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A last diagnostic possibility is to make use of Schruben s coverage function The algorithm can be described as follows, as hinted in the description of the coverage function Befored starting to run the (real) simulation, consider smaller sample sizes n and k values of , the rarity parameter, { j ; 1 j k} with 1 > > k For each value j , m independent blocks of data X = (Xi ( ))1 i n are then used, giving independent realizations of From those m realizations, the empirical distribution of can be obtained and compared with the uniform distribution Then one can see if there is a trend: if the empirical distribution gets farther from the uniform as j decreases, the current estimator can be considered as non-robust (unreliable), and a better one should be chosen Otherwise, the estimator is not rejected by the test An important remark is that, in order to apply this diagnostic, the exact value (or at least an equivalent as 0) has to be known for computing As a consequence, the diagnostic can only be used for small instances of the problem For example, when estimating the probability in an M/M/1 queue that the occupancy exceeds a value B (with B large), the exact value can be estimated for smaller values of B, and a trend can be derived The same applies when dealing with a Markov chain on a small state space, but looking at long simulation times T (such as in Example 5 above), by looking at smaller values of T The case of large Markov chains where rarity comes from rare transitions is more dif cult But one can try to construct a smaller instance of the model, with similar topology or properties (we do not care about the result being the same) and for which the exact value is known, and look to see whether the coverage function does not deviate as critical transition probabilities decrease Our three examples describe those three situations and are detailed now Figure 47 displays the coverage function for the M/M/1 queue, looking at the probability that B is reached before returning to 0 This is done for sample sizes n = 1000 and repeated k = 500 times in order to get the empricial distribution function (smoothed thanks to interpolation) In the numerical experiments, p = 03 and we chose p = 05 (not the optimal value, but to illustrate the behavior) It can be seen that as B increases, the coverage function gets worse and worse, so the estimator is not good here Look now at the case of the 2 2 matrix of Example 5, with transition matrices 02 08 05 05 P = , P = 02 08 05 05 Again, we take n = 1000 and k = 500 From Figure 48, it can be checked that as the length K of the simulation path increases, the coverage function gets worse and worse, illustrating the bad estimation We close our numerical illustrations with Example 1 Figure 49 displays the empirical coverage function for different values of , still with n = 1000 and k = 500 Again, as decreases, the coverage function gets farther from the uniform, denoting an undesirable behavior.

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