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Suppose I have a large sample drawn from a continuous distribution, size $n$, and $2 < m\ll n$ moments from that sample. Alternatively, suppose I have been given those moments by an angel, oracle, theory or fiat. Is there a named distribution, or a natural method of constructing distributions, that lets you exactly match m moments for an arbitrary m?

I know that a finite number of moments will not uniquely determine a distribution. To narrow it down a little bit, let's limit it to distributions that are differentiable (and I presume analytic) and that simplify to the normal distribution for functions with arbitrary first and second moments and skewness, excess kurtosis and higher moments = 0.

I would also strongly prefer a distribution that can be strictly positive for some set of moments and/or coefficients of the distribution function.

Instead of the usual central moments, I would accept a solution posed in terms of raw moments, or normalized moments, or cumulants if that makes the problem easier or produces a more interesting result (with the skewness and kurtosis analogs chosen in an analogous manner to those above).

It would also be a nice property, though I do not know if it is even possible, if the distribution for m equal to some specific value m* could be reduced to the standard normal distribution by some explicitly statable invertible function of $x$ and the moments -- opening the possibility of some kind of higher-order analogue of the t-statistic, or some other useful normalization.

Please note that I am not asking how to fit a predetermined distribution, nor how to select a distribution from some larger family. I am looking for the most simple and natural ways to fit a set of moments or moment-like entities exactly, like fitting an $n$th degree polynomial to $n+1$ points.

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    $\begingroup$ Of interest is Mead and Papanicolaou, Maximum entropy in the problem of moments. J. Math. Phys. 25, 2404 (1984). I don't follow the meaning of anything in your penultimate paragraph, though. $\endgroup$ – whuber Mar 13 '15 at 19:30
  • $\begingroup$ I'm interested in systems of distributions in which n-1 parameter distributions are related to n-parameter distributions by nesting, and in which distributions are built up from zero-parameter distributions like the standard normal or the standard gamma by invertible transformations of the variable and the additional parameter: x+c, x*c, x^c, e^cx, and the like. I find the ability to elaborate or simplify distributions in this way an appealing property. I think of this problem analogously, since a distribution that can take arbitrary values for the m moments will require m coefficients. $\endgroup$ – andrewH Mar 15 '15 at 6:24
  • $\begingroup$ Most of your requirements/preferences could be satisfied by Headrick's (2002) 5th order polynomial family, but it will only work for m < 7. I'm curious to know if you find a satisfactory solution there or elsewhere. Reference: Headrick, T. (2002). Fast fifth-order polynomial transforms for generating univariate and multivariate nonnormal distributions. Computational Statistics & Data Analysis, 40, 685-711. $\endgroup$ – Anthony Mar 26 '15 at 19:09
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I do not know about any exact matching technique. If an approximated technique could work for you (meaning getting a density which approximately matches m given moments), then you could consider using an orthogonal polynomial series approach. You would choose a polynomial basis (Laguerre, Hermite, etc) depending on the range of your data. I describe below a technique that I have used in Arbel et al. (1) for a compactly supported distribution (see details in Section 3).

In order to set the notation, let us consider a generic continuous random variable $X$ on $[0,1]$, and denote by $f$ its density (to be approximated), and its raw moments by $\gamma_r=\mathbb{E}\big[X^r\big]$, with $r\in\mathbb{N}$. Denote the basis of Jacobi polynomials by $$G_i(s) = \sum_{r=0}^i G_{i,r}s^r,\,i\geq 1.$$ Such polynomials are orthogonal with respect to the $L^2$-product $$\langle F,G \rangle=\int_0^1 F(s) G(s) w_{a,b}(s)d s,$$ where $w_{a,b}(s)=s^{a-1}(1-s)^{b-1}$ is named the weight function of the basis and is proportional to a beta density in the case of Jacobi polynomials. Any univariate density $f$ supported on $[0,1]$ can be uniquely decomposed on such a basis and therefore there is a unique sequence of real numbers $(\lambda_i)_{i \geq 0}$ such that $$f(s)=w_{a,b}(s)\sum_{i=0}^\infty \lambda_i G_i(s).$$ From the evaluation of $\int_0^1 f(s)\, G_i(s)\,d s$ it follows that each $\lambda_i$ coincides with a linear combination of the first $i$ moments of $S$, specifically $\lambda_i=\sum_{r=0}^i G_{i,r}\gamma_r$. Then, truncate the representation of $f$ in the Jacobi basis at a given level $N$, providing the approximation $$f_N(s)=w_{a,b}(s)\sum_{i=0}^N \left(\sum_{r=0}^i G_{i,r}\mu_r\right) G_i(s).$$ That polynomial approximation is not necessarily a density as it might fail to be positive or to integrate to 1. In order to overcome this problem, one can consider the density $\pi$ proportional to its positive part, thus defined by $\pi(s)\propto\max(f_N(s),0)$. If sampling from $\pi$ is needed, one can resort to the rejection sampler, see for instance Robert & Casella (2).

There is a companion R package called momentify that provides the approximated distribution given moments, and allows to sample from it, available at this link: https://dl.dropboxusercontent.com/u/1391912/Rpackages/momentify_1.0.tar.gz.

Below are two examples with increasing the number of moments involved. Note that the fit is much better for the unimodal density than the multimodal one.

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References

(1) Julyan Arbel, Antonio Lijoi, and Bernardo Nipoti. Full Bayesian inference with hazard mixture models. Computational Statistics & Data Analysis 93 (2016): 359-372. arXiv link, journal link.

(2) Christian Robert, and George Casella. "Monte Carlo Statistical Methods Springer-Verlag." New York (2004).

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  • $\begingroup$ (+1) Are you considering putting the package on CRAN? $\endgroup$ – Tim Nov 21 '16 at 13:25

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