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386

Obenchain

EE = (ER)²/VR (18]

Ali that is needed to confirm that EE is both the mean and the variance of EN is a little bit of very simple algebra:

E(EN) = [ER / VR]-E(R) = (ER)² / VR

and

V(EN) = [ER / VR]² • V(R) = (ER)² / VR

Poisson distributions (for non-negative, integer valued random variables) are inde\ed by a single parameter, commonly called the "intensity" and denoted by the Greek letter X. Here, that single parameter is our EE of eąuation [18]. Notę that EE thus determines the "shape" and the Fisherian "information content" of a Poisson distribution as well as its mean and variance.

Poisson Distribution: Prob(Z=k) = EE^ • exp(-EE) / T (k+1) [19]

[k = 0,1,2,...; EE>0]

Gamma distributions (for non-negative, continuous random variables) are usually indexed by two parameters: a "shape" parameter, denoted here by EE, and a "scalę" parameter, b.

Gamma Distribution: Density (z) = b^EE -z^EE"' exp(-b-z)/r(EE) [20] [0 < z <oo;EE>0, b > 0]

Standardized gamma distributions are those with scalę parameter b=l. EE is both that the mean and variance of a standardized gamma distribution. The exponential distribution is the special case of a standardized gamma distribution with EE=1; chi-squared distributions are un-standardized gamma distributions with b = 1/2 and integer degrees-of-freedom EE/2.

Previously, we divided observed regret by ER to defrne the index, I, of eąuation [8], In Poissonization, we instead multiply observed regret by ER and then divide by VR to define EN of [17]. This rescaling eąuates both the mean and variance of EN to EE of [ 18] and increases the chances that a sample of EN rescaled regrets will be well approximated by either a Poisson or standardized gamma distribution. Of course, we will usually want to confirm the adeąuacy

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