# On Methods for Real Time Sampling and Distributions in Sampling

University dissertation from Umeå : Matematisk statistik

Abstract: This thesis is composed of six papers, all dealing with the issue of sampling from a finite population. We consider two different topics: real time sampling and distributions in sampling. The main focus is on Papers A–C, where a somewhat special sampling situation referred to as real time sampling is studied. Here a finite population passes or is passed by the sampler. There is no list of the population units available and for every unit the sampler should decide whether or not to sample it when he/she meets the unit. We focus on the problem of finding suitable sampling methods for the described situation and some new methods are proposed. In all, we try not to sample units close to each other so often, i.e. we sample with negative dependencies. Here the correlations between the inclusion indicators, called sampling correlations, play an important role. Some evaluation of the new methods are made by using a simulation study and asymptotic calculations. We study new methods mainly in comparison to standard Bernoulli sampling while having the sample mean as an estimator for the population mean. Assuming a stationary population model with decreasing autocorrelations, we have found the form for the nearly optimal sampling correlations by using asymptotic calculations. Here some restrictions on the sampling correlations are used. We gain most in efficiency using methods that give negatively correlated indicator variables, such that the correlation sum is small and the sampling correlations are equal for units up to lag m apart and zero afterwards. Since the proposed methods are based on sequences of dependent Bernoulli variables, an important part of the study is devoted to the problem of how to generate such sequences. The correlation structure of these sequences is also studied.The remainder of the thesis consists of three diverse papers, Papers D–F, where distributional properties in survey sampling are considered. In Paper D the concern is with unified statistical inference. Here both the model for the population and the sampling design are taken into account when considering the properties of an estimator. In this paper the framework of the sampling design as a multivariate distribution is used to outline two-phase sampling. In Paper E, we give probability functions for different sampling designs such as conditional Poisson, Sampford and Pareto designs. Methods to sample by using the probability function of a sampling design are discussed. Paper F focuses on the design-based distributional characteristics of the ?-estimator and its variance estimator. We give formulae for the higher-order moments and cumulants of the ?-estimator. Formulae of the design-based variance of the variance estimator, and covariance of the ?-estimator and its variance estimator are presented.