In this paper, we develop a non-asymptotic local normal approximation for multinomial probabilities. First, we use it to find non-asymptotic total variation bounds between the measures induced by uniformly jittered multinomials and the multivariate normals with the same means and covariances. From the total variation bounds, we also derive a comparison of the cumulative distribution functions and quantile coupling inequalities between Pearson's chi-square statistic (written as the normalized quadratic form of a multinomial vector) and its multivariate normal analogue. We apply our results to find confidence intervals for the negative entropy of discrete distributions. Our method can be applied more generally to find confidence intervals for strictly convex functions of the weights of discrete distributions.

翻译：本文针对多项分布概率建立了一种非渐近局部正态近似方法。首先，利用该方法得出了均匀抖动多项分布与具有相同均值与协方差的多变量正态分布所诱导测度之间的非渐近全变差界。基于全变差界，进一步推导了皮尔逊卡方统计量（表示为多项向量归一化二次型）与其多变量正态模拟量之间的累积分布函数比较与分位数耦合不等式。我们将所得结果应用于离散分布负熵的置信区间估计。该方法可更广泛地应用于离散分布权重严格凸函数的置信区间构建。