Probability Statistics

A History of Parametric Statistical Inference from Bernoulli by Anders Hald

By Anders Hald

This publication bargains a close background of parametric statistical inference. protecting the interval among James Bernoulli and R.A. Fisher, it examines: binomial statistical inference; statistical inference by means of inverse chance; the crucial restrict theorem and linear minimal variance estimation through Laplace and Gauss; mistakes thought, skew distributions, correlation, sampling distributions; and the Fisherian Revolution. vigorous biographical sketches of the various major characters are featured all through, together with Laplace, Gauss, Edgeworth, Fisher, and Karl Pearson. additionally tested are the jobs performed by way of DeMoivre, James Bernoulli, and Lagrange.

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19) from 4 to  ˜  he finds   1 ma1 1 ma2 1 ˜ ˜  e , a1 > a2 .  =   x1 = a1 + ln 1 + e m 3 3 This is the first disappointment: ˜  diers from the arithmetic mean. Laplace notes that 1 limm$0 ˜  = x1 + (2a1 + a2 ) = x, 3 so the arithmetic mean is obtained only in the unrealistic case where the observed errors are uniformly distributed on the whole real line. 19), which gives   1 1 h(m|a1 , a2 ) 2 m2 em(a1 +a2 ) 1  ema1  ema2 . 3 3 He does not discuss how to use this result for estimating m.

17) = c aa+1/2 bb+1/2 (n + m)n+m+3/2 / see Laplace ([149], Art. 17). We have used the notation a[x] = a(a + 1) · · · (a + x  1). 16) the analogy to the binomial is obvious. 16) by means of Stirling’s formula. 15) is the beta-binomial or the inverse hypergeometric distribution which is also known in the form      n+m+1 b+d a+c . cient in the 1774 paper where he considers a specified sequence of successes and failure. 17), Laplace ([159], II, Art. 30) keeps h = a/n fixed, as n $ 4. Assuming that m is large, but at most of the order of n, he proves that c is asymptotically normal with mean mh and variance m mh(1  h)(1 + ).

27, where yi is the observed latitude of Manilius in relation to the moon’s apparent equator and x2i2 + x2i3 = 1, because xi2 and xi3 are the sine and the cosine of the same observed angle. The observations were planned to obtain a large variation of x2 and thus also of x1 . To estimate the parameters Mayer first uses the method of selected points. 2 The Method of Averages by Mayer, 1750, and Laplace, 1788 49 determination of the unknowns. He solves the three equations by successive elimination of the unknowns.

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