# Questions tagged [stable-distribution]

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### Inference on latent variable with observation of its convolution with itself

Problem I have an inference problem where the data observed are univariate random numbers whose distribution is obtained as follows. A latent random variable X is first sampled from a parametric ...
1 vote
40 views

### Non-negative fat-tailed "almost stable" family of distribution with finite mean?

I am looking for a finite-dimensional family of distributions $F_X(x)$ with all the following properties: Supported on $[0, +\infty)$, Fat tailed, i.e. $(1-F_X(x)) \sim x^{-\alpha}$ for $x\to +\infty$...
• 237
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### Distribution closed under convolution and truncation followed by convolution

Let $D(\theta)$ denote an absolutely continuous distribution on $\mathbb{R}$. (The finite dimensional vector $\theta$ collects the parameters of the distribution.) Assume that the p.d.f. of $D(\theta)$...
• 535
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### Partial first moments of stable distributions?

Given a stable distribution with parameters $(\alpha, \beta), \alpha>1$ is it true that its all first partial moments (i.e. the integrals of the form $\int_a^b x f(x) dx$, where $a$ and $b$ could ...
1 vote
62 views

### Estimation of ARMA model with alpha-stable power GARCH errors

I am trying to write the following code for estimating parameters for the ARMA(1,1) with alpha-stable power-GARCH(1,1) (Mittnik et al. (2002)). My code has 2 problems: 1) estimated parameters have a ...
• 11
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### Is sample mean an 'efficient' estimator for alpha stable distribution?

The question is just like the title. But...$\alpha$-stable distribution (for $\alpha\in (1,2)$) does not have the second moment, so the sample mean doesn't have variance well defined. Then for such a ...
71 views

### Show That $\sum_{K=1}^{n}\frac{X_k}{n^{\frac{1}{\alpha }}}$ If ${X_n}$ is $X_k$s Same Distribution

Let ${\{X_n}\}$ be a sequence of independent random variables and the stable distribution of order alpha $(0\le\alpha\le2)$. Show that $$\sum_{k=1}^{n}\frac{X_k}{n^{\frac{1}{\alpha }}}$$ if ${X_n}$ is ...
• 11
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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 ...
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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 ...
• 240
1 vote
26 views

### "Stable" distributions with integer support?

My understanding of stable distributions is that in order for a distribution to be stable, linear combinations of independent random variables from a given distribution (for example, a Gaussian) must ...
486 views

### 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 ...
73 views

### why are stable distribution called stable?

I know what a stable distribution is but I don’t know why do they call it Stable I am not aware of anybody who is known by stable. Is it possible because sum of a random variables end up with ...
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1 vote
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### 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 ...
• 357
2k 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 - ...
• 357
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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 ...
• 664
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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 ...
• 2,098
488 views

### 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 ...
• 303
1 vote
177 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: ...
• 31
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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 ...
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### Interpreting definition of stable distributions

I am trying to interpret the following definition: ...
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### 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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### 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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### 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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1 vote
486 views

### 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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### 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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### 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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### Questions about Nolan's stable.exe program

I'm using J. Nolan's free stable.exe program to fit the pareto-levy stable model to some income data. This is option 3 "fit a bivariate sample" in the program. In the instruction text file it is ...
• 515
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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-...
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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 ...