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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How many components of a gaussian mixtures do I need to match moments up to the $r$-th order?

Suppose I have a ($k$-dimensional) random variable $X \sim D$ and I want to find a Gaussian Mixture $GM \sim \sum_{i=1}^C \pi_i \mathcal{N}(\mu_i, \Sigma_i)$ such that the moments of order $r'$, for $...
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Find asymptotic variance of the moment estimator

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $$f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$$ and we ...
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Synthasize data given mean, variance, skew, and kurtosis in python

I would like to generate synthetic data by specifying their mean, variance, skew, and kurtosis. However, I only know how to generate synthetic data with mean and var. Here is an example with mean and ...
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Characterizing "aggregatable statistic" with moments

I'm working on a project where I have statistics for several disjoint sets of data and need to use them to compute aggregate statistics for the whole population. An example may help here. Say we have ...
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Does zero cumulants imply independence?

Question: Suppose we have two random variable $X$, $Y$ that follow non-Gaussian distribution, and we are given that: $$\operatorname { cum }(X, Y)=\operatorname { cum }(X, X, Y)=\operatorname { cum }(...
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Do the principal values of higher moments exist for the Cauchy distribution?

It is known that the Cauchy distribution has undefined moments, and that the expectation has a principal Cauchy value $\operatorname{PV}\left( \mathbb{E} [X] \right)$ of zero. I wonder if $\...
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Existence of Moments for Linear Regression With Pareto Error

Suppose I have the following model linear regression model: $y = \beta_0 + x_1i\beta_1 + x_2i\beta_2 + e_i$ with $e_i \sim Pareto(k,\alpha)$ Now if $1< \alpha < 2$, I would suppose that the ...
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Standard normal moments [duplicate]

My textbook says, without proof, that $E(X^4)=3$, where $X\sim N(0,1)$. Is it so simple to obtain?
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Meaning of asymmetric third moment skewness

Given the formula of the third moment that define the skewness: $$skewness=E\Bigg[\bigg(\frac{x_i−\bar{X}}{σ}\bigg)^3 \Bigg] = \frac{\mu_3}{\sigma^3} $$ I understand that this formula calculates the ...
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For $r>s\geq1$, convergence in $s^{\text{th}}$ mean does not imply convergence in $r^{\text{th}}$ mean

I need a counterexample for the problem: if $r>s\geq1$, convergence in $s^{\text{th}}$ mean does not imply convergence in $r^{\text{th}}$ mean. The definition for convergence in mean is as follows: ...
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If the goal is to estimate $\beta$ in $Y = X\beta + g(Z) + \epsilon$, why is $E[X(Y - X\beta - g(Z))] = 0$?

Suppose I have a model $$Y = X\beta + g(Z) + \epsilon, \qquad E[\epsilon|X, Z] = 0$$ where $Y$ is the outcome, $X$ is a binary covariate of interest, and $Z$ is a vector of covariates. The goal is to ...
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Under what conditions are there pairwise monotonic relationships between mean, variance, and (positive) skewness of a lower-bounded distribution?

I am dealing with empirical data, integer- and continuous-valued, with a lower bound (at zero) that are often positively skewed, and seem to be following either the Poisson, $\chi^2$, binomial, or ...
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Distribution where the variance diverges for some parameter values but not others

I'm wondering whether there's a standard / nice / tractable family of distributions where the variance is defined for some parameter values but not others, while the mean stays finite. I'm imagining a ...
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The use of a pseudo variance $\int_{-\infty}^\infty \text{sign}(x) (x-\mu)^2 f(x) dx$ in place of $\int_{-\infty}^\infty (x-\mu)^2 f(x) dx$

Difference between odd central moments and even central moments When we compute $n$-th central moments $$\mu_n = \int_{-\infty}^\infty (x-\mu)^nf(x) dx$$ then there is a difference in interpretation ...
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Variance of a vector-valued random variable along a unit vector

Let $X$ be a vector-valued random variable with variance $\mathbb{V}[X] < \infty$. How is the variance of $X$ along a unit-vector $\hat{v}$ defined? Can we say that in general it is $\hat{v}^\top \...
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What are the first four moments of a linear function of IID random variables?

Consider a sequence of IID random variables $X_1,X_2,X_3,...$ from a common distribution with mean $\mu$, variance $\sigma^2$, skewness $\gamma$ and kurtosis $\kappa$ (all finite). Given a sequence ...
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Is the third moment of an AR(1) dependent on $t$?

Given an AR(1) process: $$ X_t = \phi X_{t-1}+ \epsilon_t, \quad \epsilon\sim WN(0, \sigma^2) $$ I know that if $|\phi|<1$, then the process is stationary (weakly). Thus, the first and second ...
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Question about sample normalization for distribution moments [duplicate]

I am having trouble understanding how the sample formulas for distribution moments are derived, for example, the third standardized central moment is: $$ \frac{1}{n}\frac{\sum{(x - \mu)^3}}{\sigma^3} $...
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Is $E[XX^T]$ also called a product moment?

I've found a related question which was for product moment $E[XY]$. Is $E[XX^T]$ also called product moment or is there a special case name for it? This is the first term of the covariance matrix $Cov(...
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Determine the joint moment generating function $M_{V,W} (s1, s2)$ of V and W

So the initial question was: Let X and Y be independent random variables with common moment generating function $m_X(s) = m_Y (s) = e^{s^2/2}$. a) Determine the moment generating function $M_V (s)$ of ...
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What are the skewness and kurtosis of the sample mean?

Consider a sequence of IID random variables $X_1,X_2,X_3,...$ from a common distribution with mean $\mu$, variance $\sigma^2$, skewness $\gamma$ and kurtosis $\kappa$ (all finite). The mean and ...
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Moments of $\text{exp}(-|x|^{1/2})$

I'm supposed to show that all of the moments of the density $\text{exp}(-|x|^{1/2})$ are finite. I'm not convinced this is true though. The $p$th moment is \begin{align*} \mathbb{E}[X^p] &= \int_{-...
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Does skewness decrease standard deviation ceteris paribus?

For a given probability distribution, probability mass must sum to 1, thus by increasing a parameter corresponding to skewness do you shift probability away from the second central moment (variance) ...
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7 votes
1 answer
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Problem of computing standard error of higher moments

currently, I am trying to understand how one calculates the standard errors on higher moments using Rao's book [1]. On page 437, he defines $$ O_r = \frac{1}{n} \sum x_i^r, \, v_r = E[x^r], \, \mathrm{...
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Fourth moment of arch(1) process

I have an ARCH(1) process \begin{align*} Y_t &= \sigma_t \epsilon_t, \\ \sigma_t^2 &= \omega + \alpha Y_{t-1}^2, \end{align*} and I am trying to express the fourth moment $\mathbb{E}[Y_t^4]$ ...
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How to intepret the second raw moment

I know the second raw moment is defined as $E(X^2)$, for $X$ being a random variable drawn from any distribution or a given large data sample. I also know that based on variance decomposition, ...
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Adding a lognormal distribution into a model of shot noise for a Poisson process to calculate moments of the process

I'm working with a simple 1D model of some objects propagating through a domain, $x$. The setup is like shot noise, and is actually taken from a paper by Militello, F., and Omotani, J. T, 2016 Nucl. ...
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1 answer
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Find the expected value of X given other moments

Given the variance $\mathbb{V}(X)=7$ and the expected square $\mathbb{E}(X^2)=16$, what is the expected value of $X$?
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Calculate skewness in base R

I'm working on my literacy in math notation and statistics in general, using the R language as a means of translating formulas as I read through textbooks and teach myself. I'm trying to write Pearson'...
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Are the mean of a joint distribution and its marginals equal?

I know that for a multivariate normal distribution the mean of the parameters is the same as the mean of the marginal distributions for the respective parameters. But is this always the case?
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11 votes
2 answers
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Westfall says, "the proportion of the kurtosis that is determined by the central $\mu\pm\sigma$ range is usually quite small" but is the reverse true?

In his article that debunks the notion of kurtosis as measuring distribution peakedness, Peter Westfall writes, "[T]he proportion of the kurtosis that is determined by the central $\mu\pm\sigma$ ...
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Expectation of x/√(x²+2px+1) under Normal distribution

I'm need to find (or at least approximate) as a function of $p$, the expectation under $x \sim Normal(0,1)$ of: $$f(x) = \frac{x}{\sqrt{x^2+2px+1}},\hspace{1em}\textrm{where}-1<p<1$$ Wolfram ...
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What is the expected value of the log-log-normal distribution (aka the LLN distribution)?

Question 387180 discusses the pdf of the log-log-normal distribution. I'd like to know if there's an expression for the mean of this distribution. I'm trying to work it out with pencil and paper, but ...
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why the following implication relationship is true

In a paper(unpublished), I read about the following implication relationship. $g(x)$ is a smooth function. The assumption is that there exist a constant (scalar) $a>0$ such that $E[sup_{|b|\leq a}(...
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Multivariate Lindeberg-Levy CLT, why assuming $E(||X||^2)<\infty$ instead of $E(XX')<\infty$?

Multivariate Lindeberg-Levy CLT(demeaned version) states that "Let $\{X_1,...,X_N\}$ be a random sequence with mean zero, if $E(||X_i||^2)<\infty$, then $\frac{1}{\sqrt{N}}\sum_{i=1}^NX_i$ ...
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Integral of the error function

Assume a pair of normal variables $a,b\sim N(\mu, \Sigma)$, with $\rho_{ab}\neq0$. We know their joint distribution in the (shorthand) form: $$f_X(a,b)=\frac{1}{2\pi\sqrt{|\Sigma|}}exp(-\frac{1}{2}(x-\...
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Skewness of the sum of dependent variables [closed]

I have two random variables $X$ and $Y$ which are both time series of observations. The two series are not independent. I want to find the skewness, kurtosis, hyper-skewness and hyper-kurtosis of $X+...
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2 votes
1 answer
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How can a 'tall-and-skinny' property related to kurtosis be stated rigorously in terms of its (smooth) probability functions?

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[ ...
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1 vote
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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 ...
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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 ...
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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 ...
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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 ...
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1 vote
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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]=\...
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  • 561
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1 answer
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Analytically derive standard deviation of parent normal distribution (with known mean) from moments of the truncated normal distribution

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 ...
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  • 153
1 vote
1 answer
227 views

AR(1) - Stationarity condition

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 ...
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