Strong convergence in nonparametric estimation of regression functions

The nonparametric regression problem has the objective of estimating conditional expectation. Consider the model $$Y = R(X) + Z$$ , where the random variableZ has mean zero and is independent ofX. The regression functionR(x) is the conditional expectation ofY givenX = x. For an estimator of the form $$R_n (x) = sumlimits_{i = 1}^n {Y_i K{{left[ {{{left( {x - X_i } right)} mathord{left/ {vphantom {{left( {x - X_i } right)} {c_n }}} right. kern-nulldelimiterspace} {c_n }}} right]} mathord{left/ {vphantom {{left[ {{{left( {x - X_i } right)} mathord{left/ {vphantom {{left( {x - X_i } right)} {c_n }}} right. kern-nulldelimiterspace} {c_n }}} right]} {sumlimits_{i = 1}^n {Kleft[ {{{left( {x - X_i } right)} mathord{left/ {vphantom {{left( {x - X_i } right)} {c_n }}} right. kern-nulldelimiterspace} {c_n }}} right]} }}} right. kern-nulldelimiterspace} {sumlimits_{i = 1}^n {Kleft[ {{{left( {x - X_i } right)} mathord{left/ {vphantom {{left( {x - X_i } right)} {c_n }}} right. kern-nulldelimiterspace} {c_n }}} right]} }}} $$ , we obtain the rate of strong uniform convergence $$mathop {sup }limits_{xvarepsilon C} left| {R_n (x) - R(x)} right|mathop {w cdot p cdot 1}limits_ = o({{n^{{1 mathord{left/ {vphantom {1 {(2 + d)}}} right. kern-nulldelimiterspace} {(2 + d)}}} } mathord{left/ {vphantom {{n^{{1 mathord{left/ {vphantom {1 {(2 + d)}}} right. kern-nulldelimiterspace} {(2 + d)}}} } {beta _n log n}}} right. kern-nulldelimiterspace} {beta _n log n}}),beta _n to infty $$ . HereX is ad-dimensional variable andC is a suitable subset ofR ^d .