We introduce nonparametric estimators of the autocovariance of a stationary random field. One of our estimators has the property that it is itself an autocovariance. This feature enables the estimator to be used as the basis of simulation studies such as those which are necessary when constructing bootstrap confidence intervals for unknown parameters. Unlike estimators proposed recently by other authors, our own do not require assumptions such as isotropy or monotonicity. Indeed, like nonparametric function estimators considered more widely in the context of curve estimation, our approach demands only smoothness and tail conditions on the underlying curve or surface (here, the autocovariance), and moment and mixing conditions on the random field. We show that by imposing the condition that the estimator be a covariance function we actually reduce the numerical value of integrated squared error.

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Concise asymptotic theory is developed for non-linear wavelet estimators of regression means, in the context of general error distributions, general designs, general normalizations in the case of stochastic design, and non-structural assumptions about the mean. The influence of the tail weight of the error distribution is addressed in the setting of choosing threshold and truncation parameters. Mainly, the tail weight is described in an extremely simple way, by a moment condition; previous work on this topic has generally imposed the much more stringent assumption that the error distribution be normal. Different approaches to correction for stochastic design are suggested. These include conventional kernel estimation of the design density, in which case the interaction between the smoothing parameters of the non-linear wavelet estimator and the linear kernel method is described.HALL AND PATIL [No.2, = n-Ip JiffV + r"1\;2(1 -2-2r )-1J f-2(g~r) -gfr)iw + op(n-2r/(2r+I»).The same asymptotic argument applies to Jiin, the linear wavelet estimator. In the case where f is a kernel estimator the first step in the argument changes tõwhere and so J(ig)2 w = n-Ip J{t r(hp) + if}fV +p-2r1\;2(1 -2-2r )-1J{gr) -,(hp)gf r)}2 V + op(n-2r/(2r+I»). ACKNOWLEDGEMENTWe are grateful to two reviewers, whose constructive comments encouraged this succinct version of the paper.

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