Let Continuous Outcome Variables Remain Continuous

The complementary log-log is an alternative to logistic model. In many areas of research, the outcome data are continuous. We aim to provide a procedure that allows the researcher to estimate the coefficients of the complementary log-log model without dichotomizing and without loss of information. We show that the sample size required for a specific power of the proposed approach is substantially smaller than the dichotomizing method. We find that estimators derived from proposed method are consistently more efficient than dichotomizing method. To illustrate the use of proposed method, we employ the data arising from the NHSI

Learning about a Categorical Latent Variable under Prior Near-Ignorance

It is well known that complete prior ignorance is not compatible with learning, at least in a coherent theory of (epistemic) uncertainty. What is less widely known, is that there is a state similar to full ignorance, that Walley calls near-ignorance, that permits learning to take place. In this paper we provide new and substantial evidence that also near-ignorance cannot be really regarded as a way out of the problem of starting statistical inference in conditions of very weak beliefs. The key to this result is focusing on a setting characterized by a variable of interest that is latent. We argue that such a setting is by far the most common case in practice, and we show, for the case of categorical latent variables (and general manifest variables) that there is a sufficient condition that, if satisfied, prevents learning to take place under prior near-ignorance. This condition is shown to be easily satisfied in the most common statistical problems.Comment: 15 LaTeX page

Limits of Learning about a Categorical Latent Variable under Prior Near-Ignorance

In this paper, we consider the coherent theory of (epistemic) uncertainty of Walley, in which beliefs are represented through sets of probability distributions, and we focus on the problem of modeling prior ignorance about a categorical random variable. In this setting, it is a known result that a state of prior ignorance is not compatible with learning. To overcome this problem, another state of beliefs, called \emph{near-ignorance}, has been proposed. Near-ignorance resembles ignorance very closely, by satisfying some principles that can arguably be regarded as necessary in a state of ignorance, and allows learning to take place. What this paper does, is to provide new and substantial evidence that also near-ignorance cannot be really regarded as a way out of the problem of starting statistical inference in conditions of very weak beliefs. The key to this result is focusing on a setting characterized by a variable of interest that is \emph{latent}. We argue that such a setting is by far the most common case in practice, and we provide, for the case of categorical latent variables (and general \emph{manifest} variables) a condition that, if satisfied, prevents learning to take place under prior near-ignorance. This condition is shown to be easily satisfied even in the most common statistical problems. We regard these results as a strong form of evidence against the possibility to adopt a condition of prior near-ignorance in real statistical problems.Comment: 27 LaTeX page

Purification of quantum trajectories

We prove that the quantum trajectory of repeated perfect measurement on a finite quantum system either asymptotically purifies, or hits upon a family of `dark' subspaces, where the time evolution is unitary.Comment: 10 page