8 votes

Sampling from $P(x) \propto \cosh^{m}(a x) e^{-x^{2}/2}$

The distribution that the OP seeks is a straightforward extension of the formula for the case $m=1$ that Xi'an derived in the linked answer. Suppose that $P(x) \propto (\cosh(ax))^m \exp(-x^2/2)$. ...
Dilip Sarwate's user avatar
5 votes

Sampling from $P(x) \propto \cosh^{m}(a x) e^{-x^{2}/2}$

Building on the answer of Dilip, for non-integer $m$, letting $\{m\}=m-\lfloor m\rfloor$ denote the fractional part of $m$, and using the fact that $\cosh x\le \exp|x|$, the (unnormalized) target ...
Jarle Tufto's user avatar
  • 10.7k
4 votes

Density Forecasts with GAMLSS

GAMLSS uses distributional assumptions, and estimates and forecasts conditional distribution parameters. So you would need to use the what parameter to ...
Stephan Kolassa's user avatar
4 votes

Zero variance but non-zero skewness

If the variance is 0 then all the values have to be the same and there can't be any skew.
Peter Flom's user avatar
  • 117k
2 votes

A problem on bivariate random variables

Remember, when $\mathbf X\mapsto \mathbf Y$ via a one-one onto transformation, say $y_i:= g_i(x_1, \ldots, x_n), $ and $h_i:=g_i^{-1}, $ then if $\frac{\partial h_i}{\partial y_j}$ are continuous for ...
User1865345's user avatar
  • 7,887
1 vote

comparing pdf in log scale

The big advantage of using a log-scale here is that it allows you to see the relative comparison properly in the tails where the relevant probability density is extremely low. If you looked at this ...
Ben's user avatar
  • 123k
1 vote

Understanding the multivariate normal density proportional

$\mu^\top \Sigma^{-1} x$ is a scalar (which you can think of as a $1 \times 1$ matrix), so it is equal to its transpose: $$\mu^\top \Sigma^{-1} x = (\mu^\top \Sigma^{-1} x)^\top = x^\top (\Sigma^{-1})^...
angryavian's user avatar
  • 2,188
1 vote

Sampling from $P(x) \propto \cosh^{m}(a x) e^{-x^{2}/2}$

This is just an extended comment. With the restriction of the values of $m$ being positive integers there is an analytic solution of a mixture of normals each with variance 1 as shown by @DilipSarwate ...
JimB's user avatar
  • 3,684

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