Maximum likelihood estimation in weighted sampling
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Weighted sampling is appropriate when an observation enters to sample only with probability proportional to a nonnegative weight function. Weighted sample is not a random sample from the original distribution which we are interested in performing inference about it but also a random sample from a new distribution called weighted distribution which is related to the original distribution via weight function. In this paper maximum likelihood estimation for parameters of normal distribution is discussed under weighted sampling. The weight function is assumed to be specified and may or may not be involved the unknown parameters of interest. Maximum likelihood estimations are compared for weighted and unweighted sampling under some common weight function. We show that under the flexible linear and quadratic exponential weight functions, Maximum likelihood estimators have closed form and are distributed as normal but are not under power weight functions. However, for some weight functions we need to use numerical approaches. Newton-Raphson iterative method and scoring method are explained for a special weigh function.
Abstract
Weighted sampling is appropriate when an observation enters to sample only with probability proportional to a nonnegative weight function. Weighted sample is not a random sample from the original distribution which we are interested in performing inference about it but also a random sample from a new distribution called weighted distribution which is related to the original distribution via weight function. In this paper maximum likelihood estimation for parameters of normal distribution is discussed under weighted sampling. The weight function is assumed to be specified and may or may not be involved the unknown parameters of interest. Maximum likelihood estimations are compared for weighted and unweighted sampling under some common weight function. We show that under the flexible linear and quadratic exponential weight functions, Maximum likelihood estimators have closed form and are distributed as normal but are not under power weight functions. However, for some weight functions we need to use numerical approaches. Newton-Raphson iterative method and scoring method are explained for a special weigh function.
참고문헌 (17)
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