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

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Estimate betabinomial distribution parameters from weighted observations

Suppose we observe $N$ independent Bernoulli sequences of different lengths. Let $k_i$ be the number of trials in each observation, and $o_i$ be the number of "successes": ...
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Strictly Stationary Time Series with Infinite Moments

Can someone give me an example of a strictly stationary time series with infinite moments? I am reading a book on Time Series by Wayne A. Fuller where it is said that a strictly stationary time series ...
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How to reconstruct moments from aggregated data?

Let $X_{1\ldots n}$ be a stochastic variable that is log-normal distributed, with parameters $\mu$ and $\sigma$. Now suppose all $X_i$ are aggregated into $Y_{1\ldots m}$ where $Y_1$ is the mean of ...
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If the $(n-1)$th moment exists does the $n$th moment necessarily exist?

Let's suppose the distribution is unknown but that the second moment is known to be finite. Doesn't this imply that the distribution should fall off exponentially fast and therefore higher moments ...
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Shifted log-normal distribution and moments

I already know how to get the moments if $X$ is log-normally distributed. But what happens when $X$ is being shifted: $Y=aX+b$, $a>0$ and $b>0$. How to compute the moments of $Y$?
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Estimate variance from multiple distribution

Suppose I have $N$ feature vector and for each feature I have calculated the mean and variance. Now suppose I concatenate all the feature vector (appending one behind the other) and now if I want to ...
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Let $\mathbf{Y}$ be a random vector. Are $k$th moments of $\mathbf{Y}$ considered?

I am self-learning on linear model theory right now, and one thing I find surprising is that although $\mathbb{E}[\mathbf{Y}]$ is defined for a random vector $\mathbf{Y} = \begin{bmatrix} y_1 \\ y_2 ...
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Non-normal distributions with zero skewness and zero excess kurtosis?

Mostly theoretical question. Are there any examples of non-normal distributions that has first four moment equal to those of normal? Could they exist in theory?
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L-Kurtosis calculation

I am doing statistical analysis on some signal data and after some reading was thinking that L-kurtosis would be a good numerical value to use in differentiating delta trains and sine waves with ...
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33 views

3D Zernike moments vs. Spherical Harmonics. Which one has higher discriminative power as shape descriptor?

I am looking for a comprehensive study that has performed comparison of different 3D shape descriptors for classification/clustering problems. Particularly, I am interested in 3D Zernike moments vs. ...
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1answer
60 views

Sample variance from second moment

if I have a sample composed by values lesser than 1 and i want to compute the sample variance with $ \frac{n}{n-1}(\langle x_i^2 \rangle - \langle x_i \rangle^2)$ how can i do? Because the mean of the ...
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Looking for a distribution where: Mean=0, variance is variable, Skew=0 and kurtosis is variable

I am aiming to run simulations in order to estimate the influence of the distribution of $Y$ (independent variable) on a certain binary outcome $X$ (dependent variable). $Y$ must always has a mean of ...
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68 views

Tail of the inverse cdf

I am almost sure I have already seen the following result in statistics but I can't remember where. If $X$ is a positive random variable and $E(X)<\infty$ then $\epsilon F^{-1}(1-\epsilon) \to 0$ ...
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39 views

Why moments of expectation are known as “moments” [duplicate]

I am studying moments of expectation, and seen the formulas for computing the moments. There is one thing I am not clear of, and not getting answer for that. Why moments are named as moments? To my ...
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60 views

How to calculate the absolute central moment of a Binomial distribution?

There is an experiment. The coin is tossed $n$ times with $p = 0.5$. The experiment is repeated $k$ times. I need to calculate the average central moment. For example, let $n = 5$ and $k = 3$. $[0, ...
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32 views

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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52 views

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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23 views

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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203 views

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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82 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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31 views

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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64 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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78 views

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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350 views

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? [duplicate]

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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1answer
145 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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54 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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31 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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488 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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35 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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119 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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79 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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121 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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144 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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85 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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57 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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1answer
87 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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124 views

$E[e^{cX}]$ where $c < 0$ and $X$ is lognormally distributed

I am trying to calculate the expectation $$E[e^{cX}]$$ for arbitrary $c<0$ (for $c>0$ the expectation is infinite) if $X$ is lognormally distributed, i.e. $\log(X) \sim N(\mu, \sigma)$. My idea ...
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273 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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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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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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339 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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289 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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147 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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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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181 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 ...