2020/12/15 by Bryan S. Graham, Graham, Bryan S., Fengshi Niu +3 · 1 citation
Computer Science · Mathematics · Medicine · #49K35 #62G08 #91D30 #Bone and Joint Diseases #Econometrics (econ.EM) #FOS: Economics and business #FOS: Mathematics #Machine Learning and Algorithms #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2012.08444
openalex publication_date 2020/12/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let i=1,\…,N index a simple random sample of units drawn from some\nlarge population. For each unit we observe the vector of regressors Xi\nand, for each of the N\(N-1\) ordered pairs of units, an outcome\nYij. The outcomes Yij and Ykl are independent if their indices\nare disjoint, but dependent otherwise (i.e., "dyadically dependent"). Let\nWij=\(Xi',Xj'\)'; using the sampled data we seek to\nconstruct a nonparametric estimate of the mean regression function\ng\(Wij\) oversetdef\≡\𝔼\[\.Yij\|Xi,Xj\].\n We present two sets of results. First, we calculate lower bounds on the\nminimax risk for estimating the regression function at (i) a point and (ii)\nunder the infinity norm. Second, we calculate (i) pointwise and (ii) uniform\nconvergence rates for the dyadic analog of the familiar Nadaraya-Watson (NW)\nkernel regression estimator. We show that the NW kernel regression estimator\nachieves the optimal rates suggested by our risk bounds when an appropriate\nbandwidth sequence is chosen. This optimal rate differs from the one available\nunder iid data: the effective sample size is smaller and\ndW=dim(Wij) influences the rate differently.\n