Proof of Proposition 1 in : “ Matching on the Estimated Propensity Score ” ( Abadie and Imbens , 2012 )
نویسندگان
چکیده
Lemma A.1 Consider two independent samples of sizes n0 and n1 from continuous distributions F0 and F1 with common support: X0,1, . . . , X0,n0 ∼ i.i.d. F0 and X1,1, . . . , X1,n1 ∼ i.i.d. F1. Let N = n0 + n1. Assume that the support of F0 and F1 is an interval inside [0, 1]. Let f0 and f1 be the densities of F0 and F1, respectively. Suppose that for any x in the supports of F0 and F1, f1(x)/f0(x) ≤ r̄. For 1 ≤ i ≤ n1 and 1 ≤ m ≤ M ≤ n0, let |Un0,n1,i|(m) be the m-th order statistic of {|X1,i −X0,1|, . . . , |X1,i −X0,n0 |}. Then, for n0 ≥ 3:
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