论文标题
中央和非中央卡方分布的精制正常近似以及某些应用
Refined normal approximations for the central and noncentral chi-square distributions and some applications
论文作者
论文摘要
在本文中，我们证明了$ r> 0 $ r> 0 $自由度和非中心参数$λ\ geq 0 $的局部限制定理。我们使用它来为生存函数开发精致的正常近似值。我们的最大错误降至$ r^{ - 2} $的订单，该订单明显小于Horgan＆Murphy（2013）和Seri（2015）最近发现的订单$ r^{ - 1/2} $的最大错误界限。我们的结果使我们能够大幅度地减少能源检测问题中获得可忽略的错误所需的观察次数，从Urkowitz（1967）的开创性工作中建议的250美元，到这里仅$ 8 $，在这里使用我们的新近似值。我们还获得了中央和非中心卡方分布与标准正态分布之间的几个概率指标的上限，并且我们获得了中位数的近似值，该近似值改善了Robert（1990）先前获得的下限。
In this paper, we prove a local limit theorem for the chi-square distribution with $r > 0$ degrees of freedom and noncentrality parameter $λ\geq 0$. We use it to develop refined normal approximations for the survival function. Our maximal errors go down to an order of $r^{-2}$, which is significantly smaller than the maximal error bounds of order $r^{-1/2}$ recently found by Horgan & Murphy (2013) and Seri (2015). Our results allow us to drastically reduce the number of observations required to obtain negligible errors in the energy detection problem, from $250$, as recommended in the seminal work of Urkowitz (1967), to only $8$ here with our new approximations. We also obtain an upper bound on several probability metrics between the central and noncentral chi-square distributions and the standard normal distribution, and we obtain an approximation for the median that improves the lower bound previously obtained by Robert (1990).
