Metrika 2002) 55: 209±214 > Springer-Verlag 2002 In¯uence diagnostic in survey sampling: Estimating the conditional bias J. L. Moreno-Rebollo, A. MunÄoz-Reyes, M. D. JimeÂnez-Gamero and J. MunÄoz-Pichardo Departamento de Estadõ Âstica e InvestigacioÂn Operativa, Facultad de MatemaÂticas, Universidad de Sevilla, C/Tar®a s.n., 41012 Sevilla, Spain. e-mail: moreno@cica.es anamun@cica.es dolores@cica.es pichardo@cica.es) Received October 2000 Abstract. The conditional bias has been proposed by Moreno Rebollo et al. 1999) as an in¯uence diagnostic in survey sampling, when the inference is based on the randomization distribution generated by a random sampling. The conditional bias is a population parameter. So, from an applied point of view, it must be estimated. In this paper, we propose an estimator of the conditional bias and we study conditions that guarantee its unbiasedness. The results are applied in a Simple Random Sampling and in a Proportional Probability Aggregated Size Sampling, when the ratio estimator is used. Key words: In¯uence diagnostic; Conditional bias; Survey Sampling 1 Introduction Although in the literature many in¯uence diagnostics have been proposed for assessing the impact that individual observations have on statistical conclu- sions, there are no many references to in¯uence diagnostics in sample survey, specially within a design-based approach. In this context, we can cite the papers by Smith 1987), Gwet and Rivest 1992), and Hulliger 1995). Moreno-Rebollo et al. 1999) have proposed the conditional bias as an in¯uence measure that it can be used for arbitrary estimators and sampling designs. Let U fu 1 ; ... ; u N g be a ®nite population and D fS; pg a sampling design de®ned on U, being S the sample space and p the probability distribution on S. Let Y be a characteristic of the population, Y  fY 1 ; ... ; Y N g, y  yY  the parameter of interest and ^ y  ^ ys an estimator of y, s A S. Denoting by I i s, i  1; ... ; N, the random variables, I i s 1 if u i A s; I i s 0 otherwise, and by p i the ®rst-order inclusion probabilities, p i  prI i  1, i  1; ... ; N, the conditional bias of ^ y caused by the presence of u i ,0 < p i < 1, in the sample, SI i  1; ^ y, is given by