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"... Likelihood-based Approaches Let U = {1, 2, · · · , N } be the set of units in the finite population and y i and x i be respectively the values of the study variable y and the vector of the auxiliary variables x attached to the ith unit. In this chapter we restrict our discussion to the estimation w ..."

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Likelihood-based Approaches Let U = {1, 2, · · · , N } be the set of units in the finite population and y i and x i be respectively the values of the study variable y and the vector of the auxiliary variables x attached to the ith unit. In this chapter we restrict our discussion to the estimation where s is the set of sample units selected by the probability sampling design, p(s), and I(y ≤ t) is the indicator function. The population totals X = N i=1 x i or means X = X/N may also be available and can be used at the estimation stage. Likelihood-based estimation methods in survey sampling do not follow as special cases from classical parametric likelihood inferences. Under the conventional designbased framework, values of the study variable for the finite population, {y 1 , y 2 , · · · , y N }, are viewed as fixed. The only randomization is induced by the probability sampling selection of units. In the design-based setup, an unbiased minimum variance estimator or even an unbiased minimum variance linear estimator of Y does not exist is the first order inclusion probability for unit i. The HT estimator therefore is often treated as a baseline estimator for inferences concerning Y .