※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. (這是Δ-method的應用 )
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.
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Why do statisticians often look for functions of statistics whose
variances do not depend upon the parameter?
這個Example裡，u(Y/n), whose variance is essentially free of p 有什麼用途？
P
還有，不太懂為什麼要 Y/n -----> p，才能用泰勒展開式展開u(Y/n)？
Δ-method的主要應用就是做上述這些事嗎？
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