most recent 30 from http://mathoverflow.net 2013-05-20T19:53:21Z http://mathoverflow.net/feeds/user/28187 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112703/deciding-whether-or-not-an-exponentially-distributed-random-variable-exists-in-a unknown (yahoo) 2012-11-17T18:47:37Z 2012-12-19T03:22:00Z

<p>I have some set of known size but with unknown elements, $(x_1, ..., x_N) \in X$, where the elements of $X$ are exponentially distributed random variables with unknown rate parameters, $(\lambda_1, ..., \lambda_N) \in R$. I also have a "black box" function $f$ that samples an element from $X$ with uniform probability, and then returns a randomly sampled value from the chosen element's exponential distribution (corresponding, perhaps, to the time until the first instance of an event governed by the chosen variable). </p> <p>I'm looking to use $f$ to discern whether or not an exponentially distributed random variable, $x_q$, with known rate parameter, $\lambda_q$, exists in the set $X$. I also know that $\lambda_q$ is smaller then all other rate parameters in the set $X$ by at least a multiplicative factor $w$. Said another way, $\lambda_q \leq Min[(R-\lambda_q)]*w$, where $w &lt; 1$. </p> <p>Provided $w$, how many times must I use $f$ to sample from $X$ to decide whether $x_q \in X$ with some threshold confidence? </p> <p>Note - If this problem is too open ended as things stand, please feel free to suggest additional restrictions or clarifications!</p> <p>Note 2 - We can specify that $N \leq 100$, where $N$ is a positive integer, and that $w \leq \frac{1}{2}$, though we cannot say that $w &lt;&lt; 1$.</p>

http://mathoverflow.net/questions/112703/deciding-whether-or-not-an-exponentially-distributed-random-variable-exists-in-a 2012-11-19T04:35:48Z 2012-11-19T04:35:48Z

I was thinking to use some sort of Bayesian inference scheme, but if there is a simpler method...

http://mathoverflow.net/questions/112703/deciding-whether-or-not-an-exponentially-distributed-random-variable-exists-in-a 2012-11-19T04:27:08Z 2012-11-19T04:27:08Z

@fedja Provided a sufficient number of samplings, is there a known &quot;best&quot; method of deciding whether the element $x_q$ exists in $X$?

http://mathoverflow.net/questions/112703/deciding-whether-or-not-an-exponentially-distributed-random-variable-exists-in-a 2012-11-19T04:13:52Z 2012-11-19T04:13:52Z

@fedja I can probably afford something like ~10^5 samplings, though I'd be interested in what theory has to say regardless of feasibility.

http://mathoverflow.net/questions/112703/deciding-whether-or-not-an-exponentially-distributed-random-variable-exists-in-a 2012-11-19T02:00:42Z 2012-11-19T02:00:42Z

@fedja Ah, $R$ is defined earlier as the set of rate parameters associated with the exponentially distributed random variables in $X$.

http://mathoverflow.net/questions/112703/deciding-whether-or-not-an-exponentially-distributed-random-variable-exists-in-a 2012-11-19T01:56:53Z 2012-11-19T01:56:53Z

@fedja I have added some specifications for $N$ and $w$ in Note 2. I can tighten them as needed. $R - \lambda_q$ is meant to be the set $R$ without the element $\lambda_q$ (perhaps this notation is incorrect?)

http://mathoverflow.net/questions/112703/deciding-whether-or-not-an-exponentially-distributed-random-variable-exists-in-a 2012-11-18T07:20:21Z 2012-11-18T07:20:21Z

@Anthony Quas Fair point. I am looking for a bound in terms of $N$, and I have changed the question to specify that we know $N$.