Katakanlah kita memiliki variabel acak dengan varian dan mean yang diketahui. Pertanyaannya adalah: apa varian untuk beberapa fungsi yang diberikan f. Satu-satunya metode umum yang saya ketahui adalah metode delta, tetapi hanya memberikan aproximation. Sekarang saya tertarik pada , tetapi akan menyenangkan juga mengetahui beberapa metode umum.
Sunting 29.12.2010
Saya telah melakukan beberapa perhitungan menggunakan seri Taylor, tapi saya tidak yakin apakah itu benar, jadi saya akan senang jika seseorang dapat mengonfirmasinya .
Now we can approximate
Using the approximation of we know that
Using this we get:
variance
random-variable
delta-method
Tomek Tarczynski
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Jawaban:
Update
I've underestimated Taylor expansions. They actually work. I assumed that integral of the remainder term can be unbounded, but with a little work it can be shown that this is not the case.
The Taylor expansion works for functions in bounded closed interval. For random variables with finite variance Chebyshev inequality gives
So for anyε>0 we can find large enough c so that
First let us estimateEf(X) . We have
Since the domain of the first integral is interval[EX−c,EX+c] which is bounded closed interval we can apply Taylor expansion:
Substituting this formula to the previous one we get
Now for the variance we can use Taylor approximation forf(x) , subtract the formula for Ef(x) and square the difference. Then
whereT3 involves moments E(X−EX)k for k=4,5,6 . We can arrive at this formula also by using only first-order Taylor expansion, i.e. using only the first and second derivatives. The error term would be similar.
Other way is to expandf2(x) :
Similarly we get then
The formula for variance then becomes
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To know the first two moments of X (mean and variance) is not enough, if the function f(x) is arbitrary (non linear). Not only for computing the variance of the transformed variable Y, but also for its mean. To see this -and perhaps to attack your problem- you can assume that your transformation function has a Taylor expansion around the mean of X and work from there.
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