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If this relative ef ciency is less than 1, we would conclude that 1 is a more ef cient estimator of than 2 , in the sense that it has a smaller mean square error. Sometimes we nd that biased estimators are preferable to unbiased estimators because they have smaller mean square error. That is, we may be able to reduce the variance of the estimator considerably by introducing a relatively small amount of bias. As long as the reduction in variance is greater than the squared bias, an improved estimator from a mean square error viewpoint will result. For example, Fig. 7-2 shows the probability distribution of a biased estimator 1 that has a smaller variance than the unbiased estimator 2 . An estimate based on 1 would more likely be close to the true value of than would an estimate based on 2 . Linear regression analysis (s 11 and 12) is an area in which biased estimators are occasionally used. An estimator that has a mean square error that is less than or equal to the mean square error of any other estimator, for all values of the parameter , is called an optimal estimator of . Optimal estimators rarely exist. EXERCISES FOR SECTION 7-2

The museum s research department has a network that they use for messaging and collaboration with various universities around the world. Among the information exchanged and stored are details about the location of various natural gemstone mines. Samuel the museum system engineer wants the museum immediately to detect unauthorized and successful attempts to gain access to the network and to any hosts that contain sensitive information. Once alerted, Samuel would like information that can be used to hold accountable the individual(s) that have breached their perimeter. In addition, Samuel would like to have information recorded and available on unsuccessful attempts to gain access. Samuel understands that trade-offs are involved, because stopping intruders and capturing information about attempted intrusions can require significant resources that degrade system performance, and which may make legitimate access more difficult. Privacy considerations are also a constraint on intrusion detection efforts. Samuel needs to identify requirements for an IDS service that will help the museum achieve the goals while balancing the constraints.

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7-1. Suppose we have a random sample of size 2n from a 2 population denoted by X, and E1X 2 and V1X 2 . Let X1 1 2n X and 2n ia i 1 X2 1 n n ia Xi 1 (a) Is either estimator unbiased (b) Which estimator is best In what sense is it best 7-3. Suppose that 1 and 2 are unbiased estimators of the parameter . We know that V1 1 2 10 and V1 2 2 4 . Which estimator is best and in what sense is it best 7-4. Calculate the relative ef ciency of the two estimators in Exercise 7-2. 7-5. Calculate the relative ef ciency of the two estimators in Exercise 7-3. 7-6. Suppose that 1 and 2 are estimators of the parameter . We know that E1 1 2 , E1 2 2 2, V 1 1 2 10, 2 V1 2 4 . Which estimator is best In what sense is it best 7-7. Suppose that 1, 2 , and 3 are estimators of . We know that E1 1 2 E1 2 2 , E 1 32 , V 1 1 2 12, 2 2 V1 2 10 , and E1 3 2 6 . Compare these three estimators. Which do you prefer Why

Research on the relationships between aging and intelligence had its inception nearly 100 years ago in comparisons between adults and children (Kirkpatrick, 1903). The topic has captivated researchers in theoretical and clinical disciplines for over half a century (Jones & Conrad, 1933; Lorge, 1936; Miles & Miles, 1932; Willoughby, 1927). Whether intelligence declines with increasing age has long been the subject of research and debate by experts in the field (Baltes & Schaie, 1976; Botwinick, 1977; Horn & Donaldson, 1976), with both the research investigations and the controversies continuing to the present (Bengtson & Schaie, 1999; Birren, Schaie, Abeles, Gatz, & Salthouse, 1996; Craik & Salthouse, 2000; Kaufman, 2000a, 2001; Lawton & Salthouse, 1998; Lindenberger & Baltes, 1997; McArdle, Prescott, Hamagami, & Horn, 1998; Park & Schwarz, 2000). The nature of the complex relationship between aging and changes in intellectual functioning is of prime concern to clinicians who test clients across a wide age span, 127

be two estimators of . Which is the better estimator of Explain your choice. 7-2. Let X1, X2, p , X7 denote a random sample from a population having mean and variance 2 . Consider the following estimators of :

Accounting requirements and their relative importance are understood. The requirements might have been selected by applying SECURITY ACCOUNTING REQUIREMENTS (360). The planned uses of IDS are understood.

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