1. The measurement errors made by a measuring device operating under two different conditions, i =...

 

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1. The measurement errors made by a measuring device operating under two different conditions, i = 1 and i = 2, are Normally distributed with mean 0 and variance o?. You may assume that all measurements are independent. (a) Based on a single sample of measurements made under condition 1, the measure- ment errors x1, x2,. . , n were recorded. Similarly, under condition 2, measure- ment errors Y1, Y2, . , Ym were recorded. Write down the combined likelihood in terms of o1 and o2. If o1 = 02, write down the combined likelihood in terms of 01. (-) [6 marks] (b) Derive the test statistic for a maximum likelihood ratio test of Ho : ơ1 = 02 (Hint: the probability density function for N(u, o²) is f(x) : exp 202 versus H1 : o1 7 02 and show that it takes the form m/2 (n + m)(n+m)/2 (D a?)"/² (E, v}) = n"/2mm/2 (n+m)/2 (You may assume that the maximum likelihood estimators for o1 and 02 over the entire parameter space are , and VE , respectively.) [16 marks] n Li=1 (c) Given that n = 10, m - 8, ΣL 6 and Σι, 2, use an asymptotic result to carry out the maximum likelihood ratio test at the 5% significance [8 marks] 3D1 level and report your conclusion. 1. The measurement errors made by a measuring device operating under two different conditions, i = 1 and i = 2, are Normally distributed with mean 0 and variance o?. You may assume that all measurements are independent. (a) Based on a single sample of measurements made under condition 1, the measure- ment errors x1, x2,. . , n were recorded. Similarly, under condition 2, measure- ment errors Y1, Y2, . , Ym were recorded. Write down the combined likelihood in terms of o1 and o2. If o1 = 02, write down the combined likelihood in terms of 01. (-) [6 marks] (b) Derive the test statistic for a maximum likelihood ratio test of Ho : ơ1 = 02 (Hint: the probability density function for N(u, o²) is f(x) : exp 202 versus H1 : o1 7 02 and show that it takes the form m/2 (n + m)(n+m)/2 (D a?)"/² (E, v}) = n"/2mm/2 (n+m)/2 (You may assume that the maximum likelihood estimators for o1 and 02 over the entire parameter space are , and VE , respectively.) [16 marks] n Li=1 (c) Given that n = 10, m - 8, ΣL 6 and Σι, 2, use an asymptotic result to carry out the maximum likelihood ratio test at the 5% significance [8 marks] 3D1 level and report your conclusion.