Moments are summaries of random variables' characteristics (e.g., location, scale).

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Expectation Propagation - Computing mean and variance of error function

I'm still trying to wrap my head around computing the moments for the expectation propagation algorithm and whether I can use it for the following example: say i have a product of distributions which ...
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Moments and density tails

Assume that the first $n$ moments $m_1,\dots\,m_n$ of a random variable $X\in\mathbb{R}$ are known, but not its probability density function $p(x)$. Does there exist a methodology to characterize ...
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Constructing a continuous distribution to match m moments

Suppose I have a large sample drawn from a continuous distribution, size n, and 2 < m << n moments from that sample. Alternatively, suppose I have been given those moments by an angel, ...
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Determining approximate prediction interval using moments

Given the values of mean, and the next 3 central moments of a continuous random variable X > 0 with unknown pdf, is it possible to derive an approximate interval $(a,b)$ within which X will fall with ...
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Some basic questions related to the moments of a probability distribution

The available text books in most cases avoid greater details regarding some of the topics related to the moments of a probability distribution and I feel some of those topics are not really clear to ...
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53 views

Normal method of moments derivation explanation of Algebra step

In deriving normal estimators using method of moments, why does the below equality hold? $$ \frac{1}{n} \sum X_i^2 - \bar{X}^2 = \frac{1}{n} \sum (X_i - \bar{X})^2 $$ This is from Example 7.2.1 from ...
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Choosing among PDFs

This is a pretty broad question. I just learned that two random variables can have the same moments but different PDFs. Take $\mathbb{E}[X_i] = \mu$ and $\mathbb{Var}[X_i] = \sigma^2$. Since there are ...
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47 views

Transform sample to achieve target mean, skewness, etc

I have a sample of data with N values from which I calculate basic moments such as mean, standard deviation and skewness. I will then change these moments to different values according to my own ...
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Higher Moments from Factor Models

Suppose that we are fitting a linear factor model to our data $$r=\alpha+Bf+\epsilon$$ where we assume the factors, $f$, are orthogonal. Using this structure we can estimate the mean vector as ...
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Mean Preserving PDF Spreading

I have a histogram representing the PDF of an unknown discrete RV. The histogram is asymmetrical. To be clear, all I have is the histogram. Is there a known way to increase/decrease the variance of ...
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Relationship between skew and kurtosis in a sample

It is well known that $\text{excess kurtosis} \geq \text{skew}^2 - 2$, at least in a population. However, what is the relationship between skew and excess kurtosis in a finite sample? Define excess ...
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what are the moments of a multivariate distribution?

For a random variable $X \sim f(x)$, the n'th moment is defined to be $E_f[X^n]$. But for a multivariate distribution, $X$ is a vector so this definition doesn't make sense since you can't raise a ...
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Just what exactly are moments? How are they derived?

We are typically introduced to method of moments estimators by "equating population moments to their sample counterpart" until we have estimated all of the population's parameters; so that, in the ...
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95 views

Moment-generating function or characteristic function of univariate skew-t distribution

Is there a moment-generating function or a characteristic function for a univariate skew-t distribution $y\sim ST\left(\xi,\omega^2,\alpha,\nu\right)$ as defined by Azzalini?
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51 views

Asymptotic distribution for moments of gaussian distribution

Is there a way to find the asymptotic distribution for the moments of Gaussian distribution? More specifically, say you have $X_1, ..., X_n \sim N(\mu, \sigma^2)$. For a moment $m_{n, k} ...
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Higher order robust moments

Is it possible to calculate the 5th, 6th, 7th, 8th and higher-order central robust moments? Is there any other methodology and implementation? How are these comparable to regular sample moments?
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171 views

Why kurtosis of a normal distribution is 3 instead of 0

What is meant by the statement that the kurtosis of a normal distribution is 3. Does it mean that on the horizontal line, the value of 3 corresponds to the peak probability, i.e. 3 is the mode of the ...
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30 views

Fast way to compute central moments of a Poisson random variable?

I am looking for a way to quickly compute the central moments of a Poisson random variable. I've found a couple of resources on how to compute the central moments, but I'm still trying to figure out ...
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intuition for moments about the mean of a distribution?

can someone provide an intuition on why the higher moments of a probability distribution p(x) like the third and fourth moments correspond to skewness and kurtosis, ...
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Point estimation MLE and MME

Consider the family of probability mass functions given by f(x;k) = 3(4^(k-x)) x = k + 1, k + 2,.... and indexed by parameter k E Z. For a random sample of size n, derive with justification: a) ...
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If $X < a$, $EX < a$?

If a r.v. $X < a$, does it imply $EX < a$? If not, why is it different from what I know: If a r.v. $X \leq a$, it implies $EX \leq a$, proved by replacing $X$ with $a$ as the integrand. Note ...
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91 views

Why the first moment is standardized before computing higher moments, but higher moments are not?

Wikipedia says: For the second and higher moments, the central moments (moments about the mean, with c being the mean) are usually used rather than the moments about zero, because they provide ...
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Calculation of Higher-Order Cross-moments

How can I calculate standardized central cross-moments for 2 time-series? The 4th-order standardized central moment, kurtosis, is; ...
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How to compute moments from an MGF explicitly?

Suppose the mgf $M_x(t)$ is defined, for a suitable neighbourhood of zero $(-\delta, \delta)$, as $$M_X(t) = \frac{9 e^{-t}}{(3 + 2t)^2}.$$ Find an expression for $\mathbb{E}_X[X^r]$ for ...
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49 views

Ways to calculate the moments of a function of sample moments?

Are there ways to analytically derive the moments of a function of sample moments? For example, my recent question here hasn't been addressed satisfactorily yet here: link Do the moments of a ...
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Moment generating function - strange fact to check

I'm reading the lecture notes of a course in statistics and I found the next: The moment generating function is defined as $$M_X(t)=E(e^{tX})$$ Check that : $$M^{(n)}_X(t)=E(X^n) $$ Any idea where ...
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179 views

Moments of the two-parameter generalized Pareto distribution (GPD) needed

In this thread the first two moments of the two-parameter GPD are given, where the distribution might be defined as $G(y)= \begin{cases} 1-\left(1+ \frac{\xi y}{\beta} \right)^{-\frac{1}{\xi}} & ...
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Calculating the joint moment generating function

I have data of two random variables X and Y (each of size n x 1) and their joint pdf (probability density function) (of size m x m where m is NOT EQUAL TO n). How can I evaluate their joint moment ...
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63 views

Moment generating function if the PDF is $f_z= Cz^{k-1}(1-\frac{z}{d})^bF(-a+k+1,b;b+1;1-\frac{z}{d})$

Let $z$ a random variable with PDF : $f_z= Cz^{k-1}(1-\frac{z}{d})^bF(-a+k+1,b;b+1;1-\frac{z}{d})$, where $0\leq z \leq d$, $F$ is the Hypergeometric function, $k$ is a positive integer, $-a+k+1 ...
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67 views

Moment generating function of a distribution

I want to find the moment generating function (mfg) and mean deviation of this distribution: $$f(x,\epsilon,k,\theta) = k\theta^{(1+1/k+\epsilon/k)}x^{(k+\epsilon)}\exp{(-\theta x^k ...
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269 views

Robust estimation of kurtosis?

I am using the usual estimator for kurtosis, $\hat{K}=\frac{\hat{\mu}_4}{\hat{\sigma}^4}$, but I notice that even small 'outliers' in my empirical distribution, i.e. small peaks far from the center, ...
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232 views

Moment Generating Functions and Fourier Transforms?

Is a moment-generating function a Fourier transform of a probability density function? In other words, is a moment generating function just the spectral resolution of a probability density ...
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necessary condition for the existence of the posterior mean

I wonder if there exists results relating properties of the prior and the likelihood to the existence of the posterior mean (or more generally to the posterior moments). Any hint or sufficiant ...
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Third order central moment of a positive linear combination of log-normal random variables

I asked this question on MathOverflow a couple of days ago. What is the sign ($\pm$) of the third order central moment of a positive linear combination of log-normal random variables? It seems to be ...
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Estimation of Conditional Moments, and Regression Analysis

Suppose I have observations $(X_i,Y_i)$ for $i=1,..., n$ and I wish estimate a function $g$ such that $Y_i = g(X_i) +V_i$ where $V_i$ is independent of $X_i$ and $E(V_i) = 0$, $E(V_i^2) = \sigma_V^2 ...
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Moment Generating Function for Gamma Distribution

Let $X$ be a Gamma random variable with shape parameter $\alpha=2$ and scale parameter $\theta=1$. Then the moment generating function of $X$ is $$m_X(t) = \frac{1}{(1-t)^2}, t<1.$$ It is clear ...
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computing KL divergence: M projections for arbitrary distributions

Background I have a generative model for a process that can be described as follows: $$ y = t(x, w) + e $$ where $x$ and $y$ observations of a set of random variables which are related by a ...
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Moment of random variable on a integral form

Let $X_1,\dots,X_n$ are i.i.d. samples from uniform distribution on $(0,1)$. Let $\hat F_n$ be their modified empirical distribution function defined by $$ \hat ...
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Fitting moments to a distribution

I have data on the first to fourth moments of a continuous random variable and I am trying to find what distribution best fits the data. Wikipedia has a list of about 20 distributions that could fit ...
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Simplified table for computing higher order cumulants

I'm looking for a simplified table that shows how to compute the cumulant cum(x1, x2, ... xn) for multiple variables, n > 4. I know that the 2nd and 3rd order cumulants are equivalent to the compound ...
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How to improve simulation efficiency of method of simulated moments with random coefficient?

I have a model with two layers. There are 4 variables with random coefficients in the second layer of the model, which is a multivariate Logit. I assume the four random coefficients follow ...
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Example of CLT when moments do not exist

Consider $X_n = \begin{cases} 1 & w.p (1 - 2^{-n})/2\\ -1 &w.p~ (1 - 2^{-n})/2\\ 2^{k} &w.p~ 2^{-k} \text{ for } k > n\\ \end{cases}$ I need to show that even though this has ...
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Is there a generalization of trimean to $n$-th order (central) moments?

I think trimean is the cat's meow. Is there a generalization of this idea to $n$-th order (central) moments? Basically I live in a world where the pain of outliers vastly exceeds the pain of ...
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Joint distribution of two related sample variances

Consider three time series x, v, w The distribution of the values that x,v,w take are zero-mean Gaussian, stationary and independent of time (no temporal colleration) and of each other. We know the ...
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Estimate probability of event using moments of a distribution or a Taylor expansion involving the moments

Let's say we have four moments $(\mu_1, \mu_2, \mu_3, \mu_4)$ of a probabilty distribution of a random variable $X$ and the goal is to get the probability $\rm{P}(X \leq t)$ for a certain value of ...
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Do all these estimates of kurtosis and skewness have the same (asymptotic) distribution under normal sample distribution?

I have seen five types of estimates of kurtosis and skewness: three from http://stats.stackexchange.com/a/84057/1005 one from page 9 of Analysis of Financial Time Series by Ruey S. Tsay one from ...
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Whether distributions with the same moments are identical

Following are similar to but different from previous posts here and here Given two distributions which admit moments of all orders, if all the moments of two distributions are the same, then are ...
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Estimates of variance from an iid sample [duplicate]

There are two kinds of estimates of variance from an iid sample $X1, \dots, X_n$ $1/n * \sum_i (X_i - \bar{X})^2$, which is MLE $1/(n-1) * \sum_i (X_i - \bar{X})^2$, which is unbiased. The ...
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unbiased estimates and MLE of central moments and of standardized moments?

I have heard of unbiased estimate and MLE of variance, and some about those of kurtosis. Are there general results about unbiased estimates of k-th order central moments? MLE of k-th order central ...
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Definition of sample excess kurtosis?

Wikipedia http://en.wikipedia.org/wiki/Kurtosis#Sample_kurtosis calculates the sample excess kurtosis as ...