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#1
12-13-2011, 03:41 PM
 Ionic Order Member CAS SOA Join Date: Jan 2011 Location: Seattle Posts: 1,259
ASM, exercise 19.11 (Review of Mathematical Statistics)

I have the 11th edition. The question reads (with simplification):

A population contains the values 1, 2, 4, 9. A sample of 3 is withdrawn without replacement from this variable. Calculate the mean square error of the median of this sample as an estimator of the population mean.

The solution is: half the time the sample median is 2 and the other half the time it is 4. The mean of the population is 4. So the MSE is 1/2*(2-4)^2=2. I don't understand the last step. Is there a Bernoulli shortcut involved.
#2
12-13-2011, 04:00 PM
 actuary_aspire Member CAS Join Date: Sep 2009 Studying for OC1 Posts: 2,231

MSE = 1/2*(2-4)^2 + 1/2*(4-4)^2 = 1/2*(2-4)^2 = 2.
#3
12-13-2011, 04:04 PM
 Ionic Order Member CAS SOA Join Date: Jan 2011 Location: Seattle Posts: 1,259

Why can't I say:

MSE = (1/2*4+1/2*2-4)^2 = 1?
#4
12-13-2011, 04:12 PM
 actuary_aspire Member CAS Join Date: Sep 2009 Studying for OC1 Posts: 2,231

(1/2*4+1/2*2-4) is the bias. MSE = Variance + bias^2.
#5
12-13-2011, 04:17 PM
 Ionic Order Member CAS SOA Join Date: Jan 2011 Location: Seattle Posts: 1,259

Oh I see. I got the two formulas for MSE mixed up. Thanks a lot!

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