Local influence on the ratio and Horvitz-Thompson estimators through the second order approach

Since Cook introduced the notion of local influence in 1986 a great number of papers have appeared in statistics handling this topic in a wide variety of areas. However, until we know, there is no paper focussed on local influence in survey sampling. This paper assesses the local influence through the curvature of the perturbation-formed surface of the ratio estimator in a simple random sampling without replacement and the Horvitz-Thompson estimator in an inclusion proportional to size sampling design. Both estimators use auxiliary information (x) and can be obtained through a model-assisted reasoning where the ratio model with variance proportional to x plays an important role. Two perturbation schemes are considered, a model perturbation and an auxiliary variable perturbation. We illustrate the effectiveness of the method in order to identify influential points with two real examples.