Statistical inference on the Hilbert sphere with application to random densities
Statistical inference on the Hilbert sphere with application to random densities
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DOI：
10.1214/21-ejs1942
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发表时间：
2022-01-01
影响因子：
1.1
通讯作者：
Dai, Xiongtao
中科院分区：
文献类型：
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作者：
Dai, Xiongtao
无限维希尔伯特球s -∞已被广泛应用于密度函数和形状的建模，扩展了有限维的对应体。我们认为Frechet均值是s∞上数据集中趋势的内在总结。对于可靠的统计推断，我们通过建立Frechet均值在s -∞上的存在唯一性，以及样本版本的根n中心极限定理（CLT），得到了估计切向量和协方差算子的固有CLT，克服了s -∞上的无穷维性和紧性不足的障碍。然后提出了基于投影和范数的Frechet均值的渐近和自举假设检验，并证明了它们是一致的。提出的两样本检验应用于推断曼哈顿的每日出租车需求模式，建模为密度，其中的平方根密度在希尔伯特球上分析。在实际数据应用和模拟中研究了利用球面几何的假设检验的数值性质，证明了基于内在几何的检验优于基于外在几何或平面几何的检验。
The infinite-dimensional Hilbert sphere S-infinity has been widely employed to model density functions and shapes, extending the finitedimensional counterpart. We consider the Frechet mean as an intrinsic summary of the central tendency of data lying on S-infinity. For sound statistical inference, we derive properties of the Frechet mean on S-infinity by establishing its existence and uniqueness as well as a root-n central limit theorem (CLT) for the sample version, overcoming obstructions from infinite-dimensionality and lack of compactness on S-infinity Intrinsic CLTs for the estimated tangent vectors and covariance operator are also obtained. Asymptotic and bootstrap hypothesis tests for the Frechet mean based on projection and norm are then proposed and are shown to be consistent. The proposed two-sample tests are applied to make inference for daily taxi demand patterns over Manhattan, modeled as densities, of which the square root densities are analyzed on the Hilbert sphere. Numerical properties of the proposed hypothesis tests which utilize the spherical geometry are studied in the real data application and simulations, where we demonstrate that the tests based on the intrinsic geometry compare favorably to those based on an extrinsic or flat geometry.