Linked Questions

37
votes
4answers
18k views

Generic sum of Gamma random variables

I have read that the sum of Gamma random variables with the same scale parameter is another Gamma random variable. I've also seen the paper by Moschopoulos describing a method for the summation of a ...
14
votes
4answers
612 views

Do third order asymptotics exist?

Most asymptotic results in statistics prove that as $n \rightarrow \infty$ an estimator (such as the MLE) converges to a normal distribution based on a second-order taylor expansion of the likelihood ...
8
votes
2answers
4k views

Real life uses of Moment generating functions

In most basic probability theory courses your told moment generating functions (m.g.f) are useful for calculating the moments of a random variable. In particular the expectation and variance. Now in ...
6
votes
3answers
975 views

Series expansion of a density function

Here's something I've wondered about for a while, but haven't been able to discover the correct terminology. Say you have a relatively complicated density function that you suspect might have a close ...
10
votes
1answer
2k views

Why Use the Cornish-Fisher Expansion Instead of Sample Quantile?

The Cornish-Fisher Expansion provides a way to estimate the quantiles of a distribution based on moments. (In this sense, I see it as a complement to the Edgeworth Expansion, which gives an estimate ...
6
votes
2answers
803 views

Finding the distribution of iid variables X, Y given distribution of X-Y

Say I know the distribution of $X-Y$, but I do not know the distributino of $X$ (or $Y$), but I know that they are statistically independent, and I know they have the same distribution. Is the problem ...
6
votes
2answers
575 views

Elementary approach to higher order asymptotics

I am trying to understand “higher order asymptotics”. I find several texts on Likelihood asymptotics, nothing’s easy to read... if you have any nice pointers on this direction, I’ll be interested; ...
6
votes
1answer
2k views

Generalized Chi-Squared Distribution PDF

Let $\mathbf{X} \sim \mathcal{N}_n( \mathbf{m}, \mathbf{C})$ be an $n$-dimensional gaussian vector, where $\mathbf{C} \in \mathbb{R}^{n \times n}$ is not diagonal, but it is positive-definitive, $\...
5
votes
1answer
572 views

Bound for weighted sum of Poisson random variables

Suppose I have some independent Poisson-distributed random variables $X_1 \ldots X_N$ with parameters $\lambda_1 \ldots \lambda_N$. These can be thought of as processes where each arrival/event ...
7
votes
1answer
548 views

Is applying the CLT to the sum of random variables a good approximation?

I use $(\mu, \sigma^2)$ to mean a distribution with mean $\mu$ and variance $\sigma^2$, $\mathcal{N}$ added to mean the normal distribution. Let's suppose $X_1, \dots, X_n\overset{\text{iid}}{\sim}(\...
3
votes
3answers
158 views

Literature on nonparametric density estimation

I am about to write my bachelor thesis about non-parametric density estimation, especially kernel density estimators and their application in classification. As I am quite new to looking for academic ...
5
votes
0answers
829 views

How to find the distribution of the weighted sum of independent Bernoulli random variables for positive non-integer weights

How do I find the distribution of the weighted sum of independent Bernoulli random variables if the weights are non-negative real numbers? I have N number of independent Bernoulli distributed random ...
8
votes
1answer
328 views

Sum of linear combination of product of exponentials is exponential

This problem has arisen in my research: suppose that $V_i \sim \text{ED}$ are iid exponential distributions (ED) with mean $1$ and let $\lambda$ be a nonnegative number. Is it true that $$ \sum_{k=0}^...
3
votes
1answer
194 views

Moment Generating Function

why we use Moment Generating Function in probability theory ? I want to know 1-the main reasons of using (m.g.f.) instead of probability mass function(p.m.f.) and probability density function (p.d.f....
2
votes
3answers
223 views

Generating random nos based on 'k' moments

How do I generate random nos based on say k moments? (no other constraints on support) When k = 2, we generate random nos. from a normal distribution defined by the 2 moments. Can we generalize this ...

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