We introduce a general method to prove uniform in bandwidth consistency of kernel-type function estimators. Examples include the kernel density estimator, the Nadaraya-Watson regression estimator and the conditional empirical process. Our results may be useful to establish uniform consistency of data-driven bandwidth kernel-type function estimators.Comment: Published at http://dx.doi.org/10.1214/009053605000000129 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Let X; X i ; i2N, be independent identically distributed random variables. It is shown that the Student t{statistic based upon the sample fX i g n i=1 is asymptotically N(0;1) if and only if X is in the domain of attraction of the normal law. It is also shown that, for any X, if the self{normalized sums U n := P n i=1 X i ? P n i=1 X 2 i 1=2 ; n2N, are stochastically bounded then they are uniformly subgaussian that is, sup n E exp( U 2 n )<1 for some >0.

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Zirconia stabilized with 8 mole per cent of scandia has a very high oxygen ion conductivity and can “pump” oxygen from oxygen‐bearing gases thus decomposing them. In this study the rate of decomposition of the air‐pollutant species, nitric oxide, to harmless species (via the reaction: 2NO→N2+O2 ) was found to be markedly catalyzed when a potential above 1V was applied across a zirconia disk coated with either a porous platinum or porous gold electrode. It is known that platinum can both form oxides and catalyze the decomposition of NO ; whereas, gold does neither. The catalytic decomposition of NO on platinum metal is inhibited by O2 , such behavior being attributed to preferential chemisorption of O2 over NO as well as the possible formation of an inhibiting platinum oxide surface. The original rationale for this investigation was the possibility that the decomposition of NO might be enhanced if O2 were electrolytically “pumped” away from a platinum electrode deposited on zirconia, keeping the platinum oxygen‐free. However it was not anticipated that at high potentials dissociation rates a thousandfold that on nonporous platinum electrodes occur in the presence of either a platinum or gold porous electrode. No reactivity at all was observed on a nonporous gold electrode. These results suggest that catalysis occurs mainly on a surface other than the platinum or gold, namely, on the zirconia surface itself. It is proposed that F‐centers on the zirconia surface formed by the applied potential are primarily responsible for the observed enhanced catalysis.