# Thread: MATH2901 Higher Theory of Statistics

1. ## MATH2901 Higher Theory of Statistics

$f,g\text{ are increasing functions and }X,Y\text{ are i.i.d. r.v.s}$

$\text{RTP: }(f(X) - f(Y))(g(X) - g(Y)) \ge 0$

2. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by leehuan
$f,g\text{ are increasing functions and }X,Y\text{ are i.i.d. r.v.s}$

$\text{RTP: }(f(X) - f(Y))(g(X) - g(Y)) \ge 0$
We can just prove it by cases: either X ≥ Y occurs or Y > X occurs, and in either case the LHS is ≥ 0, since f and g are increasing functions.

Same is true (replacing X by x and Y by y) if x and y are just real numbers in the domain of f and g if f and g are functions defined on a subset of the reals.

3. ## Re: MATH2901 Higher Theory of Statistics

Whoops. That went over my head.

4. ## Re: MATH2901 Higher Theory of Statistics

I can't make sense out of the question.

Is it meant to be (RTP):

$P( (f(X) - f(Y))(g(X) - g(Y))\geq 0) = 1$

5. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by He-Mann
I can't make sense out of the question.

Is it meant to be (RTP):

$P( (f(X) - f(Y))(g(X) - g(Y))\geq 0) = 1$
No, it was a function of a random variable. But InteGrand's solution worked.
_______________________________________

Whilst I get why chisq is the result of Z^2 (Z~N(0,1)), where in practice does it actually get used? What's so powerful about the square of the standard normal?

6. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by leehuan
No, it was a function of a random variable. But InteGrand's solution worked.
The point I'm making is that how can a random variable exist alone without specification of it's probability?

Example, X is gamma random variable with whatever parameters. Prove that X > 0. How does this make sense? It should be prove that P(X > 0) = 1.

7. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by He-Mann
The point I'm making is that how can a random variable exist alone without specification of it's probability?

Example, X is gamma random variable with whatever parameters. Prove that X > 0. How does this make sense? It should be prove that P(X > 0) = 1.
Look at the 2014 finals. I'm sure you still have it from back when you did the course.

I don't even care about the distribution of X in this question. I just care that X is a random variable. And I want to find something about f(X), which is what the function does to the random variable (given that f is monotonic increasing).

8. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by leehuan
No, it was a function of a random variable. But InteGrand's solution worked.
_______________________________________

Whilst I get why chisq is the result of Z^2 (Z~N(0,1)), where in practice does it actually get used? What's so powerful about the square of the standard normal?
It's used in many hypothesis tests, for example.

9. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by leehuan
Look at the 2014 finals. I'm sure you still have it from back when you did the course.

I don't even care about the distribution of X in this question. I just care that X is a random variable. And I want to find something about f(X), which is what the function does to the random variable (given that f is monotonic increasing).
Look at another simple example.

Define f: R -> R such that f(x) = x^2 and let X be a random variable with gamma distribution, say. Prove that f(X) = X^2 > 0.

It's obvious that this is true because you're just squaring positive random variables and they stay positive. But how does it make sense alone? It should be prove that P(X^2 > 0) = 1.

The point is, random variables cannot exist alone. It needs to be associated with probabilities.

10. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by leehuan
Whilst I get why chisq is the result of Z^2 (Z~N(0,1)), where in practice does it actually get used? What's so powerful about the square of the standard normal?
It gets used a lot to understand variances and forms the basis of other distributions such as the F-distribution (which has obvious applications in ANOVA, regression etc)

11. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by He-Mann
Look at another simple example.

Define f: R -> R such that f(x) = x^2 and let X be a random variable with gamma distribution, say. Prove that f(X) = X^2 > 0.

It's obvious that this is true because you're just squaring positive random variables and they stay positive. But how does it make sense alone? It should be prove that P(X^2 > 0) = 1.

The point is, random variables cannot exist alone. It needs to be associated with probabilities.
It makes perfect sense to me after seeing the earlier question. And again, I couldn't care less if it's normal or gamma or exponential or a discrete r.v.

You've done the course before. If you're dissatisfied, pick up the exam paper and take it up with the lecturer.

It might be an abuse of notation. But otherwise I don't see what's wrong with it.

12. ## Re: MATH2901 Higher Theory of Statistics

MInd blanking.

$f_X(x) = \alpha x^{-(\alpha+1)}, 11$

$\text{The 80th percentile of }X\text{ is }x_{0.8}=5^{\frac{1}{\alpha}}$

$\text{iii) Find an expression for }\mathbb{E}[X\mid X > x_{0.8}]$

13. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by leehuan
MInd blanking.

$f_X(x) = \alpha x^{-(\alpha+1)}, 11$

$\text{The 80th percentile of }X\text{ is }x_{0.8}=5^{\frac{1}{\alpha}}$

$\text{iii) Find an expression for }\mathbb{E}[X\mid X > x_{0.8}]$
By the way, this is a Pareto Distribution with scale parameter 1 (and shape parameter alpha). This distribution is related to the "80-20" law or "Pareto principle", which you may have heard of.

A fact about Pareto distributions is that the conditional distribution of a Pareto r.v. X given the event that X is greater than a given number b (where b is greater than or equal to the scale parameter of the distribution) is still a Pareto distribution, with the same shape parameter but with new scale parameter b. Using this fact and the formula for the mean of a Pareto distribution, the desired conditional mean can be deduced.

(These facts are facts that should be proved before being used for this Q. I suppose, and can be all proven as an exercise using standard methods for dealing with truncated distributions. For this particular Q., you wouldn't need to know these facts, you could just do the Q. normally, but they are interesting and it is what the Q. was probably getting at (particularly the 80-20 law, which is maybe why they chose the 80-th percentile), so I included them here.)

14. ## Re: MATH2901 Higher Theory of Statistics

$f_X(x) = \sqrt{\frac{\beta}{2\pi x^3}}\exp \left[-\frac{\beta}{2x}+\alpha - \frac{\alpha^2 x}{2\beta}\right]. x> 0\\ \text{and it is given that }\mathbb{E}[X] = \frac{\beta}{\alpha}$

$\text{Some info: MLE of }\alpha\text{ is }\hat{\alpha} = \frac{\beta n}{\sum_{i=1}^n X_i}$

$\text{The Fisher information matrix is }n \begin{pmatrix} \frac{1}{\alpha} & -\frac{1}{\beta} \\ -\frac{1}{\beta} & \frac{1}{2\beta^2}+\frac{\alpha}{\beta^2} \end{pmatrix}$

$\text{iv) Hence show that the approximate variance for }\hat{\alpha}\text{ for large }n\text{ is}\\ \text{Var}(\hat{\alpha})=\frac{2\alpha^2+\alpha}{n }$

15. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by leehuan
$f_X(x) = \sqrt{\frac{\beta}{2\pi x^3}}\exp \left[-\frac{\beta}{2x}+\alpha - \frac{\alpha^2 x}{2\beta}\right]. x> 0\\ \text{and it is given that }\mathbb{E}[X] = \frac{\beta}{\alpha}$

$\text{Some info: MLE of }\alpha\text{ is }\hat{\alpha} = \frac{\beta n}{\sum_{i=1}^n X_i}$

$\text{The Fisher information matrix is }n \begin{pmatrix} \frac{1}{\alpha} & -\frac{1}{\beta} \\ -\frac{1}{\beta} & \frac{1}{2\beta^2}+\frac{\alpha}{\beta^2} \end{pmatrix}$

$\text{iv) Hence show that the approximate variance for }\hat{\alpha}\text{ for large }n\text{ is}\\ \text{Var}(\hat{\alpha})=\frac{2\alpha^2+\alpha}{n }$
$\noindent First step would be to find the 1,1 entry of the inverse of the Fisher information matrix (this would be the approximate variance of \widehat{\alpha} for large n). To eliminate \beta's from your answer, use the relationship between \widehat{\alpha} and \beta from the MLE formula, and make sure to replace \frac{1}{n}\sum_{i=1}^{n}X_{i}'s with \frac{\beta}{\alpha}, which is the mean.$

16. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by InteGrand
$\noindent First step would be to find the 1,1 entry of the inverse of the Fisher information matrix (this would be the approximate variance of \widehat{\alpha} for large n). To eliminate \beta's from your answer, use the relationship between \widehat{\alpha} and \beta from the MLE formula, and make sure to replace \frac{1}{n}\sum_{i=1}^{n}X_{i}'s with \frac{\beta}{\alpha}, which is the mean.$
For my reference sake, does the 2,2 entry of the inverse approximate Var(\hat{\beta})?

Edit: Oops.

17. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by leehuan
For my reference sake, does the 2,2 entry of the inverse approximate \hat{\beta}?
$\noindent It is the approximate \textbf{variance} of \widehat{\beta} (for large n), yes.$

18. ## Re: MATH2901 Higher Theory of Statistics

Hang on, I'm not sure if I'm doing something wrong because I run into a circular argument.

$\text{I get } \text{Var}( \hat{a} ) = \frac{n}{2\alpha\beta^2} \left( \frac{1}{2\beta^2} + \frac{\alpha}{\beta^2} \right)=\frac{n}{2\beta^4}\left( \frac1{2\alpha} +1 \right)\\ \text{which shouldn't be a problem}$

$\text{I rearrange the definition of the MLE into }\beta = \frac{\hat{a}}{n}\sum_{i=1}^nX_i\\ \text{but then to get rid of }\sum_{i=1}^n X_i\text{ I have to put }\beta\text{ back in there?}$

19. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by leehuan
Hang on, I'm not sure if I'm doing something wrong because I run into a circular argument.

$\text{I get } \text{Var}( \hat{a} ) = \frac{n}{2\alpha\beta^2} \left( \frac{1}{2\beta^2} + \frac{\alpha}{\beta^2} \right)=\frac{n}{2\beta^4}\left( \frac1{2\alpha} +1 \right)\\ \text{which shouldn't be a problem}$

$\text{I rearrange the definition of the MLE into }\beta = \frac{\hat{a}}{n}\sum_{i=1}^nX_i\\ \text{but then to get rid of }\sum_{i=1}^n X_i\text{ I have to put }\beta\text{ back in there?}$
I think double check your Var, the beta terms should end up cancelling out I think.

20. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by InteGrand
I think double check your Var, the beta terms should end up cancelling out I think.
Oh. My bad. I didn't do 1/det

21. ## Re: MATH2901 Higher Theory of Statistics

I think you forgot to divide by the determinant of the Fisher Information Matrix. You appear to have multiplied by it (or something).

22. ## Re: MATH2901 Higher Theory of Statistics

Also I think the final answer's denominator should be n^2 (not n).

23. ## Re: MATH2901 Higher Theory of Statistics

Any tips on what test statistic to use for a hypothesis test?

24. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by Flop21
Any tips on what test statistic to use for a hypothesis test?
It depends on the hypothesis test and the situation at hand. Common ones are when you have asymptotic normality under a null hypothesis (which is often the case from the Central Limit Theorem), you can use a Z-test ( https://en.wikipedia.org/wiki/Z-test ), or you could use a Chi-Squared test sometimes. Depends on what purpose the test is serving.

25. ## Re: MATH2901 Higher Theory of Statistics

Originally Posted by InteGrand
Also I think the final answer's denominator should be n^2 (not n).
Oh I double checked that. I put the n back in there, got a determinant of (2alpha beta^2)/n^2 * (a matrix with n's appearing everywhere). That's fine I reckon

Originally Posted by Flop21
Any tips on what test statistic to use for a hypothesis test?
When we were taught it, we were honestly taught "educated guess". Normally I try to exploit the CLT.

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