Questions tagged [moments]

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

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2answers
195 views

Independence of Mean and Variance of Discrete Uniform Distributions

In the comments below a post of mine, Glen_b and I were discussing how discrete distributions necessarily have dependent mean and variance. For a normal distribution it makes sense. If I tell you $\...
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23 views

Moments of the Dirichlet distribution [duplicate]

I was reading the Wikipedia article of the Dirichlet distribution which gives a general equation for the moments of a Dirichlet distributed random variable $X=(X_1,\cdots,X_K) \sim Dir(\boldsymbol{\...
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1answer
33 views

What is the cumulants of a whole data in terms of the cumulants of its parts?

I have around 8 billion data points, and I need to calculate the distribution and the cumulants of this distribution. However, due to technical restrictions, and time constraints, I can only ...
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1answer
37 views

Where does the constant “B” come from in moment generating fuctions?

In my book "Mathematical Statistics with Applications" by Dennis Wackerly it's stated that the moment generating function exists if The moment-generating function m(t) for a random variable Y is ...
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1answer
91 views

How Many Moments Uniquely Define a Distribution with Finite Support?

Simple question, but one to which I could not find the exact answer elsewhere. How many moments of a discrete probability distribution with finite support are required to uniquely identify the exact ...
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25 views

Computing the fourth moment of a compound Gaussian distibution

I am working with a physical application, in which time-evolution is studied by monte-carlo evolution of Gaussian states (that is, the mean and variance of these states evolve according to coupled ...
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1answer
56 views

Need help to understand Feller's statement “whenever $r$th moment exists so do all preceding moments”

I am reading the book of Feller called "An Introduction to Probability Theory and Its Applications, Vol I" (third edition, page 227) and am stuck at the moment he explains the notion of variance of a ...
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1answer
16 views

Bound for density of random variable with finite second moment

Let $\mathbf{X}$ be a vector-valued random variable with finite second moment and density $\rho$. Assume that $\rho$ is bounded and continuous. As $\mathbf{X}$ has finite second moment, I hope to find ...
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21 views

Finite second moments inhertitable to conditional variables?

Assume a random vector $\mathbf{x}=(x_1,\ldots,x_n)^\top$ that has finite second moments, i.e., $$\int\mathbf{x}\mathbf{x}^\top\rho(\mathbf{x})\,\text{d}\mathbf{x} < \infty.$$ Does it follow that ...
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8 views

Estimation of variance in GMM

In GMM, the efficient weight matrix minimizes the asymptotic variance of the GMM estimator by setting: $$ W_T^{opt} = S_T^{-1}$$ where $S_T$ is an estimator of the asymptotic variance of the moments,...
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139 views

Exponential family and geometric distribution: how do we prove the sum of independent geometric random variables has negative binomial distribution?

Let $X$ be the number of failures before the first success in a sequence of Bernoulli trials with probability of success $\theta$. Then $P_{\theta}[X = k] = (1-\theta)^{k}\theta$, $k = 1,2,\ldots$ ...
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1answer
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“Appropriate conditions” for method of moments estimator to exist, be consistent, and asymptotically normal?

My statistics text has the following theorem, and alludes to "appropriate conditions on the model", but never specifies what those conditions are. What conditions are necessary? Let $\hat{\theta}_n$...
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1answer
77 views

Why does $\mathrm{E}[e^{-X}] = 0$ imply $\mathrm{P}(X = \infty)=1$?

Came across the following problem: with the following solution: I can follow the whole solution, except for the last statement: $\mathrm{E}[e^{-X}] = 0, $ equivalently, $\mathrm{P}(X = \infty)=1.$ ...
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14 views

Are there any distributions where higher moments are defined but lower ones are not? [duplicate]

My assumption is that there is not (unless there is some playing around with inverse distributions). But are there any examples of distributions where ,say, the second moment exists but the first ...
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1answer
26 views

Theoretical dependency of moments on parameter of a Boltzmann distribution

Assume $$X\sim \frac{e^{-\beta Nf(x)}}{Z_{\beta}}$$ where $$ f(x) = -hx -x^2 + \frac{1}{\beta N}(1-x)\ln(1-x) + (1+x)\ln(1+x) $$ and $Z_{\beta}$ is the appropriate normalization factor. The support ...
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1answer
43 views

Fitting Log logistic and calculating its mean

I am trying to fit a log logistic curve to my set of data library(MASS) library(survival) library(fitdistrplus) library("actuar") ...
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14 views

Moment problem on bounded interval, sufficient and necessary conditions

On the unbounded interval (the Hamburger moment problem), we know that given the sequence of moments $m_k, \forall k = 0, 1, \dots, n$ the necessary and sufficient conditions that the density function ...
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1answer
46 views

How to estimate the probability mass function of a discrete variable from moments

Consider a bounded, discrete random variable $X$ whose range is $(0,1,\ldots, M)$. We are given the first $k$ moments of this distribution, call them $m_1, \ldots, m_k$. We are interested in ...
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1answer
142 views

Higher order moments of a multivariate Gaussian rv

Let $X~N_d(\mu,\Sigma)$ be a multivariate Gaussian random vector. Is there a convenient formula for each of $$ \mu_p\triangleq \mathbb{E}\left[\sum_{i=1}^d |X_i|^p\right], $$ in terms of $\mu$ and ...
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23 views

Measure modality of a distribution

In statistics, different orders of moments are tools to characterize a distribution, for example mean, covariance, skewness etc., which also gives an intuitive way to visualize the distribution. But ...
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64 views

Getting skew normal parameters from its moments

I have data on expenditure in dollars, and for set of countries i know average expenditure, sd, skewness. For example in country A mean=200\$, sd=100, skewness=1.5 and i want to estimate probability ...
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1answer
43 views

Determine expected value for continuous random variable [duplicate]

I know that $E[X^n]$ is found by $$\displaystyle\int_{0}^\infty{x^nf_x(x)dx}$$ I simplified this to $$\displaystyle\int_{0}^\infty{ \frac{x^{\frac{v}{2}-1+n}e^{\frac{-x}{2}}}{\displaystyle\int_{0}^\...
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Higher moments of linear regression residuals?

I previously asked this on Math StackExchange, with no success, but this post will add to that with some simulations. Background In the following linear regression with i.i.d $\epsilon_i$ $(i = 1, \...
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1answer
45 views

Relationship between exponential families and moment generating functions

I have recently been playing around with some change of measure arguments for shifting the mean of a sub-gaussian distribution. It occurred to me however, that sub-gaussianity might not be the natural ...
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1answer
61 views

Are there always two degrees of freedom in any probability distribution?

Take any random variable $X$ that follows some distribution $P(X)$. I was looking at this Wikipedia page and I'm trying to get some intuition for why we choose to define standard moments the way we do....
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Getting parameters of new distribution after moment matching using expectation propagation

In this paper, the authors apply Expectation Propagation for feature selection. However, I don't understand how to get the analytical expression from equation 25. They say: The right-hand side of (...
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27 views

What's the name of the ratio between successive moment of random variable?

Let $X$ be a random variable. Is there a name to the ratio: $\frac{E[X^{n+1}]}{E[X^{n}]}$ where $E[\cdot]$ is the expectation of $\cdot$. We let $n\ge0$ and for $n=0$ it's simply the expectation ...
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1answer
48 views

Gaussian distribution: moments, independence and rotation

I have a few questions with respect to the gaussian distribution, its moments and independence. So a gaussian distribution is fully specified by its first two moments, the mean and variance (or ...
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33 views

Minimum possible correlation for monotonic data

Suppose we have a dataset $(x_i, y_i)$ where both $x$ and $y$ are monotonically non-decreasing with $i$. So obviously the Spearman rank correlation between $x$ and $y$ is 1. However, what is the ...
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1answer
100 views

Estimate Information Entropy from Moments

I hope I'm using the right terminology below. I have access to the moments statistics of a large sample. That is, I have $\sum(x)$, $\sum(x^2)$, ..., $\sum(x^k)$. I also have access to max and min, ...
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329 views

Relations between moments and cumulants

From the definition of KGF (cumulant generating function) we can write: \begin{align} K_x(t) &= \log_eM_x(t) \\ &= \log_e\left[1+\sum_{r=1}^{\infty}\frac{t^r}{r!}\mu_r^{'}\right] \\ &= ...
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31 views

Moments of the Generalized Dirichlet distribution

I have been trying to solve the following integral. $ \int\theta_j \sum_{k=1}^K \theta_k \beta_{k,w} \prod_{k=1}^K \frac{\Gamma(\alpha_k + \beta_k)}{\Gamma(\alpha_k)\Gamma(\beta_k)} \theta_k^{\...
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1answer
88 views

Why Does Second Order Weak Stationarity include Statement on Covariances in addition to Statement on Mean and Variance?

A stochastic process is second order weakly stationary if all random variables have same mean (first moment), and same variance (second moment?), and covariances that are time-invariant (second moment ...
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49 views

Almost Gaussian distribution with interesting moments

I have a rather funny situation here. There is this random complex number, $z$, with an unknown distribution $\rho_\lambda(z)$ which depends on a parameter, $\lambda$. I can compute the moments and ...
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Can I use moments of a distribution to sample the distribution?

I notice in statistics/machine learning methods, a distribution is often approximated by a Gaussian, and then that Gaussian is used for sampling. They start by computing the first two moments of the ...
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4answers
275 views

In general, how should we find the pmf given only the moment generating function without comparing its form to that of famous pmf?

Background It is known that moment generating function generates moments, but does it hold information about the probability of the random variable being realised at a particular value? Example ...
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58 views

How to optimize ratiometric loss function with variance term in it?

I'm training a neural network (or any ML model with non-convex gradient-based optimization) to predict a continuous outcome variable. Currently, I use the mean squared error loss function, i.e., if $y$...
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1answer
82 views

Tradeoffs of robust mean measures (trimmed, Huber, cosh, etc)

After recently having delved into the world of robust measures (for location, mean being the classical case), I have had difficulty understanding robust measures' core dynamic. Basically, what are ...
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2answers
113 views

Testing whether the conditional correlations/covariances differ between two groups

I have two samples of variables $\{y_{1i},y_{2i},x_i,s_i\}$. Where $y_1$ and $y_2$ are binary variables, $x$ is a continuous variable and $s$ is a sample indicator, taking the value 0 in one sample ...
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1answer
24 views

Are there forms of error which affect kurtosis?

I am currently taking a research methods course online. Today we talked about systematic and random error. The instructor pointed out that systematic error is expected to influence the mean of an ...
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2answers
64 views

$\mathbb{E}[\min (X_{1:n}) + \max(X_{1:n})]/2 = \mathbb{E}[\text{median}(X_{1:n})]$?

Say $X$ is continuous random variable, and we have $n$ iid samples, denoted as $X_{1:n}$. Then can we say the following $$\mathbb{E}[\min (X_{1:n}) + \max(X_{1:n})]/2 = \mathbb{E}[\mathrm{median}(X_{...
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0answers
116 views

Which concentration inequalities apply when moments are infinite?

I have 2 questions: Suppose I have a finite mean but an infinite variance for a discrete distribution w/support $\{1,2,\dots\}$. Is there any probability inequality tighter than Markov in this case? ...
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2answers
34 views

Linear Least square estimate of $x^3$ given $x$ and the moments

I have been struggling to find a direction on how to proceed with the following problem. Given that $x$ is a zero mean (non-Gaussian) random variable with moments E$(x^n)=\mu_n$. I need to find the ...
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Moments of the horseshoe prior?

Are the first two moments well defined for the horseshoe prior? I would say that the expectation is zero but the variance does not exist. Using the following argument. Let $$\beta_i \mid \lambda_i, \...
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1answer
124 views

Fourth order moments of the sum of multivariate normal distribution

Suppose $X\sim \mathcal{N}_d(\boldsymbol{0},\boldsymbol{\Sigma})$ follows a $d$-dimensional multivariate normal distribution. Let $\text{A}_i$ and $\text{A}_j$ be two arbitrary symmetric $d\times d$ ...
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1answer
39 views

Clarification on traditional notation for moments

$\newcommand{\E}{\mathrm{E}}$ I have been going over the Hill et al. (1976) paper on Johnson distributions and struggling with some notation that is undefined but I believe might have been considered ...
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65 views

How well is a power law distribution described by the first four moments?

For a normal distribution, the first two moments (mean and variance) are sufficient statistics for the entire distribution. Suppose I have a power law distribution, and I have data on the first, ...
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2answers
323 views

Is the posterior of a random variable's mean necessarily the mean of that random variable's posterior?

Let's say I have a model that's like, $$ Y \;|\; \theta_1 \sim P(Y \;|\; \theta_1) $$ $$\theta_1 \;|\; \theta_2 \sim P(\theta_1 \;|\; \theta_2) $$ $$ \theta_2 \;|\; \theta_3 \sim P(\theta_2 \;|\; \...
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58 views

Finding the distribution and its parameters with moment generating function

I am learning basic statistics and I am trying to solve the (example) problems but I can't figure out how to solve the following problem. I understand how to use the MGF to find expected value and ...
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1answer
128 views

Joint raw moments of multivariate normal

Let $X_1,X_2$ follow a bivariate standard normal distribution with some non-zero correlation coefficient, $\rho\neq 0$. Let the function $f(z) = z^k,\; k=1,...$. By Stein's lemma, we have that $$\...