Improved Asymptotic Formulae for Statistical Interpretation Based on Likelihood Ratio Tests

Abstract

In this work, we improve the asymptotic formulae to describe the probability distribution of a test statistic in G. Cowan \emph{et al.}'s paper~\cite{asimov} from a perspective totally different from last version of this arXiv entry. The starting point of this version seems more natural. The probability distribution function under the hypothesis $H$ is $f(T_\mu | \mu_H) = \sum_{n=0}^{+\infty}f(T_\mu|n,\mu_H)P(n|b+\mu_Hs)$ $= \sum_{n=0}^{n_{\text{small}}}f(T_\mu|n,\mu_H)P(n|b+\mu_Hs) + \sum_{n>n_{\text{small}}}f(T_\mu|n,\mu_H)P(n|b+\mu_Hs)$ $\approx \sum_{n=0}^{n_{\text{small}}}f_{\text{LS}}(T_\mu|n,\mu_H)P(n|b+\mu_Hs) + (1-\sum_{n=0}^{n_{\text{small}}}P(n|b+\mu_Hs))f_{\text{LS}}(T_\mu|n_{\text{small}}, \mu_H)$\. Here $P(n|\nu)$ is Poisson distribution function; $n_{\text{small}}$ is the boarder between large statistics (LS) and small statistics (SS), and has to be chosen appropriately. If the number of events is greater than $n_{\text{small}}$, the probability distribution of $T_\mu$ is described by a single function $f_{\text{LS}}$. $f_{\text{LS}}$ is basically the classic asymptotic formulae with a correction. For each possible number of events not greater than $n_{\text{small}}$, we obtain the probability distribution, $f_{\text{SS}}$, based on a simplifed 6-bin distribution of the observables. $f_{\text{SS}}(T_\mu|n,\mu_H) = \sum_{k_0+k_1+k_2+k_3+k_4+k_5=n}\frac{n!}{k_0!k_1!\cdots k_5!}\Pi_{i=0}^5(\frac{b_i+\mu_Hs_i}{b+ \mu_Hs})^{k_i} \times f_{\text{binned}}(T_\mu|n_i=k_i,i=0,1,\cdots,5;\mu_H)$ In this way, the bump structures due to small sample size can be well predicted. The new asymptotic formulae provide a much better differential description of the test statistics.Comment: 13 pages, 7 figures, a different perspective, able to describe the discrete feature in small-statistics case