallest confidence region is obtained for the location and scale parameters of the two-parameter exponential distribution. For this purpose, we use constrained optimization problems. We first provide some suitable pivotal quantities to obtain a balanced confidence region. We then obtain the smallest confidence region by minimizing the area of the confidence region using the Lagrangian method. Two numerical examples are presented to illustrate the proposed methods. Finally, some applications of proposed joint confidence regions in hypothesis testing and the construction of confidence bands are discussed.