Let $x_1, ..., x_n$ and $y_1, ..., y_n$ be two independent random samples from $X$ and $Y$. We have $µ_X = E (X ) > 0, µ_Y = E (Y ) > 0$ and $σ^2_X = Var (X )$ and $σ^2_Y = Var (Y )$.

Derive the asymptotic distribution of $\frac{\overline x_n+ \overline y_n}{\overline x_n- \overline y_n}$.

where ${\overline x_𝑛}$ is the sample average of the 𝑥𝑖

Haven't put any additional information because I am hitting a wall, really don't know how to resolve this. I have looked at the delta method as a possible route but it's too convoluted for me

  • $\begingroup$ What have you done so far? Can you edit the question and show your current work? $\endgroup$ – Brennan Nov 17 '19 at 18:22
  • $\begingroup$ Wished I could tell you more, but I really stuck. Don't really know how to proceed. I know people would rather see me find the solution myself but if I knew how I would have put it down here. Thanks for caring Brennan $\endgroup$ – James Nov 17 '19 at 18:46

I can provide you with a sketch of proof to proceed.

The quantity $\sqrt{n}(\bar{x}-\mathrm{E}(X)) \overset{d}{\to} N(0,\,\mathrm{V}(X))$ by Central Limit Theorem. Similarly, the same can be said for the other quantity, $\bar{y}$.

Since a linear combination of Gaussian variables are Gaussian, then the summation and subtraction of the two quantities are also Gaussian. Prove that $\bar{x}+\bar{y}$ and $\bar{x}-\bar{y}$ are of zero means, then the quotient/ratio of two zero-mean Gaussian random variable is a Cauchy distribution.


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