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 $\frac{\overline x_n+ \overline y_n}{\overline x_n- \overline y_n}$.
Could you guys help out.
Really looked at the other peoples questions and I am clearly over my head with this. Really need the help, wished I could put more of my steps down but just hitting a wall. I saw that many people have questions related to this so I think that could help many.
This problem is solved by a straight forward application of the delta method. To simplify notations I will use the shorthand $f(x,y)$ for the ratio $(x+y)/(x-y)$, and for the partial derivatives $f_x = \partial f/\partial x$ and $f_y=\partial f/\partial y$. Also $a$ and $b$ are shorthands for $\mu_X$ and $\mu_Y$.
What follows assumes that $a\ne b$ and that $\tau^2=f_x(a,b)^2\sigma_X^2+f_y(a,b)^2\sigma^2_Y>0$. Otherwise things become more complicated, as indicated in whuber's comment.
The OP wants to know the asymptotic distribution of $f(\bar x_n, \bar y_n)$, where $\bar x_n$ is the sample average of the $x$s and $\bar y_n$ of the $y$s. By the central limit theorem, which applies because $\sigma^2_X$ and $\sigma^2_Y$ are both finite, $z_n:=\sqrt n (\bar x_n-a)$ tends in distribution to a $N(0,\sigma_X^2)$ random variable and $w_n:=\sqrt n (\bar y_n-b)$ tends in distribution to a $N(0,\sigma_Y^2)$ random variable. So write $(\bar x_n,\bar y_n) =(a,b)+(z_n,w_n)/\sqrt n$ and expand $f(\bar x_n,\bar y_n)$ is a Taylor series, truncating with the linear term, so $$ f(\bar x_n,\bar y_n) =f(a,b)+f_x(a,b) \frac{z_n}{\sqrt{n}} + f_y(a,b) \frac{w_n}{\sqrt{n}} + \cdots$$ and $$ \sqrt n\left(f(\bar x_n,\bar y_n)-f(a,b)\right) = f_x(a,b){z_n} +f_y(a,b) {w_n}+\cdots.\tag{*}$$ As explained in the cited reference, the $+\cdots$ terms do not affect the answer. Since $z_n$ and $w_n$ are converging in distribution to a gaussian limit, the right hand side in (*) is too, to $N(0,\tau^2)$, where $\tau^2=f_x^2 \sigma_X^2 + f_y^2\sigma_Y^2.$
The idea here is that after recentering and rescaling, your random variable is approximately a fixed linear combination of $\bar x_n$ and $\bar y_n$, with coefficients $f_x(a,b)$ and $f_y(a,b)$, and so has variance $f_x(a,b)^2\text{Var}(\bar x_n) + f_y(a,b)^2\text{Var}(\bar y_n)=\tau^2/n$.