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Register a new userDensity estimation from an individual numerical sequence
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Added by
Gusztav Morvai
Publication date
2007
fields
Informatics Engineering
and research's language is
English
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This paper considers estimation of a univariate density from an individual numerical sequence. It is assumed that (i) the limiting relative frequencies of the numerical sequence are governed by an unknown density, and (ii) there is a known upper bound for the variation of the density on an increasing sequence of intervals. A simple estimation scheme is proposed, and is shown to be $L_1$ consistent when (i) and (ii) apply. In addition it is shown that there is no consistent estimation scheme for the set of individual sequences satisfying only condition (i).
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We consider univariate regression estimation from an individual (non-random) sequence $(x_1,y_1),(x_2,y_2), ... in real times real$, which is stable in the sense that for each interval $A subseteq real$, (i) the limiting relative frequency of $A$ under $x_1, x_2, ...$ is governed by an unknown probability distribution $mu$, and (ii) the limiting average of those $y_i$ with $x_i in A$ is governed by an unknown regression function $m(cdot)$. A computationally simple scheme for estimating $m(cdot)$ is exhibited, and is shown to be $L_2$ consistent for stable sequences ${(x_i,y_i)}$ such that ${y_i}$ is bounded and there is a known upper bound for the variation of $m(cdot)$ on intervals of the form $(-i,i]$, $i geq 1$. Complementing this positive result, it is shown that there is no consistent estimation scheme for the family of stable sequences whose regression functions have finite variation, even under the restriction that $x_i in [0,1]$ and $y_i$ is binary-valued.
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