Sensitivity Calculation

The observable in this analysis in the largest number of events observed in any bin during the entire year (N_large) rather than a simple number of signal events. This means that determining sensitivity is slightly more complicated because one cannot use standard Feldman-Cousins look-up tables to determine average upper limits. As in other AMANDA analyses, sensitivity was calculated using the standard

where mu₉₀-bar is the average upper limit and n_s is the expected number of signal events. The average upper limit was determined , exactly as the name implies, by the formula

where p is the probability of obtaining a given value of N_largefor the year (determined using Monte Carlo simulations) and mu₉₀ is the upper limit for that value of N_large. These upper limits were determined using a 90% confidence interval and applying the Feldman Cousins likelihood ratio ordering method to Monte Carlo with varying signal strengths. For a visual representation of this, see the plot below, which shows confidence intervals for a range of N_large and signal strengths. The x-axis is the largest number of events obtained in any bin in a year (N_large) and the y-axis is the expected average number of signal events per burst (proportional to flux). The asterisks show the upper and lower bound of a Feldman-Cousins 90% confidence interval for a given signal strength. For example, with no signal, one can expect N_large to be either 3 or 4 events over 90% of the time, while for a signal strength of 0.45 events per burst, one expects between 3 and 6 events (inclusive) over 90% of the time. One can then read the upper limit for a given value of N_large off this plot as the first signal strength which no longer contains this value of N_large in it's 90% confidence interval. The upper limit for N_large = 3 is around .65, the upper limit for N_large = 4 is around 1.22.

Based on Monte Carlo simulations, the probability of obtaining N_large of 3 is 21.3%, the probability of obtaining N_large of 4 is 75.4% and the probability of obtaining N_large of 5 is 3.2%. Thus, the average upper limit (equation 2) is approximately .213*.65 + .754*1.22 + .032*1.73 = 1.11 events per burst. (This is the probability for each N_large multiplied by its corresponding upper limit from the above plot.) This approximate calculation ignores the small contributions from N_largeof 6 and higher. As is typically done for AMANDA analyses, to obtain the sensitivity, one then simply finds the flux such that the expected number of signal events (n_s) matches the average upper limit just obtained (mu₉₀= 1.11) so that the fraction mu₉₀/n_s drops out of equation 1.