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# Mathematical Statistics with Applications, Seventh Edition

www.downloadslide.com 8.4 Evaluating the Goodness of a Point Estimator 399 It follows that [ ] E(S ′2 ) = 1 n∑ n E (Y i − Y ) 2 = 1 n (n − 1)σ 2 = i=1 ( n − 1 n ) σ 2 and that S ′2 is biased because E(S ′2 ) ≠ σ 2 . However, [ ] E(S 2 ) = 1 n∑ n − 1 E (Y i − Y ) 2 = 1 n − 1 (n − 1)σ 2 = σ 2 , i=1 so we see that S 2 is an unbiased estimator for σ 2 . Two final comments can be made concerning the point estimators of Table 8.1. First, the expected values and standard errors for Y and Y 1 − Y 2 given in the table are valid regardless of the distribution of the population(s) from which the sample(s) is (are) taken. Second, all four estimators possess probability distributions that are approximately normal for large samples. The central limit theorem justifies this statement for Y and ˆp, and similar theorems for functions of sample means justify the assertion for (Y 1 − Y 2 ) and ( ˆp 1 − ˆp 2 ). How large is “large”? For most populations, the probability distribution of Y is mound-shaped even for relatively small samples (as low as n = 5), and will tend rapidly to normality as the sample size approaches n = 30 or larger. However, you sometimes will need to select larger samples from binomial populations because the required sample size depends on p. The binomial probability distribution is perfectly symmetric about its mean when p = 1/2 and becomes more and more asymmetric as p tends to 0 or 1. As a rough rule, you can assume that the distribution of ˆp will be mound-shaped and approaching normality for sample sizes such that p ±3 √ pq/n lies in the interval (0, 1), or, as you demonstrated in Exercise 7.70, if n > 9 (larger of p and q)/(smaller of p and q). We know that Y , ˆp,(Y 1 −Y 2 ), and ( ˆp 1 − ˆp 2 ) are unbiased with near-normal (at least mound-shaped) sampling distributions for moderate-sized samples; now let us use this information to answer some practical questions. If we use an estimator once and acquire a single estimate, how good will this estimate be? How much faith can we place in the validity of our inference? The answers to these questions are provided in the next section. 8.4 Evaluating the Goodness of a Point Estimator One way to measure the goodness of any point estimation procedure is in terms of the distances between the estimates that it generates and the target parameter. This quantity, which varies randomly in repeated sampling, is called the error of estimation. Naturally we would like the error of estimation to be as small as possible.

www.downloadslide.com 400 Chapter 8 Estimation DEFINITION 8.5 The error of estimation ε is the distance between an estimator and its target parameter. That is, ε =|ˆθ − θ|. Because ˆθ is a random variable, the error of estimation is also a random quantity, and we cannot say how large or small it will be for a particular estimate. However, we can make probability statements about it. For example, suppose that ˆθ is an unbiased estimator of θ and has a sampling distribution as shown in Figure 8.4. If we select two points, (θ − b) and (θ + b), located near the tails of the probability density, the probability that the error of estimation ε is less than b is represented by the shaded area in Figure 8.4. That is, P(|ˆθ − θ| < b) = P[−b

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