<p>I can provide you with a sketch of proof to proceed.</p>

<p>The quantity <span class="math-container">$\sqrt{n}(\bar{x}-\mathrm{E}(X)) \overset{d}{\to} N(0,\,\mathrm{V}(X))$</span> by Central Limit Theorem. Similarly, the same can be said for the other quantity, <span class="math-container">$\bar{y}$</span>.</p>

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

CC BY-SA 4.0
2019-11-23T04:22:20.927Z
2019-11-23T07:55:22.727Z
2019-11-23T07:55:22.727Z
QuantStats
<p>Let <span class="math-container">$x_1, ..., x_n$</span> and <span class="math-container">$y_1, ..., y_n$</span> be two independent random samples from <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>. 
We have <span class="math-container">$µ_X = E (X ) &gt; 0, µ_Y = E (Y ) &gt; 0$</span> and <span class="math-container">$σ^2_X = Var (X )$</span> and <span class="math-container">$σ^2_Y = Var (Y )$</span>.</p>

<p>Derive the asymptotic distribution of <span class="math-container">$\frac{\overline x_n+ \overline y_n}{\overline x_n- \overline y_n}$</span>.</p>

<p>where <span class="math-container">${\overline x_}$</span> is the sample average of the </p>

<p>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</p>

CC BY-SA 4.0
2019-11-17T18:22:41.210Z
What have you done so far? Can you edit the question and show your current work?
Brennan
CC BY-SA 4.0
2019-11-17T18:46:52.577Z
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
James
<econometrics><statistics>
Derive the asymptotic distribution
James
# Derive the asymptotic distribution

(Asked by: username_0 on 2019-11-17)

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 ${\overline x\_}$ is the sample average of the

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

**Tags:** econometrics, statistics
> **Comments:**
> **username_1**: What have you done so far? Can you edit the question and show your current work?  
> **username_0**: 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  

---

## 1 answers

### Answer by username_2 on 2019-11-23. Score: 0

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.


---

