Testing Functional Connection between Two Random Variables
Two discrete random variables, ξ and η are considered. The goal is to decide whether \(\eta \) is a function of ξ. A series of tests are performed, \((\xi _{i},\eta _{i}),1\,\leq \,i\,\leq \,m\) , are independent experiences with the same distribution as
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Dedicated to Professor Yuri Vasilyevich Prokhorov for his 80th birthday
Abstract Two discrete random variables, and are considered. The goal is to decide whether is a function of . A series of tests are performed, .i ; i /; 1 i m, are independent experiences with the same distribution as .; /. The hypothesis is declined if i D j ; i ¤ j holds for some i ¤ j . A condition is given on the character of convergence of the joint distribution of and ensuring the rejection of the hypothesis with a given limiting probability p. Keywords Dependence testing • Sieve method
Mathematics Subject Classification (2010): 60F99
1 Introduction Let and be two, not necessarily independent random variables. The goal of the present paper is to study the situation when one needs to decide if is a (deterministic) function of or not, by using many independent tests. The probability Pof the event that D k and D ` is pk;` , the probability of being k is pk D ` pk` . Suppose that we have m tests. Let i .i / .1 i m/ be totally independent copies of ./. We will study the probability Pr. ! ; m/ of
G.O.H. Katona () R´enyi Institute, Budapest, Hungary e-mail: [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 20, © Springer-Verlag Berlin Heidelberg 2013
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the event that m experiments (mis)indicate that (deterministically) determines , that is, there are no i and j .1 i; j m/ such that i D j ; i 6D j . Of course, if is really a function of then Pr. ! ; m/ D 1 for every m, otherwise it is a decreasing function of m. The most practical case is when the probabilities pk;` are constant. Then the probability Pr. ! ; m/ tends to 0 when m ! 1. One could ask many questions in this case, for instance to study the rate of convergence as a function of the pk;` ’s, but we will be investigating another case, namely the one when the probabilities are very small. In the rest of the paper a series of probability distributions will be considered, that is pk;` .n/; pk .n/ where n tends to infinity. The number of possible values of and are finite, but this is also increasing with n. One can easily see that the smaller probabilities require a larger m to give a counter-example for the functional connection. Therefore m is also supposed to depend on n. For the sake of convenience we will not denote this dependence. Heuristic form of Theorem 1. If the probabilities uniformly decrease and m is increasing faster than 1 qP P 2 2 k pk k;` pk;` then a counter-example shows, with large probability, that is not a function of . On the other hand, if m is increasing slower than the quantity above then the probability of a counter-example is nearly 0. It is more convenient to use a logarithmic form in the precise formulation, this is why we introduce the following quantity: 0 1 X X 2 A H2 . ! / D log2 @ pk2 pk;` : k
(1)
k;`
Since the probabilit
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