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Abstract
Consider a sequence of observations Yk = Xk + ek, where {Xk : 1 ≦ k < ∞} are i.i.d. random variables having distribution function F and {ek : 1 ≦ k < ∞} are arbitrary random errors of observation. The stochastic geyser problem asks for conditions under which F can be uniquely determined from a knowledge of the sequence of Yk's. The objective of the present article is to show that, if F is continuous and strictly increasing and the sample quantiles of successive blocks {Xn+ 1Xn+ 2, …, Xn+ K} of particular lengths K can be a.s. estimated to within an error of size o(1) as K →∞, then we can almost surely determine F from a single realization of {Yk : 1 ≦ k < ∞}.
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2
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Abstract
Consider a sequence of observations Yk
= Xk
+ ek
, where {Xk
: 1 ≦ k < ∞} are i.i.d. random variables having distribution function F and {ek
: 1 ≦ k < ∞} are arbitrary random errors of observation. The stochastic geyser problem asks for conditions under which F can be uniquely determined from a knowledge of the sequence of Yk
's. The objective of the present article is to show that, if F is continuous and strictly increasing and the sample quantiles of successive blocks {Xn
+ 1
Xn
+ 2, …, Xn
+ K
} of particular lengths K can be a.s. estimated to within an error of size o(1) as K →∞, then we can almost surely determine F from a single realization of {Yk
: 1 ≦ k < ∞}.
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