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彩票热门号码的概率估计与均匀性检验AI 解读

We consider the problem of estimating probabilities for the different numbers in a lottery when the winning numbers are selected sequentially without replacement. Numerical methods are developed to compute maximum likelihood estimates of the probabilities and their variance-covariance matrix based on observed outcomes. The generalized likelihood ratio statistic –2 ln(λ) for testing equiprobable numbers is also given. The methods are applied to a data set from the Lotto…

具有依赖性的伯努利试验中的统计推断AI 解读

A model for Bernoulli trials with Markov dependence is developed which possesses the usual frequency parameter $p = P\lbrack X_i = 1\rbrack$ and an additional dependence parameter $\lambda = P\lbrack X_i = 1 \mid X_{i-1} = 1\rbrack$. Sufficient statistics for the model with $p$ and $\lambda$ unknown are found and an exact closed form expression for their small sample joint distribution is given. Large sample distribution theory is also given and small sample variances…

两样本 Kolmogorov–Smirnov 检验的渐近效率AI 解读

A simple derivation of asymptotic efficiency for the two sample Kolmogorov-Smirnov statistic is given and evaluated for normal location and normal scale alternatives. Using equal samples to simplify the derivation, the limiting efficiency is obtained by letting the type I error α go to zero while the type II error goes to β, 0 < β < 1. For unimodal symmetric location alternatives, the efficiency is the same as that obtained for the Mood and Brown median test. Limits of…

两样本 Kolmogorov–Smirnov 检验的渐近效率AI 解读

A simple derivation of asymptotic efficiency for the two sample Kolmogorov-Smirnov statistic is given and evaluated for normal location and normal scale alternatives. Using equal samples to simplify the derivation, the limiting efficiency is obtained by letting the type I error α go to zero while the type II error goes to β, 0 < β < 1. For unimodal symmetric location alternatives, the efficiency is the same as that obtained for the Mood and Brown median test. Limits of…