Who Invented the Delta Method, Really?
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n statistics, the delta method is a procedure for finding the means and variances of functions of random variables. In spite of the usefulness of the delta method, its history does not seem to have been properly documented. A relatively recent paper (Ver Hoef, 2012) admitted, somewhat dismayingly, that ‘‘despite the widespread use of the delta method, it is almost impossible to find who first proposed it.’’ The author pointed out that the method was first rigorously proved by Crame´r (1946, p. 353) and then suggested that the technique might have been originally proposed by Dorfman in 1938 (Dorfman, 1938). A follow-up paper (Portnoy, 2013) indicated that Gauss and Bessel were quite familiar with the delta method in the nineteenth century in the form of the law of propagation of errors. However, the 2013 paper gave no further indications regarding Gauss’s and Bessel’s use of the method. It further pointed out that Doob (1935, p. 167) had actually rigorously proved the result three years before Dorfman’s paper. However, in his paper, Doob referred to both Wright (1934, p. 211) and Kelley (1928, p. 49), who had used the method earlier. We are therefore left with the unsatisfactory situation in which the delta method seems to be ascribed to authors who, we then find out, themselves refer to earlier writers. Short of a proper reference, many statisticians prefer to ascribe the delta method to Crame´r (1946, p. 353). But, as was discussed earlier, Crame´r was far from being the first person who either used it or rigorously proved it. The question thus remains, who exactly invented the delta method, and when?
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The Delta Method and the Law of Propagation of Errors ^h Suppose is asymptotically N ðh; r2 =nÞ, i.e., pffiffiffi ^ D 2 nðh hÞ ! N ð0; r Þ. Then, for a differentiable function g such that g0 ðhÞ 6¼ 0, gð^hÞ can be written as a Taylor series expansion about h as g ^h ¼ gðhÞ þ ^h h fg0 ðhÞ þ Rn g; P where Rn ! 0 as ^h ! h. We also have that ^h ! h and P
hence Rn ! 0. By applying Slutsky’s lemma to pffiffiffi ^ nfgðhÞ gðhÞg, we obtain the result of the delta method that i oD h pffiffiffin ^ 2 ð1Þ n g h gðhÞ !N 0; fg0 ðhÞg r2 : We shall give a simple, yet historically significant, illustration. In 1898, Pearson and Filon proved that h 2 i pffiffiffi D nðr qÞ!N 0; 1 q2 (Pearson & Filon, 1898), where r and q are the sample and population correlation coefficients, and n is the sample size. It is seen that for large samples, the variance of r depends on the unknown parameter q. In 1915, Fisher suggested the transformation tanh1 q (Fisher, 1915, p. 521). Using the delta method, let us see whether this transformation stabilizes the variance when q 6¼ 0. Using gðqÞ ¼ 12 ln
1þq 1q
in (1), we have
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where the partial derivative og=o^hi is evaluated at ^ hi ¼ hi . The above may be regarded as the law of propagation of absolute errors.
pffiffiffi 1 1 þ r 1 1 þ q D n ln ln !N ð0; 1Þ; 2 1r 2 1q so that
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