Archive for Laplace transform

mean simulations

Posted in Books, Statistics with tags , , , , , , , , on May 10, 2023 by xi'an

A rather intriguing question on X validated, namely a simulation approach to sampling a bivariate distribution fully specified by one conditional p(x|y) and the symmetric conditional expectation IE[Y|X=x]. The book Conditional Specification of Statistical Models, by Arnold, Castillo and Sarabia, as referenced by and in the question, contains (§7.7) illustrations of such cases. As for instance with some power series distribution on ℕ but also for some exponential families (think Laplace transform). An example is when

P(X=x|Y=y) = c(x)y^x/c^*(y)\quad c(x)=\lambda^x/x!

which means X conditional on Y=y is exponential E(λy). The expectation IE[Y|X=x] is then sufficient to identify the joint. As I figured out before checking the book, this result is rather immediate to establish by solving a linear system, but it does not help in finding a way to simulating the joint. (I am afraid it cannot be connected to the method of simulated moments!)


an a-statistical proof of a binomial identity

Posted in Books, Statistics, University life with tags , , , , on May 15, 2014 by xi'an

When waiting for Andrew this morning, I was browsing through the arXived papers of the day and came across this “Simple Statistical Proof of a Binomial Identity” by P. Vellaisamy. The said identity is that, for all s>0,

\sum_{k=0}^n (-1)^k {n \choose k}\dfrac{s}{s+k} = \prod_{k=1}^n \dfrac{k}{k+s}

Nothing wrong with the maths in this paper (except for a minor typo using Exp(1) instead of Exp(s), p.2).  But I am perplexed by the label “statistical” used by the author, as this proof is an entirely analytic argument, based on two different integrations of the same integral. Nothing connected with data or any statistical  technique: this is sheer combinatorics, of the kind one could find in William Feller‘s volume I.

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