Improved Sequential Stopping Rule for Monte Carlo Simulation

Monte Carlo (MC) methods are widely used for estimating an unknown parameter by means of repeated trials or realizations of a random experiment. An important particular case is that in which the parameter to be estimated is the probability p of a certain event H, and realizations are independent. In this setting, the technique of negative-binomial MC (NBMC) [1] can be used. This technique applies a sequential stopping rule, which consists in carrying out as many realizations as necessary to obtain a given number N of occurrences of H. Based on this rule, an estimator is introduced in [1], and it is shown to have a number of interesting properties, in the form of respective bounds for its bias, relative precision, and confidence level for a relative interval [p/µ 2 , pµ 1 ]; µ 1 , µ 2 > 1. Specifically, regarding the latter, it is derived in [1] that the confidence level c = Pr[p/µ 2 ≤ p ≤ pµ 1 ] has an asymptotic value c as p → 0, given by 2 c = γ(N, (N -1)µ 2 ) -γ N, N -1

Furthermore, the confidence level c is assured to exceed c for

In this paper, the sufficient conditions that assure a confidence level c > c for the NBMC estimator are relaxed in two ways:

• The restriction on p given by (3) is eliminated, i.e. p can be an unconstrained value between 0 and 1.

• The condition for µ 1 given by ( 2) is weakened, while maintaining the condition for µ 2 . Thus µ 1 can be further decreased while having the same guaranteed confidence level given by ( 1).

The result is presented in Section 2, and conclusions are given in Section 3.

Consider a random experiment, and an event H associated to that experiment (more generally, there may be a set of events associated to the experiment, one of which is of interest). The probability p of event H is to be estimated from independent realizations of the experiment, using the method described in [1]. Specifically, given4 N ∈ N, with N ≥ 3, realizations are carried out until N occurrences of H are obtained. The number of realizations is thus a negative-binomial random variable5 n, from which p is estimated as [1] p

For µ 1 , µ 2 > 1 given, consider the interval [p/µ 2 , pµ 1 ], and its associated confidence c = Pr[p/µ 2 ≤ p ≤ pµ 1 ]. As shown in [1], c tends to c given by (1) as p → 0. Proposition 1. For any p ∈ (0, 1), with p given by (4), the lower bound c > c holds if

Proof. Consider N ≥ 3, µ 1 , µ 2 > 1, and p ∈ (0, 1). Let us define

The confidence c is given by 1 -c 1 -c 2 with6

Let c1 and c2 be respectively defined as lim p→0 c 1 and lim p→0 c 2 . From [1, appendix C], c1 = γ(N, (N -1)/µ 1 ) and c2 = 1 -γ(N, (N -1)µ 2 ). We will show that c1 > c 1 and c2 > c 2 for µ 1 , µ 2 as in (5). This will establish

for µ 2 as in (5) and n 2 given by (7).

In order to show that c1 > c 1 , we first note that

(10) Lemma 1 given in the Appendix implies that the right-hand side of (10) will be greater than or equal to that of (8

or equivalently

Since p > 0, (12) holds for µ 1 as in (5).

This result removes some of the restrictions that are used in [1] to assure that c > c. Specifically, p can take any value, and the minimum required value for µ 1 is lower.

For the particular case that µ 1 = µ 2 = 1 + m, where m > 0 is a relative error margin, it is easily seen that the limiting condition in (5) is that for µ 2 , i.e.

The dashed curves in Fig. 1 depict the guaranteed confidence c (given by ( 1)) as a function of N and m, for m within the allowed range (13). The solid line represents the minimum confidence cmin that can be guaranteed as a function of m. This corresponds to the lowest N permitted by (13) for a given m; increasing N gives larger guaranteed confidence levels. The achievable region in the (m, c) plane is that above the solid curve, in the following sense: for any (m, c) within this region, the confidence level associated to the error margin m can be assured to be greater than c, irrespective of p; this is accomplished by selecting N according to (1) (or, equivalently, using the curves in

In this paper, the statistical characterization of the NBMC estimator introduced in [1] has been improved by relaxing the conditions that guarantee a certain confidence level. It has been established that, for p ∈ (0, 1) arbitrary, the NBMC estimator has a confidence level better than (1) provided that µ 1 , µ 2 satisfy (5). This result extends the range of application of the NBMC estimation technique.

The following lemma, used in the proof of proposition 1, is now established.

the curve is convex the following inequality holds:

Proof. We first note that the sub-integral function is increasing for n < (N -1)/p, and is convex for (N -1 -√ N -1)/p < n < (N -1)/p. As figure 2 illustrates, each term of the sum in (15) can be identified with the area of a rectangle of unit width. Specifically, the term corresponding to a given n is associated to the rectangle that extends horizontally from n -1 to n in the figure. In addition, for the rectangles situated to the right of (N -1

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