BRS-inequality
BRS-inequality is the short name for Bruss-Robertson-Steele inequality. This inequality gives a convenient upper bound for the expected maximum number of non-negative random variables one can sum up without exceeding a given upper bound.
For example, suppose 100 random variables are all uniformly distributed on, not necessarily independent, and let, say. Let be the maximum number of one can select in such that their sum does not exceed. is a random variable, so what can one say about bounds for its expectation? How would an upper bound for behave, if one changes the size of the sample and keeps fixed, or alternatively, if one keeps fixed but varies ? What can one say about, if the uniform distribution is replaced by another continuous distribution? In all generality, what can one say if each may have its own continuous distribution function ?
General problem
Let be a sequence of non-negative random variables that are jointly continuously distributed. For and let be the maximum number of observations among that one can sum up without exceeding.Now, to obtain one may think of looking at the list of all observations, first select the smallest one, then add the second smallest, then the third and so on, as long as the accumulated sum does not exceed. Hence can be defined in terms of the increasing order statistics of, denoted by, namely by
What is the best possible general upper bound for if one requires only the continuity of the joint distribution of all variables? And then, how to compute this bound?
Identically distributed random variables.
Theorem 1 Let be identically distributed non-negative random variables with absolutely continuous distribution function. Thenwhere solves the so-called BRS-equation
As an example, here are the answers for the questions posed at the beginning.
Thus let all be uniformly distributed on. Then on, and hence on.
The BRS-equation becomes
The solution is , and thus from the inequality
Since one always has, this bound becomes trivial for.
For the example questions with this yields.
As one sees from, doubling the sample size and keeping fixed, or vice versa, yield for the uniform distribution in the non-trivial case the same upper bound.
Generalised BRS-inequality
Theorem 2. Let be positive random variables that are jointly distributed such that has an absolutely continuous distribution function.If is defined as before, then
where is the unique solution of the BRS-equation
Clearly, if all random variables
have the same marginal distribution, then recaptures, and recaptures.
Again it should be pointed out that
no independence hypothesis whatsoever is needed.
Refinements of the BRS-inequality
Depending on the type of the distributions, refinements of Theorem 2 can be of true interest. We just mention one of them.Let be the random set of those variables one can sum up to yield the maximum random number, that is,
and let denote the sum of these variables.
The so-called
residual
is by definition always non-negative, and one has:
Theorem 3. Let be jointly continuously distributed with marginal distribution functions, and let be the solution of. Then
The improvement in compared with therefore consists of
The expected residual in the numerator is typically difficult to compute or estimate, but there exist nice exceptions. For example, if all are independent exponential random variables, then the memoryless property implies the distributional symmetry of the residual and the overshoot over. For fixed one can then show that :. This improvement fluctuates around , and the convergence to, seems quick.