# Questions tagged [moments]

Moments are summaries of random variables' characteristics (e.g., location, scale). Use also for fractional moments.

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### Cross-sectional properties of an AR(1)

I have an AR(1): $$X_t = c + \phi X_{t-1} + \epsilon,$$ where $\epsilon$ is white noise. I want to estimate this AR(1) on panel data. However, my time dimension is very short, and so I thought that ...
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### GMM model understanding and J test results

Hi there, I am reading some GMM models and am wondering how can I get the total moment functions from the model description? I thought it should be 100 functions, one for each portfolio, but I am not ...
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### Expectation of the product of iid random variables

If we have iid random variables $X_1,X_2,...,X_N$ with $\mathbb{E}X_i=\mu$, is it true that $\mathbb{E}\prod X_i=\mu^N$? I had no doubt that this is true, until I tried it out with Python, using ...
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### Independence of first and second sample moments (about zero) of normally distributed random variables

If $X_1,...,X_n$ are independent normally distributed random variables with means $\mu_i$ and variances $\sigma_{i}^2$ and, \begin{align} M_1^2 &=\left(\dfrac{1}{n}\sum\limits_{i=1}^n X_i\right)^2 ...
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### What is the phenomenon I'm observing here?

I was looking at the St. Petersburg paradox and wanted to make a quick simulation to see the results. I made a quick program simulating 1 Billion "games" and the average of the gain (over ...
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Skewness is generally defined as a standardized third-order centered moment. $$S(X) \triangleq \frac{\mathbb{E}\left[ \left( X - \mathbb{E} \left[ X \right] \right)^3 \right]}{\left( \mathbb{E}\left[ ... 1 vote 1 answer 47 views ### Showing that E[X^2] = E[Y^2] for two RV following Frechet distributions with different location parameters only [closed] If random variable X follows a Fréchet distribution (https://en.wikipedia.org/wiki/Fr%C3%A9chet_distribution) with shape parameter \alpha, scale parameter s, and location parameter m, that is X ... 0 votes 3 answers 321 views ### Why Assumed Density Filtering is also called Moment Matching? I am learning about Assumed Density Filtering (ADF) and Expectation Propagation in the context of bayesian deep neural networks. I have seen in some textbooks and papers that ADF is also called moment ... 1 vote 0 answers 28 views ### Computing difficult expectation I am doing some derivations for a project and at some point the following integral shows up$$ \int x^{c} \log(1-F(x)) N f(x) [1-F(x)]^{N-1} dx $$i.e the expectation of the product of natural ... 1 vote 2 answers 202 views ### Moments (mean and skewness) of an AR(1) process with Chi2 or Gamma innovation distribution A bit of context I am looking for a lag-1 autoregressive process with non-Gaussian innovation/residual error, which is capable of producing both skewed and non-skewed marginal distributions. I am ... 1 vote 0 answers 67 views ### Delta method for third moment Suppose X_1,...,X_n is a sample from a population with mean \mu and variance \sigma^2 and third central moment of \mu_3. I want to justify that:$$E[\left( h(\bar{X})-E(h(\bar{X}))\right)^3]=\...
Given $\sigma_t$ and $\mu_t$ of a truncated normal distribution, as well as $\mu_p$ of the parent normal distribution, I would like to analytically compute the standard deviation $\sigma_p$ of the ...
Consider the well-known AR(1) model: $$x_t = \phi X_{t-1} + \epsilon_t$$ where, as usual, $\epsilon$ is an independent white noise process. I have read many sources. All of them get away saying that ...