Related papers: On combining significances. Some trivial examples
In this paper the generalization of the Poisson distribution is derived for the case when each consecutive event changes event rate. A simple formula for the probability of observing of a given number of events for the selected period of…
When measuring the value of a function to be minimized is not only expensive but also with noise, the popular simultaneous perturbation stochastic approximation (SPSA) algorithm requires only two function values in each iteration. In this…
We consider two related problems arising from a question of R. Graham on quasirandom phenomena in permutation patterns. A ``pattern'' in a permutation $\sigma$ is the order type of the restriction of $\sigma : [n] \to [n]$ to a subset $S…
We propose modified frequentist definition for the determination of confidence intervals for the case of Poisson statistics. Namely, we require that 1-\beta' \geq \sum_{n=o}^{n_{obs}+k} P(n|\lambda) \geq \alpha'. We show that this…
Positive predictive value and negative predictive value are two widely used parameters to assess the clinical usefulness of a medical diagnostic test. When there are two diagnostic tests, it is recommendable to make a comparative assessment…
Several experiments in high-energy physics and astrophysics can be treated as on/off measurements, where an observation potentially containing a new source or effect ("on" measurement) is contrasted with a background-only observation free…
We study sums of independent random variables that take values $0$, $1/2$, or $1$. We show that the probability mass function of the sum splits into two interleaved parts: one supported on the integers and the other supported on the…
This paper describes a recent mathematical method called conflation for consolidating data from independent experiments that are designed to measure the same quantity, such as Planck's constant or the mass of the top quark. Conflation is…
Let $S_n=I_1+\cdots+I_n$ be a sum of independent indicators $I_i$, with $p_i=\Pr(I_i=1)=1-\Pr(I_i=0)$, $i=1,\ldots,n$. It is well-known that the total variation distance between $S_n$ and $Z_\lambda$, where $Z_\lambda$ has a Poisson…
We consider macroscopic, mesoscopic and "S-scopic" quantum superpositions of eigenstates of an observable, and develop some signatures for their existence. We define the extent, or size $S$ of a superposition, with respect to an observable…
In randomized experiments, treatment and control groups should be roughly the same--balanced--in their distributions of pretreatment variables. But how nearly so? Can descriptive comparisons meaningfully be paired with significance tests?…
As it is well known, the standard deviation of a weighted average depends only on the individual standard deviations, but not on the dispersion of the values around the mean. This property leads sometimes to the embarrassing situation in…
We compute the moment of order n of the Poisson stochastic integral of a random process u over a metric space X as a sum that runs over all partitions of {1,...,n} and involves the addition of points to Poisson configurations. This formula…
The Poisson multinomial distribution (PMD) describes the distribution of the sum of $n$ independent but non-identically distributed random vectors, in which each random vector is of length $m$ with 0/1 valued elements and only one of its…
We present two different approaches for parameter learning in several mixture models in one dimension. Our first approach uses complex-analytic methods and applies to Gaussian mixtures with shared variance, binomial mixtures with shared…
We discuss the traditional criterion for discovery in Particle Physics of requiring a significance corresponding to at least 5 sigma; and whether a more nuanced approach might be better.
Composite likelihood has shown promise in settings where the number of parameters $p$ is large due to its ability to break down complex models into simpler components, thus enabling inference even when the full likelihood is not tractable.…
Certain monotonicity properties of the Poisson approximation to the binomial distribution are established. As a natural application of these results, exact (rather than approximate) tests of hypotheses on an unknown value of the parameter…
We give a simple development of the concentration properties of compound Poisson measures on the nonnegative integers. A new modification of the Herbst argument is applied to an appropriate modified logarithmic-Sobolev inequality to derive…
Simpson's paradox and collapsibility are two closely related concepts in the context of data analysis. While the knowledge about the occurrence of Simpson's paradox helps a statistician to draw correct and meaningful conclusions, the…