2. [12 marks] Comparing two estimators for the variance of a normal population In this question we ask you to examine th

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2 12 Marks Comparing Two Estimators For The Variance Of A Normal Population In This Question We Ask You To Examine Th 1 (113.67 KiB) Viewed 32 times

2. [12 marks] Comparing two estimators for the variance of a normal population In this question we ask you to examine the properties of two alternative estimators of the variance of a normal population. To fix the ideas, suppose we consider the distribution of heights of Dutch men; it is well known that height in most populations of a single sex) is normally distributed and that the Dutch are tall, so let's assume that the distribution of height in this population has a mean of 182cm with a standard deviation of 7cm. Now consider the statistical problem of estimating the variance of height in this population. The true value is 49, but we consider the “real-world" scenario in which we don't know this fact; instead we can study random samples from the population and estimate the variance from the data obtained in these samples. To make things precise, we introduce notation as follows. We consider a random sample of n men and denote their heights Y: (i = 1,...,n), where these values are assumed to be independent and identically distributed following a normal distribution with mean u and variance o: Y-N(4,6%), i = 1,...,n. Now consider the following two possible estimators of o2: 71 1 S3-1 τη Ση - γ)2 Q2 = (Y(0.75n) - Yo.25n) 1.349 The first of these should be familiar from previous tutorial exercises. The second is based on the interquartile range of the sample, where we have used Yo.75n) to denote the upper quartile and Yo.25n) the lower quartile, based in turn on defining Yo to represent the ith smallest observation in the sample - with 0.75n and 0.25n rounded to the nearest integer. (Note that slightly more complicated methods are used in practice to estimate sample quantiles, but with large samples the results become equivalent and these details do not need to be considered here.) a) [1 mark] Considering how they are calculated from the sample values, what advantages and disadvantages do you think each of these estimators might have? [Think in terms of ease of calculation, and how the entire set of y, values are used for each estimate. b) [1 mark] Given the true values in the population described above, what are the population upper and lower quartiles, i.e., what are the values of height below which the shortest 25% of the population are found and above which the tallest 25% of the population are found? c) [2 marks] If Y, -N(1,02), then E(Ypn)) +00-(p) as n + (where 0 < p < 1), where 0-(p) denotes the inverse normal cumulative distribution function evaluated at p. i.e., the value below which there is probability p in a standard normal distribution. In words, this says that the expected value of the sample quartile converges to the corresponding population quartile. Use this result (which you don't need to demonstrate) to prove that is an asymptotically unbiased estimator of o.