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**Extra resources for An essay on the psychology of invention in the mathematical field**

**Example text**

With some ljJ the problern of monotonicity of MI/I can be overcome by a simple modification of 1/J. For example, with I/J(u)=(1-e-")/(1 +e-"), =median (y 1 , y 2 , ••. ), we can put u, a u::::; 1 u ;?! 19. 57). arge it may be impractical to perform the exact calculations outlined in preceding sections, especially in those cases where ranking of observations is needed. One may have to resort to approximate calculations on grouped data. arge sample sizes it may weil be that data are only supplied in grouped form.

In the examples that follow, S is ONE-SAMPLE LOCATION PROBLEMS 27 compared with the appropriate likelihood procedure. :__=0 d(x;, t) at ~ L... 3 If the distribution of X is N(e, a 2 ), we have e5 (B) = (2/n) 112 ja eML(B) = 1/a giving Pitman ARE= 2/n. l: 0; this is an exponential distribution with median e and mean e/In 2. e5 (B)=C~2 ) eML(8) = e1 ARE= (ln2) 2 Since the sample mean is a commonly used location estimator, a comparison of its efficiency relative to the median is of interest. 4 the MLE of is (sample meanjln 2).

In the examples that follow, S is ONE-SAMPLE LOCATION PROBLEMS 27 compared with the appropriate likelihood procedure. :__=0 d(x;, t) at ~ L... 3 If the distribution of X is N(e, a 2 ), we have e5 (B) = (2/n) 112 ja eML(B) = 1/a giving Pitman ARE= 2/n. l: 0; this is an exponential distribution with median e and mean e/In 2. e5 (B)=C~2 ) eML(8) = e1 ARE= (ln2) 2 Since the sample mean is a commonly used location estimator, a comparison of its efficiency relative to the median is of interest. 4 the MLE of is (sample meanjln 2).