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Questions tagged [stable-distribution]

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Easy way to sample a Bayes posterior distribution of stable distributions?

I have a markov chain $P(x_{i+1}|x_i)=\rho(x_{i+1} ; \alpha,\beta,c, x_i)$, where $\rho$ is the stable distribution with mean $x_i$. I'm interested in fixing $x_1$ and $x_3$, and sampling an $x_2$ ...
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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 \...
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Difference between a translationary invariant and a stable distribution

I understand that a stable distribution is a distribution whose linear combination of two independent random variables with this distribution has the same distribution (ignoring location and scale ...
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1answer
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Are the family of stable distributions differentiable everywhere on the real line?

Are stable distributions smooth enough for each index of stability $\alpha$ between 0 and 2, and skewness parameter $\beta$ between 0 and 1? Where there any papers that mention this?
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1answer
32 views

Is there an analytical way to find the area under the probability density function of a stable distribution over a particular interval I=[a,b]

I am trying to find the area over the interval I=[a,b] for a stable distribution. As you know, in general, the densities of stable laws do not have explicit expressions via elementary functions. ...
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1answer
181 views

Distributions that being to domain of attraction of a stable law that are not unimodal?

I was wondering whether there are any distribution that belongs to the domain of attraction of a stable law that is not unimodal. It is known that distribution in that law converge to a stable ...
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41 views

Stable distribution

Let $X_1, X_2,\cdots, X_n $ be a sequence of i.i.d. random variables with common characteristic function $$f(t)=e^{-t^2-\sqrt{|t|}}.$$ $a)$ Find real numbers $\eta_n$ such that $\frac {X_1+ X_2+\...
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0answers
54 views

How can I scale the $k$-th moment of a time series to a different time frequency?

I have a time series, let's say N daily log-returns. I want to study the moments (possibly the distribution) of the weekly returns. I have two ways: 1) Using the time-additivity property of ...
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2answers
557 views

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 - ...
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2answers
194 views

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 ...
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Location parameter estimation in $\alpha-$stable distributions

Let $x$ be a $\alpha-$stable distributed random variable of parameters $\alpha,\beta,c,\mu$. When $\alpha \gt 1$ I can estimate the location parameter $\mu$ of the distribution as $\mu=E[x]$ But how ...
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1answer
144 views

Does stable distribution belong to exponential family?

According to Hougaard(1986), positive stable distribution on R+ belongs to exponential family, how about the case the support of stable distribution being less than zero? The purpose of this question ...
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1answer
93 views

Stable Distribution Log-likelihood and AIC values

I have used the stableFit function from the fBasics package to come up with parameters (alpha, beta, gamma, and delta) for a stable distribution as you can see below: ...
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2answers
306 views

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 ...
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1answer
229 views

Interpreting definition of stable distributions

I am trying to interpret the following definition: ...
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667 views

Estimating Alpha parameter in a stable distribution

I am trying to estimate the alpha parameter of a supposed $\alpha$-stable distributed set of data. I have tried from the Hill estimator to more advanced fitting method, but they are or too ...
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2answers
690 views

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 ...
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1answer
363 views

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 ...
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How does one construct the likelihood function of a distribution in the alpha stable family given non-i.i.d. data?

Taking a simple alpha-stable distribution, the Normal Inverse Gaussian distribution for example, how would one derive the likelihood function provided non-i.i.d. data, e.g. a price series?
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3answers
869 views

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 ...
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3answers
4k views

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 (...
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3answers
3k views

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-...
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1answer
191 views

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 ...