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

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Moment generating function of univariate skewed-t distribution

Is there a moment generating or a characteristic function for a univariate skewed-t distribution $y\sim ST\left(\xi,\omega^2,\alpha,\nu\right)$ as defined by Azzalini
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37 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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20 views

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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108 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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21 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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36 views

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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71 views

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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71 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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41 views

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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78 views

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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37 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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68 views

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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72 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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58 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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48 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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205 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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156 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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142 views

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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52 views

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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135 views

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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91 views

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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107 views

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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42 views

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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132 views

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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59 views

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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147 views

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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316 views

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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77 views

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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54 views

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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226 views

Definition of sample excess kurtosis?

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

Transform moments of a random variable to fit the moments of another

Let's say we have the a set $(X_i,Y_i)$, $i\in I$, $I$ is an arbitray finite set of indexes, and the model $$ Y = g(X\beta)$$ Using some method, we obtain the individual first four moments of the ...
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27 views

Finding the moments for the Mauchly sphericity statistic

Suppose we have $V \sim W_p(n-1,\sigma^2 I)$ In order to find the moments of the Mauchly sphericity statistic $\Lambda = \frac{|V|}{(tr(\frac{V}{p}))^p}$ i read one can split it defining $\Lambda_1 = ...
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84 views

Variance of sample Variance

In page 292 of Introduction to Mathematical Statistics by Hogg and Craig it is stated that in order for the variance of the sample variance, i.e. $\text{var}(S^2)$ to exist we need to assume that ...
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Calculate $E(X)$ and $\sigma(X)$ for a simple urn problem

In a box there are 4 red and 7 green balls. A random sample of 5 of the balls is made. Let the number of red balls in the sample be $X$ (a random variable). Calculate $E(X)$ and $\sigma(X)$
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57 views

tensor cross covariance calculation

I have an $N \times P$ matrix $X$ (with $\Sigma$ variance covariance matrix $P \times P$) and I am using R to calculate the tensor product $E\{(X-\mu)(X-\mu)' \otimes (X-\mu)(X-\mu)'\}$ (tensor ...
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108 views

How to prove three properties of the moment generating function? [duplicate]

The moment generating function of a random variable $X$ is defined to be the function $$M_{X}(t)=E(e^{tX})=\sum_{n=0}^{\infty}\frac{E(X^n)}{n!}t^n.$$ Let $I=\{t\in\mathbb R:M_{X}(t)<\infty\}.$ I ...
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In finding the moment generating function why do we multiply by $e^{tx}$ for each pmf term?

The moment generating function that is associated with the discrete random variable $X$ and pmf $f(x)$ is defined as: $$M(t) = E\left[e^{tX}\right] = \sum_{x \in S} e^{tx} f(x).$$ Where does this ...
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Computing non-central moments and normalizer of a quartic exponential distribution

Consider a random variable $X$ which has quartic exponential distribution: $$X \sim P(x)=\frac{1}{Z}e^{ax + bx^2 + cx^3 + dx^4}$$ How can one compute $Z$ or non-central moments $E X^k$ given that they ...
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Interpretation of a PDF squared [duplicate]

I have a problem where the crucial variable is the integral of the squared PDF of a random variable, i.e. $\int f(x)^2dx$ How should I interpret this property of a distribution? If $f(x)$ is ...
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184 views

Laplace distribution and, generally, interpreting an undefined moment

I'm studying the distributional properties of a laplace distribution, and I'm trying to get some intuition beyond plotting the distribution of what it means to have an undefined moment. In wikipedia ...
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153 views

Show that the random variable $V=X-a$ and $U=a-X$ have same distribution?

Given $X$ is a continuous random variable whose density is symmetric about a point $a$. Show that $V=X-a$ and $U=a-X$ have same distribution. $$F_U(u) = P(U \leq u) = P(X-a \leq u) = F_X(a+u)$$ ...
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Moments and exponential distribution

Stuck on how to solve this, can't seem to find the answers online at all a. The random variable $X$ has an exponential distribution with parameter and probability density function $f_X(x) = \theta ...
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59 views

How to show that $\mathrm{mgf}$ $M(s)$ and $\mathrm{pgf}$ $P(s)$ are related?

Let $X$ be an integer-valued $rv$ with $\mathrm{pgf}$ $P(s)$ (probability generating functions) and suppose that $\mathrm{mgf}$ $M(s)$ (moment generating functions) exist for $s∈(-s_0,s_0),s_0>0$. ...