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Ch7 Estimation

Ch7 Estimation - Stat1801 Probability and Statistics...

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Stat1801 Probability and Statistics: Foundations of Actuarial Science Fall 2010-2011 Chapter VII Estimation § 7.1 Point Estimation θ : unknown parameter of the population, or the corresponding pdf ( ) x f θ ) : a point estimator of 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 estimate of θ . Example 7.1 Suppose we are interested in a normal population with unknown mean μ and variance 2 σ , 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 μ and 2 σ are the unknown parameters. We may draw a random sample from this population and calculate the statistics X and 2 S , and then use their values to estimate μ and 2 σ . Hence we may denote X = μ ) , 2 2 S = σ ) as the point estimators of μ and 2 σ . § 7.2 Desirable Properties of Estimator The notations θ ˆ and θ refer to different quantities, with θ ˆ as the sample statistic that can be observed from the sample while θ is 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 , 0 U , a reasonable estimator of θ is the maximum of the sample, i.e. ( ) n X X X ,..., , max ˆ 2 1 = θ . However, since the population mean is 2 θ μ = , it is also reasonable to estimate θ by using the sample mean, i.e. X 2 ˆ = θ . P.140
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