## truncated Normal moments

Posted in Books, Kids, Statistics with tags , , , , , on May 24, 2019 by xi'an An interesting if presumably hopeless question spotted on X validated: a lower-truncated Normal distribution is parameterised by its location, scale, and truncation values, μ, σ, and α. There exist formulas to derive the mean and variance of the resulting distribution,  that is, when α=0, $\Bbb{E}_{\mu,\sigma}[X]= \mu + \frac{\varphi(\mu/\sigma)}{1-\Phi(-\mu/\sigma)}\sigma$

and $\text{var}_{\mu,\sigma}(X)=\sigma^2\left[1-\frac{\mu\varphi(\mu/\sigma)/\sigma}{1-\Phi(-\mu/\sigma)} -\left(\frac{\varphi(\mu/\sigma)}{1-\Phi(-\mu/\sigma)}\right)^2\right]$

but there is no easy way to choose (μ, σ) from these two quantities. Beyond numerical resolution of both equations. One of the issues is that ( μ, σ) is not a location-scale parameter for the truncated Normal distribution when α is fixed.

## econometrics summer masterclass at Warwick, 15 May

Posted in pictures, Statistics, Travel, University life with tags , , , , , on May 9, 2019 by xi'an There is an Econometrics Summer Masterclass taking place in the department of economics next week in Warwick, on May 15, with Don Rubin as one of the speakers and the masterclass teacher.

## QuanTA

Posted in Books, pictures, Running, Statistics, University life with tags , , , , , , , on September 17, 2018 by xi'an My Warwick colleagues Nick Tawn [who also is my most frequent accomplice to running, climbing and currying in Warwick!] and Gareth Robert have just arXived a paper on QuanTA, a new parallel tempering algorithm that Nick designed during his thesis at Warwick, which he defended last semester. Parallel tempering targets in parallel several powered (or power-tempered) versions of the target distribution. With proposed switches between adjacent targets. An improved version transforms the local values before operating the switches. Ideally, the transform should be the composition of the cdf and inverse cdf, but this is impossible. Linearising the transform is feasible, but does not agree with multimodality, which calls for local transforms. Which themselves call for the identification of the different modes. In QuanTA, they are identified by N parallel runs of the standard, or rather N/2 to avoid dependence issues, and K-means estimates. The paper covers the construction of an optimal scaling of temperatures, in that the difference between the temperatures is scaled [with order 1/√d] so that the acceptance rate for swaps is 0.234. Which in turns induces a practical if costly calibration of the temperatures, especially when the size of the jump is depending on the current temperature. However, this cost issue is addressed in the paper, resorting to the acceptance rate as a proxy for effective sample size and the acceptance rate over run time to run the comparison with regular parallel tempering, leading to strong improvements in the mixture examples examined in the paper. The use of machine learning techniques like K-means or more involved solutions is a promising thread in this exciting area of tempering, where intuition about high temperatures can be actually misleading. Because using the wrong scale means missing the area of interest, which is not the mode!