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| visits | member for | 2 years, 10 months |
| seen | Apr 16 at 21:11 | |
| stats | profile views | 74 |
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Apr 7 |
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Minimum of two exponential variates: What's wrong with this derivation? The distribution of $X$ conditional on the fact that it is less than $Y$ is not the same as the distribution of $X$. |
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Apr 1 |
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What are the distributions on the positive k-dimensional quadrant with parametrizable covariance matrix? @Xi'an You are correct, I did not process them as parameters of the exponential family. Doing so would have made the family no longer a natural family, and including them would have just muddled the algebra for comuting the moment equations (which was muddled enough to begin with). |
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Apr 1 |
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What are the distributions on the positive k-dimensional quadrant with parametrizable covariance matrix? @xian There are the 6 exponents $e_i$ and $f_i$ as well. |
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Mar 31 |
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What are the distributions on the positive k-dimensional quadrant with parametrizable covariance matrix? One way of slighly extending this is to consider the natural exponential family $$f(\mathbf{X}|\mathbf\theta)=h(\mathbf{x})e^{\mathbf\theta^T\mathbf{X}-A(\mathbf\theta)}.$$ Then the mean and covariance are the gradient and Hessian of $A$. If $h$ is a polynomial (with real exponents > -1) then $A$ is the log of a polynomial (with real exponents), and the variance and Hessian are rational functions. I think this gives enough freedom to represent any mean and covariance matrix. |
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Mar 22 |
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Results on Monte Carlo estimates produced by importance sampling @Xi'an You are quite welcome. (In case it is not clear, I meant no offense. Quite the opposite). |
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Mar 22 |
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Results on Monte Carlo estimates produced by importance sampling @BerkUstun The capital G is a typo for a small that I will fix promptly. X/Y is just a generic ratio of random variables. IIRC all this is explained in Liu's Monte Carlo book (something with scientific in the title.) |
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Mar 22 |
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What is the probability of rolling all faces of a die after n number of rolls this is crossposted to math.se. Also the fractions in brackets are flipped. |
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Mar 21 |
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Results on Monte Carlo estimates produced by importance sampling Even if this was not the OP's intention, I would be interested in some pointers on how to figure out when self-normalization will go horribly wrong. |
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Mar 21 |
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Results on Monte Carlo estimates produced by importance sampling I suspect, given that the OP is reporting small variance estimators that are biased, but seem to have small variance, that he may be asking about self normalized importance sampling. See Radford Neal's rant on the Harmonic mean estimator for a good example, that takes what would be an importance sampling estimate with 0 variance, and returns nonsense. I am not certain that this does not ever occur in regular importance sampling, but it is certainly rare. |
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Mar 4 |
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Need help on Gibbs sampling with truncated normal and gamma In your case with mean $\beta$ and variance $1/\lambda$, you truncate at 0 and $\infty$. So after translating to the standard normal your $a$ and $b$ are $-\beta\sqrt{lambda}$ and $\infty$ respectively. You will note that $1-\Phi(-\beta\sqrt{\lambda})$ occurs in the denominator of your function. Just as an aside, it would probably be well worth your while spending a week or so reviewing your probability. If you are playing in this swamp without a good knowledge of the distributions and transformations, the alligators are going to eat you alive. |
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Mar 4 |
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Need help on Gibbs sampling with truncated normal and gamma Yes, sort of. I am working with standard normals, it is just a linear transformation to get to any other normal. |
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Feb 27 |
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How to compute importance sampling? @Xi'an Sorry to bother you again, but your definition of h contain the t distribution, when it should just have the $\sqrt{|x/(1-x)|}$. This causes all the estimates to be off because you are multiplying by f^2 instead of f. The approximations should be around 1.015, while they are around 0.25. |
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Feb 25 |
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How to compute importance sampling? The problems with the variance are akin to the game where you win $2^n/n$ dollars with probability $2^{-n}$. The expected payout may be infinite, but I am not about to sell my house to play. With the folded gamma there will be large excursions, but they will be extremely rare. In practice using $\frac{\sum hf/g}{\sum f/g}$ with the folded gamma performs worse, although in theory it has finite variance. |
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Feb 25 |
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How to compute importance sampling? Here the definition of sam3 takes 5 times as many samples as the other tests. This makes the folded Gamma look better than the Cauchy-power mixture, while the reverse is true. The folded Gamma estimator has infinite variance, while the Cauchy-power mixture does not. |
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Feb 18 |
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Stats book suggestions Perhaps, you could say a bit more about the level you are interested in. Are you comfortable with measure theoretic probability? What do you not like about Wasserman's style? The fact that there are theorems and proofs, or the fact that the proofs are sketchy? What exactly are you looking for? I'm not sure what theorem style has to do with QQ-plots. You might look at Hogg McKean and Craig it does cover the Wilcoxon test. |
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Feb 16 |
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Count data and binomial distribution If you are throwing $n$ balls into $10^6$ bins where each bin is equally likely then the number of balls that end up going into a specific bin has a $\text{Binomial}(n,10^{-6})$ distribution. This can be approximated by a $\text{Poisson}(n/10^6)$ distribution, extremely well approximated if you are throwing millions of balls. |
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Feb 15 |
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How does linear discriminant analysis reduce the dimensions? The $K$ centroids lie in an at most $K-1$ dimensional affine subspace. For example, two points lie on a line, a $2-1$ dimensional subspace. This is just the definition of an affine subspace and some elementary linear algebra. |
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Feb 14 |
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Odds of drawing at least k identical values among m after n draws? I have added code that performs the normal approximation from the paper at the end of my answer (it was too big to fit here). I get $p(3000, 300, 20)\approx 0.6468276$ (an error of about 0.0005). |
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Feb 13 |
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Odds of drawing at least k identical values among m after n draws? @whuber I've been bitten by that error in the past. This is one place where having 1 based arrays is hazardous. I wish there was some way to get it to complain if out of range values were encountered. Functions that silently ignore 'bad' input are dangerous. |
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Feb 13 |
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Odds of drawing at least k identical values among m after n draws? Quoting from the help page for tabulate "There is a bin for each of the values '1, ..., nbins'; values outside that range and 'NA's are (silently) ignored." I believe that your function should read something like t <- table(replicate(nIterations, any(tabulate(floor(runif(n, min=0, max=m)) + 1) >= k))) |
