※Example Let Y_n (or Y for simplicity) be b(n,p). Thus Y is approximately N(p,p(1-p)/n). Statisticians often look for functions of statistics whose variances do not depend upon the parameter. Here the variance of Y/n depends upon p. Can we find a function, say u(Y/n), whose variance is essentially free of p? Since Y/n converges in probability to p, we can approximate u(Y/n) by the first two terms of its Taylor's expansion about p, namely by u(Y/n) ≒ v(Y/n) = u(p) + (Y/n - p)u'(p). Of course, v(Y/n) is a linear function of Y/n and thus also has an approximate normal distribution; clearly, it has mean u(p) and variance [u'(p)]^2．p(1-p)/n. (这是Δ-method的应用 ) But it is the latter that we want to be essentially free of p; thus we set it equal to a constant, obtaining the differential equation u'(p) = c/√p(1-p). A solution of this is u(p) = (2c)arcsin√p. If we take c = 1/2, we have, since u(Y/n) is approximately equal to v(Y/n), that u(Y/n) = arcsin√(Y/n) has an approximate normal distribution with mean arcsin√p and variance 1/4n, which is free of p. --------------------------------------------------------------------------- Why do statisticians often look for functions of statistics whose variances do not depend upon the parameter? 这个Example里，u(Y/n), whose variance is essentially free of p 有什麽用途？ P 还有，不太懂为什麽要 Y/n -----> p，才能用泰勒展开式展开u(Y/n)？ Δ-method的主要应用就是做上述这些事吗？ --

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