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

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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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58 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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49 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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66 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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143 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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1answer
43 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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35 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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Definition of sample excess kurtosis?

Wikipedia http://en.wikipedia.org/wiki/Kurtosis#Sample_kurtosis calculates the sample excess kurtosis as ...
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40 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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22 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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1answer
61 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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38 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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1answer
89 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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73 views

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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39 views

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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111 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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141 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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1answer
57 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 $s∈(-s_0,s_0),s_0>0$. ...
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113 views

Translating R code about treatment into Effect size. Expected mean and variance

Using R, I created groups of individuals with trait values. Then I simulated a treatment that modified their trait value (see below). Finally I run a one-way Anova on them using the individuals traits ...
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23 views

How to improve location and scatter estimation conditioning on higher order statistics?

Using sample moments, how can the mean and variance estimators be improved if e.g. skewness and kurtosis are known exactly? And what about using estimates for these instead, which should imho be of no ...
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724 views

Moment generating function of multinomial distribution

How would one find the moment generating function of the multinomial distribution, $\underline{X} \sim \mathrm{multinomial}(n, \underline{p})$? I know that by definition we have $$M_X ...
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189 views

MLE and MoM for parameters of truncated normal

I observe a r.v. $Y$ which is bounded from the right by $a = \frac{c - \mu}{\sigma}$. Hence I observe the moments: $(1) E(Y) = \mu -\sigma \frac{\phi(a)}{\Phi(a)}$ and $(2) Var(Y) = \sigma^2[1 - ...
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88 views

Parameters, Estimates

I lack some knowledge in the concepts of parameters, estimates and moment (math and stats). I can't find an online easy-to-understand source of information about these concepts. Would you help me with ...
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207 views

Difference between cumulants and moments

In particular, is the $n$th cumulant equivalent to the $n$th central moment (i.e. about the mean)? There's little difference I can see between MGFs (moment generating) and CGFs (cumulant generating), ...
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166 views

Confidence error bars and “central point”: Should we emphasize the median?

Say I want to plot summary data with a point and a 95% confidence interval around that point. What should my point really be? Mean, mode, or median? I know that mean = median for any symmetrical ...
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63 views

Distribution with fifth order moment?

I understand that the fifth moment of a distribution gives finer control of the asymmetry of the tails. Please can you give me a reference to a distribution that can handle 5 moments (such as the ...
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216 views

Do skewness and kurtosis uniquely determine type of distribution?

Inspired by this answer, I have following question: Is it enough to know just skewness and kurtosis in order to determine distribution that data comes from? Is there any theorem that implies this? ...
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1answer
61 views

Finding moments for a theoretical density function

I am working on finding higher order moments for a given theoretical function, to be used in modelling of daily log-returns. The PDF is, $f_r(x) =$ $\begin{cases} \quad ...
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51 views

Is it possible to calculate mutual information by moments generating functions?

I went to listen to a workshop and some audience asked the presenter how the moments can improve the mutual information. I am learning the MI(Mutual Information) and moments so don't have enough ...
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126 views

Difference between the two normal distributions

I have two random variables $X$ and $Y$ which follows Normal distribution , whose pdf's are given by $f(x)= \frac{1}{2 \sqrt{2 \pi} \sigma}[e^{\frac{-(x-1)^2}{2 \sigma^2}}+e^{\frac{-(x+1)^2}{2 ...
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59 views

Confusion about using moment condition in a multiple regression model

The very simple case assumes that we have a model like $y = a + bx + e$ where the condition $cov(x,e)=0$ is true. Hence one can use the relationship of the moment conditions to estimate the parameter ...
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778 views

Moment generating function of the inner product of two gaussian random vectors

Can anybody please suggest how I can compute the moment generating function of the inner product of two gaussian random vectors, each distributed as $\mathcal N(0,\sigma^2)$, independent of each ...
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766 views

Combining two covariance matrices

I'm calculating the covariance of a distribution in parallel and I need to combine the distributed results into on singular Gaussian. How do I combine the two? Linearly interpolating between the two ...
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201 views

Moments of the Kolmogorov distribution

Up to what order do the moments of the Kolmogorov distribution exist? References would be appreciated.
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What is coskewness and how can it be calculated?

I would like to calculate coskewness of two random variables. However I couldn't find even basic information on this matter. Is there a standard definition? How to calculate it? If not what are my ...
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1answer
188 views

covariance of RVs under a nonlinear transformation

I have a multivariately distributed random 3-vector ...
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147 views

Estimate the second moment of a latent variable using a conditionally unbiased proxy

The Setup: Let $X_t$ denote an unobservable stochastic sequence where the first two unconditional moments are finite constants; ie $\mathbb{E} X_t = \mu < \infty$ and $\mathbb{E} X_t^2 = \gamma ...
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121 views

Proving that central moment is finite

I'm having trouble showing that the 2nd central moment is finite. I have $X_1,\ldots,X_n \overset{iid}{\sim} f(x)$ with $E[X_1]=\mu$ and $E[X_1^k]$ exists and is finite for any integer $k \geq 1$. I ...
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313 views

Does finite kth moment imply lesser moments are finite? [duplicate]

Possible Duplicate: Proof that if higher moment exists then lower moment also exists For a random variable $X$, lets say I know $E[X^k]$ is finite and I know that $E[X]$ is finite. Can I ...
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495 views

Central Moments of Symmetric Distributions

I am trying to show that the central moment of a symmetric distribution: $${\bf f}_x{\bf (a+x)} = {\bf f}_x{\bf(a-x)}$$ is zero for odd numbers. So for instance the third central moment $${\bf ...
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155 views

Question about a derivative of the 2nd-step moments in a two-step estimator as a joint GMM-estimators approach

I'm reading Newey & McFadden - Large sample estimation and hypothesis testing (in the Handbook of Econometrics, Volume 4, 1994, page 2176). In the model I'm interestend in has some former ...
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97 views

Question about inverse in a two-step estimator as a joint GMM-estimators approach

I'm reading Newey & McFadden - Large sample estimation and hypothesis testing (in the Handbook of Econometrics, Volume 4, 1994, page 2178). My model which I'm interested in has some former ...
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93 views

Distribution function

Find (without using MGF) the mean and variance. $$f(x) = \exp(-kx)x^{(r-1)}k^r/(r-1)!\ \text{ for }\ x>=0$$ $$f(x) = 0\ \text{ for }\ x<0$$ $r$ positive integer, $k>0$
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Proving that MGF determines PDF when the PDF is defined for whole real line

If two PDFs have the same moment generating function that converges in an open set around 0, then the PDFs are same. This is a well known fact, but I can't find its proof. If the PDFs are defined ...