Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [stable-distribution]

32 questions
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Random variables $X, Z$ such that $Z$ and $\sqrt{X + Z}$ have the same distribution?

I am looking for the distribution of a random variable $Z$ defined as $$Z = \sqrt{X_1+\sqrt{X_2+\sqrt{X_3+\cdots}}} .$$ Here the $X_k$'s are i.i.d. and have same distribution as $X$. 1. Update I am looking to find a simple distribution for $X_k$,…
Vincent Granville
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Is the Student-t distribution a Lévy stable distribution?

Let $X$ have a Student-t distribution, so that \begin{align*} f_X(x|\nu ,\mu ,\beta) = \frac{\Gamma (\frac{\nu+1}{2})}{\Gamma (\frac{\nu}{2}) \sqrt{\pi \nu} \beta} \left(1+\frac{1}{\nu}\left(\frac{x - \mu}{\beta}\right)^2…
Puzzle
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Stable distributions that can be multiplied?

Stable distributions are invariant under convolutions. What sub-families $F$ of the stable distributions are also closed under multiplication? In the sense that if $f\in F$ and $g\in F $, then the product probability density function, $f \cdot g$…
becko
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The positive stable distribution in R

Positive stable distributions are described by four parameters: the skewness parameter $\beta\in[-1,1]$, the scale parameter $\sigma>0$, the location parameter $\mu\in(-\infty,\infty)$, and the so-called index parameter $\alpha\in(0,2]$. When…
ocram
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Estimating the parameters of a sum of a Gaussian and an $\alpha$-stable random variable

Let's assume I have a set of samples of a random variable $$ X = Y + Z \>, $$ where $Y$ is Gaussian (with a mean of zero and variance $\sigma^2$) and $Z$ has a symmetric $\alpha$-stable distribution with index of stability $\alpha$ and scale…
quant_dev
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Generalization of Brownian motion to $\alpha$-stable distributions

Brownian motion is constructed as a limit of the sum i.i.d. Gaussian increments. Can one use a non-Gaussian $\alpha$-stable distribution (e.g. the Cauchy distribution) instead, and still construct a process? Would the scale parameter of such process…
quant_dev
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Fitting the parameters of a stable distribution

I have a data set and I have to fit this data set with a stable distribution. The problem is that the stable distributions are known analytically only in the form of the characteristic function (Fourier transform). How can I do this?
emanuele
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Estimating parameters of sum-stable RV via L-estimators

One of the purported uses of L-estimators is the ability to 'robustly' estimate the parameters of a random variable drawn from a given class. One of the downsides of using Levy $\alpha$-stable distributions is that it is difficult to estimate the…
shabbychef
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Do Lévy α-stable distributions maximize entropy subject to a simple constraint?

Is there a simple constraint on real-valued distributions such that the maximum entropy distribution is Lévy α-stable? Special cases include the Normal and Cauchy distributions for which the answer is affirmative; see…
Fritz
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Generalisation of the notion of correlation for $\alpha$-stable distributions

Pearson correlation is defined via variance and covariance, so will not work when applied to $\alpha$-stable distributions with $\alpha \neq 2$. Is there a way to generalise the notion of correlation to such distributions, e.g. by doing some form of…
quant_dev
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Multivariate stable distribution

I know that if $\pmb{X}_1$ and $\pmb{X}_2$ are independent copies of a $n \times 1$ random vector $\pmb{X}$, then $\pmb{X}$ is said to be sum stable in $\mathbb{R}^n$ if $a\pmb{X}_1 + b\pmb{X}_2 \stackrel{D}{=}c\pmb{X} + \pmb{d}$ for any positive…
Shanks
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CLT and stable distributions

I have a few questions about generalizations of the CLT and stable distributions. I'm trying to correct my understanding and make it precise. Please forgive my naivete, I am not a professional statistician :-) If I take the sum of a large enough…
Frank
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Does stable distribution belong to exponential family?

According to Hougaard (1986), positive stable distribution on $\mathbb{R}^+$ belongs to exponential family, how about the case the support of stable distribution being less than zero? The purpose of this question is to confirm whether MLE is equal…
kurtkim
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What is the limiting posterior in the generalized Bayesian central limit theorem?

The central limit theorem characterizes the limiting distribution of the sum of increasingly many finite-variance independent random variables: the limit is Gaussian. The generalized central limit theorem relaxes the finite variance assumption: the…
Fritz
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Interpreting definition of stable distributions

I am trying to interpret the following definition: A non-degenerate distribution is a stable distribution if it satisfies the following property: Let X1 and X2 be independent copies of a random variable X. Then X is said to be stable if for any…
upabove
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