The Stack Overflow podcast is back! Listen to an interview with our new CEO.

Questions tagged [moments]

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

Filter by
Sorted by
Tagged with
1
vote
0answers
42 views

Expected value for f(x) [closed]

I am reading an article and trying to extend their case to a multivariate case. I have the function $f_{i} (x)=\frac{1}{|Σ|}f_{0}((x-μ_{i})'Σ^{-1}(x-μ_{i}))$, where $f_{0}(.)$ is a base density ...
9
votes
3answers
988 views

What‘s wrong with my proof of the Law of Total Variance?

According to the Law of Total Variance, $$\operatorname{Var}(X)=\operatorname{E}(\operatorname{Var}(X\mid Y)) + \operatorname{Var}(\operatorname{E}(X\mid Y))$$ When trying to prove it, I write $$ \...
0
votes
1answer
34 views

Better measure of tail extremity than kurtosis

According to Wikipedia, the only correct interpretation of kurtosis is "tail extremity," the logic being that datapoints within one standard deviation of the mean are raised to the fourth power and ...
2
votes
1answer
50 views

Moments of an AR(1) Process

Definition of an AR(1) process In an Autoregressive Process, a time series can be generated based on a stochastic difference equation. \begin{align} X_t = c + \phi \, X_{t-1} + \epsilon \end{align} ...
1
vote
1answer
66 views

What is the moment generating function of $P(X=x)=\alpha\theta^x$?

Let $X$ be a discrete random variable such that $P(X=x)=\alpha\theta^x$, $x=1,2,\ldots,$ and $0 < \theta < 1$ and $P(X=0)=1-\sum^\infty_{x=1}\alpha\theta^x$, where $\alpha$ is a constant. Find ...
4
votes
5answers
120 views

How do I find all even moments (and odd moments) for $f_X(x)=\frac{1}{2}e^{-|x|}$?

I was asked to find a formula for all even moments of the form $E(X^{2n})$ and all odd moments of the form $E(X^{2n+1})$ using a mgf. Can you help me find the even moments? I will attempt to solve for ...
2
votes
1answer
89 views

Monte Carlo simulation for the log-normal distribution

Consider $X$ that follows a log-normal distribution with parameters $\mu=1$ and $\sigma=1$. The moments are known: $m_n=E[X^n]=e^{n+n^2/2}$. For example, $m_{10}=e^{60}$ (of the order of $10^{26}$). I ...
2
votes
0answers
33 views

Are a distribution's higher-order features harder to estimate?

In what sense, if any, are a distribution's higher-order features (e.g., moments, cumulants) harder to estimate than its lower-order features, for at least some distributional families? For example, ...
0
votes
0answers
18 views

Measures of Uncertainty in Higher Order Moments [duplicate]

Suppose I have a sample of data and compute the mean. I am able to quantity uncertainty in this by computing the variance. I am wondering if known methods exist for evaluating the uncertainty ...
2
votes
1answer
61 views

Claims and questions regarding $n$-sphere distribution?

CONTEXT In my research, I am utilizing an $n$-sphere distributions along with two related distributions. I'd like to make certain I have a firm handle on the way to describe my three distributions. I ...
0
votes
0answers
22 views

Analytical unbiased estimates for bootstraping

I've realized that calculating average of a central statistic over all possible bootstrap samples is equivivalent to using an unbiased version of it (sample central). Meaning that given a sample $\vec{...
0
votes
0answers
28 views

How to calculate High Order Moments?

I have a raw data set for Radio Signals (type: complex) and I want to calculate the High Order Moments and Cumulants (see this paper PART: II-A) In the Paper a ...
8
votes
2answers
313 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 $\...
0
votes
0answers
25 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{\...
0
votes
1answer
34 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 ...
1
vote
1answer
40 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 ...
7
votes
1answer
101 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 ...
0
votes
0answers
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 ...
3
votes
1answer
59 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 ...
0
votes
1answer
18 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 ...
0
votes
0answers
23 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 ...
0
votes
0answers
10 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,...
0
votes
0answers
264 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 $P_{\theta}[X = k] = (1-\theta)^{k}\theta$, $k = 1,2,\ldots$ ...
3
votes
1answer
21 views

“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 $\hat{\theta}_n$...
2
votes
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, $\mathrm{P}(X = \infty)=1.$ ...
0
votes
0answers
15 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 ...
2
votes
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 ...
0
votes
1answer
50 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") ...
0
votes
0answers
16 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 ...
1
vote
1answer
48 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 ...
2
votes
1answer
206 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 ...
0
votes
0answers
41 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 ...
1
vote
0answers
70 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 ...
0
votes
1answer
45 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}^\...
1
vote
0answers
56 views

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, \...
1
vote
1answer
52 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 ...
1
vote
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....
0
votes
0answers
14 views

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 (...
1
vote
0answers
28 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 ...
1
vote
1answer
77 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 ...
0
votes
0answers
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 ...
2
votes
1answer
104 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, ...
1
vote
1answer
399 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] \\ &= ...
0
votes
0answers
33 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^{\...
3
votes
1answer
98 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 ...
1
vote
0answers
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 ...
14
votes
2answers
1k views

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 ...
5
votes
4answers
463 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 ...
1
vote
0answers
92 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 $y$...
2
votes
1answer
107 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 ...