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Let $X_1, X_2, \cdots, X_n$ be an identical and independently distributed sample from $N(\mu, \sigma^2)$, define:

$$D = \frac{1}{t}\left[\overline{X} + \frac{1-\rho}{2} S^2\right]$$

where:

$t$ and $p$ are fixed constants

$\overline{X} = \frac{1}{n}\sum_{i=1}^n X_i$, i.e, the sample mean

$S^2 = \frac{1}{n} \sum_{i=1}^n \left(X_i-\overline{X}\right)^2$, i.e, the biased sample variance

Then what is the sampling distribution of $D$?


I know that $\overline{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$ and $n\frac{S^2}{\sigma^2} \sim \chi^2(n-1)$, but how do I derive the sampling distribution of $D$?

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    $\begingroup$ The trick to solving this is a special known result that $\overline{X}$ and $S^2$ are independent (even though both are based on the same $X_i$~$N(\mu, \sigma^2)$), so your problem essentially reduces to finding the sum of a Normal random variable and a Chisquared random variable. That is the upside. The downside is that finding the sum of a Normal and a Chisquared may be much more tricky than it sounds ... $\endgroup$ – wolfies Nov 22 '13 at 17:16
  • $\begingroup$ What do you want to use the result for/what kind of an answer do you need? $\endgroup$ – Glen_b Nov 22 '13 at 22:01
  • $\begingroup$ @wolfies haha yes, I tried using mgf's to no avail... it really looks much more difficult than it seems. Also since there is a constant term in front of the chi squared random variable, wouldn't that make the entire term a Gamma distributed random variable? $\endgroup$ – TeTs Nov 23 '13 at 5:34
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    $\begingroup$ @Glen_b This is part of a bigger problem which (ideally) requires a formal proof of the sampling distribution of $D$. This is because I would like to create test statistics for $D$ and conduct hypothesis testing. $\endgroup$ – TeTs Nov 23 '13 at 5:57
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I would go with Monte Carlo simulation. Pretty straightforward and less chance of mistakes. There may be some slick math-stats solution too, but Monte Carlo is probably faster.

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  • $\begingroup$ Thanks for the suggestion. I might actually try that first just to see the results. To clarify, do I just generate random values according to the distribution of $\overline{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$ and $n\frac{S^2}{\sigma^2} \sim \chi^2(n-1)$, then substitute back into the expression to calculate the corresponding value for $D$, then apply a kernel density estimation technique to estimate the distribution for $D$? $\endgroup$ – TeTs Nov 23 '13 at 5:41
  • $\begingroup$ Yea that would work fine $\endgroup$ – user31668 Nov 23 '13 at 15:48

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