Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011Chapter VII Estimation § 7.1 Point Estimationθ: unknown parameter of the population, or the corresponding pdf ( )xfθ): a point estimatorof θ. It is a statistic (function of the items of a random sample) and is a random variable. An observed value of θ)is a point estimateof θ. Example 7.1Suppose we are interested in a normal population with unknown mean μand variance 2σ, i.e. we only know (or assume) that the population distribution resembles the normal bell-shaped curve, but have no idea in the location and spread of the distribution. Then μand 2σare the unknown parameters. We may draw a random sample from this population and calculate the statistics Xand 2S, and then use their values to estimate μand 2σ. Hence we may denote X=μ), 22S=σ)as the point estimators of μand 2σ. § 7.2 Desirable Properties of EstimatorThe notations θˆand θrefer to different quantities, with θˆas the sample statistic that can be observed from the sample while θis the population parameter that is unobservable. Usually for any parameter, there exist many different methods to do the estimation. For example, based on a random sample from ()θ,0U, a reasonable estimator of θis the maximum of the sample, i.e. ()nXXX,...,,maxˆ21=θ. However, since the population mean is 2θμ=, it is also reasonable to estimate θby using the sample mean, i.e. X2ˆ=θ. P.140
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