In this paper two problems are considered, both involving the nonparametric estimation of the support of a random vector from a sequence of independent identically distributed observations. In the first problem, after observing n independent random vectors with a common unknown distribution µ, we are given one new measurement and we wish to know whether or not it belongs to the support of µ. In the second problem, after observing the n independent random vectors with a common unknown distribution µ, we then observe n additional independent random vectors with a common unknown distribution v. In this case we wish to know whether or not the support of v is completely contained within the support of µ. Decision schemes are presented and then convergence properties are established .
Asymptotic properties of the rth power distortion measure associated with quantized k-dimensional random variables are considered. Subject only to a moment condition, it is shown that the infimum over all N level quantizers of the quantity N r'k times the rth power average distortion converges to a finite constant as N + cc
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