Contributions to Survey Sampling and Applied Statistics by H. O. Hartley, H. A. David

By H. O. Hartley, H. A. David

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M y problem is n o w to m a k e an inference about the number in the envelope that I have not seen. Then, surely, I am back in the previous case. I have to m a k e an inference about the number in the other envelope without any information. It is obvious, I think, that I have n o recourse but to refuse to m a k e an inference. " Suppose that the piece of paper is returned to the envelope, and I have not marked it in any way. " Suppose I find that the number is 2003. Then from my two observations I know that the two numbers are 7 and 2003.

4. THE FINITE P O P U L A T I O N P R O B L E M The preceding views have deep relevance to the problem of inference about a finite population. I would first like to claim that the logical nature of the problem may be stripped out by considering the simplest cases. Given a mode of approach, say, the frequentist one, a large a m o u n t of theoretical work on approximation for large finite populations may be, and has been, undertaken. But I take the view that one can see the logical problems merely by considering the case of populations of size one and of size two.

A. Fisher, which almost the whole profession rejects. 5. 1 can n o w talk about observing envelope 1 with probability p and envelope 2 with probabil­ ity 1 — p. Then our observation is a r a n d o m variable X, which equals 9 with known probability p and 6 with known probability p . N o w consider unbiasedness. W e shall, indeed we must, use as estimator of 9 + 0 , a function that depends only on X. So we shall consider ^ ( ^ i ) and g {9 ) as estimators for the two cases. Then we must have X x 2 2 X 2 2 2 P i 0 i ( 0 i ) + P 0 2 ( 0 2 ) = 0i + 02 and this must hold for all (0 , 9 ) e R .

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