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Winter, 2008 Tuesday, March 4Stat 321 – Lecture 24Point Estimation (6.1)Example 1: Caplehorn and Bell (1991) investigated the times that heroin addicts remained in a clinic for methadone maintenance treatment. The data in heroin.mtw include the amount of time that subjects stayed in facility until treatment terminated. About 37% of subjects were still in the facility when the study ended.(a) What would you like to estimate about the population?(b) Pretend this was our population and we only had a sample of 5 subjects. Can we apply a t-interval procedure for the population median? What would we like to be true about the samplingdistribution of the sample median?Def: A point estimator is a statistic, a function of the random variables in a random sample, used to estimate a parameter (of a population or of a probability distribution). A point estimate is a calculated value of a point estimator for given sample data. In general, the parameter we are trying to estimate can be denoted by , and the estimator by θˆ. An estimator is unbiased if its expected value is equal to the value we are trying to estimate: E(θˆ) =  . We can determine whether this property holds either through simulation (is the mean of the empirical sampling distribution equal to the parameter value of interest) or use rules for expected value. Usually, if we are selecting among several estimators, we choose the unbiased estimator. If there is more than one unbiased estimator, we choose the one with the smallest standard deviation.Recall:X is an unbiased estimator for . In fact, has the smallest variance of all unbiased estimators for  when sampling from a normal population.pˆ is an unbiased estimator for p. Lab 7: Is p~an unbiased estimator for p?sample 5 c4 c71Winter, 2008 Tuesday, March 4let c8(k1)=mean(c7)let c9(k1)=median(c7) let k1=k1+1Example 2: Suppose we had a normal population:Scottish militiamen:Example 3: Suppose we want to compare the following two estimators for  =2:1)(θˆ211nXXniinXXnii212)(θˆ(a) Is either of these an unbiased estimator for 2? What about ?(b) What is the direction of the bias?(c) See text p. 233 for formal proof that S2 is an unbiased estimator of


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Cal Poly STAT 321 - Lecture 24

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