Let X_1, ..., X_n be a random sample from a pdf that is symmetric about mu. An e
ID: 3181017 • Letter: L
Question
Let X_1, ..., X_n be a random sample from a pdf that is symmetric about mu. An estimator for mu that has been found to perform well for a variety of underlying distributions is the Hodges-Lehmann estimator. To define it, first compute for each i lessthanorequalto j and each j = 1, 2, ..., n the pairwise average X bar_i, j = (X_i + X_j)/2. Then the estimator is mu cap = the median of the X bar_i, j 's. Compute the value of this estimate using the data below. 26.0 30.3 29.1 49.0 33.6 27.7 29.2 27.9 23.2 31.3 mu cap =Explanation / Answer
Following table shows all the possible samples and corresponding mean:
Following is the ordered data set of means:
Since there are 55 data values so median will be 28th data value. That is required median is 29.2.
j i Samples Sample mean 1 1 26 26 26 2 1 30.3 26 28.15 2 2 30.3 30.3 30.3 3 1 29.1 26 27.55 3 2 29.1 30.3 29.7 3 3 29.1 29.1 29.1 4 1 49 26 37.5 4 2 49 30.3 39.65 4 3 49 29.1 39.05 4 4 49 49 49 5 1 33.6 26 29.8 5 2 33.6 30.3 31.95 5 3 33.6 29.1 31.35 5 4 33.6 49 41.3 5 5 33.6 33.6 33.6 6 1 27.7 26 26.85 6 2 27.7 30.3 29 6 3 27.7 29.1 28.4 6 4 27.7 49 38.35 6 5 27.7 33.6 30.65 6 6 27.7 27.7 27.7 7 1 29.2 26 27.6 7 2 29.2 30.3 29.75 7 3 29.2 29.1 29.15 7 4 29.2 49 39.1 7 5 29.2 33.6 31.4 7 6 29.2 27.7 28.45 7 7 29.2 29.2 29.2 8 1 27.9 26 26.95 8 2 27.9 30.3 29.1 8 3 27.9 29.1 28.5 8 4 27.9 49 38.45 8 5 27.9 33.6 30.75 8 6 27.9 27.7 27.8 8 7 27.9 29.2 28.55 8 8 27.9 27.9 27.9 9 1 23.2 26 24.6 9 2 23.2 30.3 26.75 9 3 23.2 29.1 26.15 9 4 23.2 49 36.1 9 5 23.2 33.6 28.4 9 6 23.2 27.7 25.45 9 7 23.2 29.2 26.2 9 8 23.2 27.9 25.55 9 9 23.2 23.2 23.2 10 1 31.3 26 28.65 10 2 31.3 30.3 30.8 10 3 31.3 29.1 30.2 10 4 31.3 49 40.15 10 5 31.3 33.6 32.45 10 6 31.3 27.7 29.5 10 7 31.3 29.2 30.25 10 8 31.3 27.9 29.6 10 9 31.3 23.2 27.25 10 10 31.3 31.3 31.3Related Questions
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