arXiv subjects

S. Y. Novak

Publications and source records attributed to S. Y. Novak.

3 recordsLinked to original sources

arXiv2020

On the T-test

The $T$-test is probably the most popular statistical test; it is routinely recommended by the textbooks. The applicability of the test relies upon the validity of normal or Student's approximation to the distribution of Student's statistic $\,t_n$. However, the latter assumption is not valid as often as assumed. We show that normal or Student's approximation to $\,Ł(t_n)\,$ does not hold uniformly even in the class $\,{\cal P}_n\,$ of samples from zero-mean unit-variance bounded distributions. We present lower bounds to the corresponding error. The fact that a non-parametric test is not applicable uniformly to samples from the class $\,{\cal P}_n\,$ seems to be established for the first time. It means the $T$-test can be misleading, and should not be recommended in its present form. We suggest a generalisation of the test that allows for variability of possible limiting/approximating distributions to $\,Ł(t_n)$.

math.ST↗

arXiv2019

Poisson approximation

We overview results on the topic of Poisson approximation that are missed in existing surveys. The topic of Poisson approximation to the distribution of a sum of integer-valued random variables is presented as well. We do not restrict ourselves to a particular method, and overview the whole range of issues including the general limit theorem, estimates of the accuracy of approximation, asymptotic expansions, etc. Related results on the accuracy of compound Poisson approximation are presented as well. We indicate a number of open problems and discuss directions of further research.

math.PR↗

arXiv2014

Lower bounds to the accuracy of inference on heavy tails

The paper suggests a simple method of deriving minimax lower bounds to the accuracy of statistical inference on heavy tails. A well-known result by Hall and Welsh (Ann. Statist. 12 (1984) 1079-1084) states that if $\hatα_n$ is an estimator of the tail index $α_P$ and $\{z_n\}$ is a sequence of positive numbers such that $\sup_{P\in{\mathcal{D}}_r}\mathbb{P}(|\hatα_n-α_P|\ge z_n)\to0$, where ${\mathcal{D}}_r$ is a certain class of heavy-tailed distributions, then $z_n\gg n^{-r}$. The paper presents a non-asymptotic lower bound to the probabilities $\mathbb{P}(|\hatα_n-α_P|\ge z_n)$. We also establish non-uniform lower bounds to the accuracy of tail constant and extreme quantiles estimation. The results reveal that normalising sequences of robust estimators should depend in a specific way on the tail index and the tail constant.

math.ST↗