Grzegorz A. Rempała, Jacek Wesołowski
We consider distributional limit of the Pearson chi-square statistic when the number of classes m increases with the sample size n in such way that $n/\sqrt{m} \to λ$. Under mild moment conditions, the limit is Gaussian for $λ = \infty$, Poisson for finite $λ > 0$, and degenerate for $λ = 0$.
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In this paper, we prove that, in dimension one, the Poincaré inequality is equivalent to a new transport-chi-square inequality linking the square of the quadratic Wasserstein distance with the chi-square pseudo-distance. We also check tensorization of this transport-chi-square inequality.
Chi-square processes with trend appear naturally as limiting processes in various statistical models. In this paper we are concerned with the exact tail asymptotics of the supremum taken over (0; 1) of a class of locally stationary chi-square processes with particular admissible trends. An important tool for establishing our results is a weak version of Slepian's lemma for chi-square processes. Some special cases including squared Brownian bridge and Bessel process are discussed.
This paper studies the supremum of a chi-square process with trend over a threshold-dependent-time horizon. Under the assumption that the chi-square process is generated from a centered self-similar Gaussian process and the trend function is modeled by a polynomial function, we obtain the exact tail asymptotics of the supremum of the chi-square process with trend. These results are of interest in applications in engineering, insurance, queueing and statistics, etc. Some possible extensions of our results are also discussed.
Two--sided bounds are constructed for a probability density function of a weighted sum of chi-square variables. Both cases of central and non-central chi-square variables are considered. The upper and lower bounds have the same dependence on the parameters of the sum and differ only in absolute constants. The estimates obtained will be useful, in particular, when comparing two Gaussian random elements in a Hilbert space and in multidimensional central limit theorems, including the infinite-dimensional case.
This paper concerns the development of Stein's method for chi-square approximation and its application to problems in statistics. New bounds for the derivatives of the solution of the gamma Stein equation are obtained. These bounds involve both the shape parameter and the order of the derivative. Subsequently Stein's method for chi-square approximation is applied to bound the distributional distance between Pearson's statistic and its limiting chi-square distribution, measured using smooth test functions. In combination with the use of symmetry arguments, Stein' method yields explicit bounds o…
On the product of correlated normal random variables and the noncentral chi-square difference distribution
该研究将相关正态随机变量乘积及其独立副本之和表示为非中心卡方差分布，推导了其概率密度、矩、取负概率等精确公式，给出Stein刻画与Lévy-Khintchine表示，证明其自可分解性。
For $X_i(t), i=1,\ldots, n, t\in [0,T]$ centered Gaussian processes, the chi-square process $\sum_{i=1}^{n}X_i^2(t)$ appears naturally as limiting processes in various statistical models. In this paper, we are concerned with the exact tail asymptotics of the supremum taken over discrete grids of a class of locally stationary chi-square processes where $X_i(t),\ 1\leq i\leq n$ are not identical. An important tool for establishing our results is a generalisation of Pickands lemma under the discrete scenario. An application related to the change-point problem is discussed.
In this paper, we analyze a multivariate counterpart of the generalized weighted Kolmogorov-Smirnov statistic, which is the supremum of weighted locally stationary chi-square process over non-compact interval. The boundedness and the exact tail asymptotic behavior of the statistics are derived. We illustrate our findings by two examples where the statistic is defined by Brownian bridge and fractional Brownian motion respectively.
There are many information and divergence measures exist in the literature on information theory and statistics. The most famous among them are Kullback-Leiber relative information and Jeffreys J-divergence. The measures like, Bhattacharya distance, Hellinger discrimination, Chi-square divergence, triangular discrimination and harmonic mean divergence are also famous in the literature on statistics. In this paper we have obtained bounds on triangular discrimination and symmetric chi-square divergence in terms of relative information of type s using Csiszar's f-divergence. A relationship among …
This article presents a new proof of the rate of convergence to the normal distribution of sums of independent, identically distributed random variables in chi-square distance, which was also recently studied in \cite{BobkovRenyi}. Our method consists of taking advantage of the underlying time non-homogeneous Markovian structure and studying the spectral properties of the non-reversible transition operator, which allows to find the optimal rate in the convergence above under matching moments assumptions. Our main assumption is that the random variables involved in the sum are independent and h…
The cumulants and moments of the log of the non-central chi-square distribution are derived. For example, the expected log of a chi-square random variable with v degrees of freedom is log(2) + psi(v/2). Applications to modeling probability distributions are discussed.
We prove distributional limit theorems for random walk adic transformations obtaining ergodic distributional limits of exponential chi squared form.
In this paper we show that the conditional distribution of perturbed chi-quare risks can be approximated by certain distributions including the Gaussian ones. Our results are of interest for conditional extreme value models and multivariate extremes as shown in three applications.
Moderate deviation principle for the chi-square statistics
本文研究样本量增大时有限字母表动态变化下的Pearson卡方统计量，建立了其中偏差原理。
A new large deviation results for the Pearson chi-square and Log-likelihood ratio statistics are obtained. Here attention is focused on the case when the number of groups increases to infinity and the probabilities of groups decreases to zero, as the sample size tends to infinity.
A Bessel-zero obstruction to hyperbolic complete monotonicity of noncentral chi-square densities
通过分析密度函数$p(uv)p(u/v)$的小$u$展开符号，利用完全贝尔多项式和贝塞尔函数零点，证明了非中心卡方密度为双曲完全单调当且仅当非中心参数为零。
Density Function of Weighted Sum of Chi-Square Variables with Trigonometric Weights
研究了权重为角度余弦的卡方变量加权和分布，求得了其密度函数的精确表达式及大自由度下的近似，并与高斯分布进行了比较。
The Geometry of Chi-Square Degrees of Freedom
本文提出并证明了卡方分布自由度的一个简单几何解释，该思想可追溯至Fisher，但确切结果此前未被明确陈述或证明。
We find asymptotic formulas for error probabilities of two-fold Pearson goodness-of-fit test as functions of two critical levels. These results may be reformulated in terms of tails of two-dimensional distributions of the Bessel process. Necessary properties of the Infeld function are obtained.
Friedman's chi-square test is a non-parametric statistical test for $r\geq2$ treatments across $n\ge1$ trials to assess the null hypothesis that there is no treatment effect. We use Stein's method with an exchangeable pair coupling to derive an explicit bound on the distance between the distribution of Friedman's statistic and its limiting chi-square distribution, measured using smooth test functions. Our bound is of the optimal order $n^{-1}$, and also has an optimal dependence on the parameter $r$, in that the bound tends to zero if and only if $r/n\rightarrow0$. From this bound, we deduce a…
Linear combinations of chi square random variables occur in a wide range of fields. Unfortunately, a closed, analytic expression for the pdf is not yet known. As a first result of this work, an explicit analytic expression for the density of the sum of two gamma random variables is derived. Then a computationally efficient algorithm to numerically calculate the linear combination of chi square random variables is developed. An explicit expression for the error bound is obtained. The proposed technique is shown to be computationally efficient, i.e. only polynomial in growth in the number of ter…
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 …
Chi-square simulation of the CIR process and the Heston model
提出基于广义高斯变量幂和的中心卡方密度新表示，结合Marsaglia极坐标法扩展与Beasley-Springer-Moro直接反演法，实现高精度、鲁棒且高效的卡方采样，并应用于Heston模型的非中心卡方方差模拟。
Direct Inversion for Chi-Square Sampling, the Squared Bessel Process, and Applications
提出基于二维切比雪夫展开的直接反演方法模拟平方贝塞尔过程及非中心卡方变量，解决了小自由度计算难题，并高效应用于CIR过程模拟。
It is well-known that each statistic in the family of power divergence statistics, across $n$ trials and $r$ classifications with index parameter $λ\in\mathbb{R}$ (the Pearson, likelihood ratio and Freeman-Tukey statistics correspond to $λ=1,0,-1/2$, respectively) is asymptotically chi-square distributed as the sample size tends to infinity. In this paper, we obtain explicit bounds on this distributional approximation, measured using smooth test functions, that hold for a given finite sample $n$, and all index parameters ($λ>-1$) for which such finite sample bounds are meaningful. We obtain bo…
A branch cut approach to the probability density and distribution functions of a linear combination of central and non-central Chi-square random variables
本文用支割线方法研究中心与非中心卡方随机变量线性组合的分布，聚焦密度函数，通过有限单变量积分提供新表征，数值验证有效。
Non-asymptotic approximations for Pearson's chi-square statistic and its application to confidence intervals for strictly convex functions of the probability weights of discrete distributions
本文提出非渐近局部正态近似，用于Pearson卡方统计量，并推导全变差界及分位数耦合不等式，进而为离散分布的负熵等严格凸函数构建置信区间。
The generalized Marcum functions $Q_μ(x,y)$ and $P_μ(x,y)$ have as particular cases the non-central $χ^2$ and gamma cumulative distributions, which become central distributions (incomplete gamma function ratios) when the non-centrality parameter $x$ is set to zero. We analyze monotonicity and convexity properties for the generalized Marcum functions and for ratios of Marcum functions of consecutive parameters (differing in one unity) and we obtain upper and lower bounds for the Marcum functions. These bounds are proven to be sharper than previous estimations for a wide range of the parameters.…
We study the dimer model on special subgraphs of the square hexagon lattice called "tower graphs" of size $N$. Using integrable probability techniques, we confirm that as $N \rightarrow \infty$, the local statistics are translation invariant Gibbs measures, as conjectured by Kenyon-Okounkov-Sheffield. We also present a 2+1-dimensional discrete time growth process, whose time $N$ distribution is exactly the dimer model on the size $N$ tower, and we compute the current of this growth process and confirm that the model belongs to the Anisotropic KPZ universality class.
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