This paper presents an improved result on the negative-binomial Monte Carlo
technique analyzed in a previous paper for the estimation of an unknown
probability p. Specifically, the confidence level associated to a relative
interval [p/\mu_2, p\mu_1], with \mu_1, \mu_2 > 1, is proved to exceed its
asymptotic value for a broader range of intervals than that given in the
referred paper, and for any value of p. This extends the applicability of the
estimator, relaxing the conditions that guarantee a given confidence level.Comment: 2 figures. Paper accepted in IEEE Transactions on Communication

Hernando, Jose M.

Mendo, Luis

English

arXiv

Luis Mendo

Jose M. Hernando

Crossref

Improved Sequential Stopping Rule for Monte Carlo Simulation

This letter presents an improved result on the negative-binomial Monte Carlo technique analyzed in a previous paper (L. Mendo and J. M. Hernando, ``A simple sequential stopping rule for Monte Carlo simulation,'' IEEE Trans. Commun., vol. 54, no. 2, pp. 231-241, Feb. 2006) for the estimation of an unknown probability p. Specifically, the confidence level associated to a relative interval [p/mu_2,\, pmu_1], with mu_1, mu_2>1, is proved to exceed its asymptotic value for a broader range of intervals than that given in the referred paper, and for any value of p. This extends the applicability of the estimator, relaxing the conditions that guarantee a given confidence level

Mendo Tomás, Luis

Hernando Rábanos, José María

Archivo Digital UPM

A simple sequential stopping rule for Monte Carlo simulation,&quot;

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 11, NOVEMBER 2008 1761 
Transactions Letters 
Improved Sequential Stopping Rule for Monte Carlo Simulation 
Luis Mendo and José M. Hernando, Member, IEEE 
Abstract—This letter presents an improved result on the 
negative-binomial Monte Carlo technique analyzed in a pre-
vious paper1 for the estimation of an unknown probability 
p. Specifically, the confidence level associated to a relative 
interval \p/fi2, Pl^i], with fii, ¡12 > 1, is proved to exceed its 
asymptotic value for a broader range of intervals than that 
given in the referred paper, and for any value of p. This extends 
the applicability of the estimator, relaxing the conditions that 
guarantee a given confidence level. 
Index Terms—Simulation, Monte Carlo methods, sequential 
stopping rule. 
I. INTRODUCTION 
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//x2, Pi»i]; Mi> M2 > L Specifically, regarding the 
latter, it is derived in [1] that the confidence level c = 
Pi[p//J2 < P < PÍ»I] has an asymptotic value c as p —> 0, 
given by2 
c = 7 ( ^ , ( ^ - 1 ) ^ ) - 7 AT, N - Í 
Mi 
(1) 
Furthermore, the confidence level c is assured to exceed c for 
M2 > 
N + VN 
N - 1 Mi > 
N-Í 
N • IN 
(2) 
Paper approved by F. Santucci, the Editor for Wireless System Performance 
of the IEEE Communications Society. Manuscript received February 5, 2007; 
revised May 31, 2007. 
The authors are with the Signals, Systems and Radio Communications 
Department, Polytechnic University, 28040 Madrid, Spain (e-mail: {lmendo, 
hernando}@grc.ssr.upm.es). 
Digital Object Identifier 10.1109/TCOMM.2008.070015 
!L. Mendo and J. M. Hernando, "A simple sequential stopping rule for 
Monte Carlo simulation," IEEE Trans. Commun., vol. 54, no. 2, pp. 231-241, 
Feb. 2006. 
2f(r, x) denotes the incomplete gamma function, defined as j(r, x) = 
l / r ( r ) - ^e-Hr-1dt. 
provided that3 
p < 
N -Í 
1 
(3) [ p v _ 1 | M l 
In this letter, the sufficient conditions that assure a confi-
dence 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 \i\ given by (2) is weakened, while 
maintaining the condition for ¡J,^. Thus \i\ can be further 
decreased while having the same guaranteed confidence 
level given by (1). 
The result is presented in Section II, and conclusions are given 
in Section III. 
II. RESULT 
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 e 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 ] 
(4) P = 
n 
For MI, M2 > 1 given, consider the interval [p/¡J,2,P¡J,I], and 
its associated confidence c = Pr[p//X2 <p< PJJ-I]- As shown 
in [1], c tends to c given by (1) as p —> 0. 
Proposition 1: For any p e (0, 1), with p given by (4), the 
lower bound c> c holds if 
M2 > 
N- N 
N - Í Mi > 
N-Í 
N • N • 
(5) 
Proof: Consider N > 3, /xi, /x2 > 1, and p e (0, 1). Let 
us define 
"AT-1" 
n-i 
«-2 
i?Mi 
(N - 1)M2 
p 
(6) 
(7) 
3
 [-J and ["•] respectively denote rounding to the nearest integer towards 
-oo and towards oo. 
4N denotes the set of natural numbers, {1, 2, ... }. 
Random variables are denoted in boldface throughout the letter. 
0090-6778/08$25.00 © 2008 IEEE 
1762 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 11, NOVEMBER 2008 
The confidence c is given by 1 
c\ = Pr[n < n-i — 1] 
c\ — C2 with6 
P N 
(N-
Til — 1 
I)! ¿^ 
' n=N 
(n-1)^-^(1-p)r -N 
c2 = Pr[n > n2 + 1] 
E (n-l)^-1^! p J (N-iy. •p) 
i-N 
(8) 
(9) 
n = n 2 + l 
Let ci and c2 be respectively defined as limp^o c\ and 
limp^o C2- From [1, appendix C], ci = j(N, (N — l)//xi) 
and C2 = 1 — 7(Ar, (AT — 1)/X2)- We will show that ci > c\ 
and C2 > C2 for ¡J,\, ¡J,^ as in (5). This will establish7 that 
c > c. 
The inequality c-2 > c-2 is equivalent to Pr[n < n^] > 
limp^o Pr[n < n-2], which is established in [1, appendix C] 
for ¡J,2 as in (5) and n-2 given by (7). 
In order to show that ci > ci, we first note that 
ci = 7 N, 
N-l 
Mi 
P N 
(N-iy.Jo 
N-l 
PH-1 
t N-l, -
ptdt 
> 
P N 
n\ — 1 (10) 
t N - l -
ptdt. ( A T - l ) ! ^ _ i 
Lemma 1 given in the Appendix implies that the right-hand 
side of (10) will be greater than or equal to that of (8) if 
ni - 1 < (N - 1/2 - y/N - l/2)/p + 1/2. From (6), m is 
upper bounded by (N — l)/(p/xi) + 1. Therefore, in order to 
assure that c\ > c\ it is sufficient that 
(11) N-l 
W i 
or equivalently 
Mi > 
AT 
< 
AT-
— 
1 
2 
h-^-i 1 
p + 2 ' 
AT-1 
->-Hf 
(12) 
Since p > 0, (12) holds for /xi 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 /xi is lower. 
For the particular case that \i\ = ¡J,^ = 1 + m, where m > 0 
is a relative error margin, it is easily seen that the limiting 
condition in (5) is that for ¡J,^, i.e. 
m > 
N+Í (13) 
AT-1 
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 cm¡n 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 
6 The following notation is used: fcW = k(k— 1) • • • (k — i + 1), fc(0' = 1. 
7
 It should be noted that, although [1, appendix C] considers «2 > a, the 
actual range of values for «2 is «-2 > a — 1. Nonetheless, the proofs in [1, 
appendix C] can be readily generalized to «2 > a, — 1. 
N=100 / : N=30, : N=10 ' 
Fig. 1. Guaranteed confidence c and minimum confidence that can be 
guaranteed c m ¡ n . 
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 [1, fig. 5(a)]). Comparing with 
[1, fig. 4], it is seen that Proposition 1 enlarges the achievable 
region, specially for low m; besides, it removes the restriction 
on p. Fig. 1 thus replaces [1, fig. 4(a)]; and [1, figures 4(b) 
and 5(b)] are no longer necessary. 
As an example, consider the following problem: given an 
NBMC estimator p of an unknown p with N = 30, find the 
minimum m such that Pr[p/(l + m) <p< p(l + m)] > 75%. 
From Proposition 1 and (1), m = 23.7%. The results in [1], on 
the contrary, can only give m = 24.5%, since (23.7%, 75%) is 
not in the achievable region according to [1, fig. 4(a)]; besides, 
p < 0.224 is required. 
III. CONCLUSIONS 
In this letter, 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 G (0, 1) arbitrary, the NBMC estimator 
has a confidence level better than (1) provided that MI> M2 
satisfy (5). This result extends the range of application of the 
NBMC estimation technique. 
APPENDIX 
The following lemma, used in the proof of proposition 1, 
is now established. 
Lemma 1: Given N, n* £ N with N > 3 and 
N-h 
n < 
N-h i (14) 
the following inequality holds: 
n* 
-
l
e-
pndn> YJ(n-l)(-N'1)(l-p)n'N- (15) -N-
N-l 
=N 
MEND0 and HERNANDO: IMPROVED SEQUENTIAL STOPPING RULE FOR MONTE CARLO SIMULATION 1763 
the curve is convex 
<w-1>M-rt"-w (n-1) ( n M ,(1-p) 
N ( N - 1 - V N - 1 ) / p 
Fig. 2. Graphical representation for the proof of Lemma 1. 
Proof: We first note that the sub-integral function is 
increasing for n < (N — l)/p, and is convex for (N — 1 — 
^JN — l)/p < n < (N — l)/p. As figure 2 illustrates, each 
term of the sum in (15) can be identified with the area of a rect-
angle 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 — \JN — l)/p (where the 
sub-integral function is convex), the fiat tops are replaced by 
straight lines joining the centers, without altering the total area. 
The inequality (15) will hold if the area below the curve in the 
interval (N — Í, n*) is larger than the shaded area in the figure. 
We divide this interval in two: (N - 1, (N-Í- VN - l)/p) 
and {{N — 1 — \JN — l)/p, n*), and require that in each of 
these intervals the area of the curve be larger than the part 
of the shaded area corresponding to that interval. In the first 
interval, since the curve is increasing, it suffices that the curve 
be above the square marks for n < [(N — 1 — ^JN — l)/p\, 
as shown in the figure. In the second interval, since the curve 
is increasing and convex, it suffices that the curve be above 
the square mark located at [(N — 1 — \JN — l)/p\ + 1 and 
above the triangle marks. As a result, to establish (15) it is 
sufficient that 
(i) the sub-integral function be above the square marks for 
N <n<(N - Í - VN - l)/p + 1; and 
(ii) the sub-integral function be above the triangle marks for 
(N - 1 - VN - l)/p + 1 <n<n*. 
We analyze these conditions separately, 
(i) With x defined as 
1 , ( n - l ) " - ^ - ^ - 1 ) ? 
- I n -
p (n-W-Vil-p)»-" 
- 1 l ^ 1 
P ¿ = i 
In 1 
n • 1 
n N l n ( l - r i - ( n - l ) , 
(16) 
condition (i) is expressed as x > 0 for n < (N — 1 -
y/N - l)/p + 1. Defining M = N-landv=(n- l)p, 
l^nfl-fr-^Vlfg-MW-P) " 
P i=i P \P P 
(17) 
Using the Taylor expansion ln( l — t) = — J27Li ^/J, \t\ < 1, 
(17) is transformed into x = J27Lo xiP^ w i t n 
N-l 
(j 
' i=l 
M 
3+2 j+l 
(18) 
The term xo is easily seen to be nonnegative for v < M—vM, 
i.e. for n < (N — 1 — \JN — l)/p + 1. We now prove that 
the remaining coefficients Xj, j > 1 are also nonnegative for 
v < M - VM. 
We begin with the case N > 5, j > 2. Using the inequality 
N-l „N-1 
5> 
¿=i 
i w+i > (i - l)j+1dl 
_(N- 2)J+2 _ ( M - 1)J'+2 
3 + 2 ~ 3 + 2 
in (18), we can bound 
( M - 1)J+2 + (j + i y + 2 - (j + 2)Mz/J+ 1 
Xj
 > Ü + 1)Ü + 2 ) ^ + 1 ' 
(19) 
(20) 
The denominator in (20) is positive. Let yj denote the 
numerator. We compute 
^ = (Í + 2)Ü + 1 ) ( ¡ / - M y ' < 0 , (21) 
VÚ\V=M-VM = M 3 
s to consider v = M — ^ /M. Expressing 
p+2 (i
 /LV+1 
V VMJ [o-^X1^) <2> {> + íÁl-Jn)-{->+i 
and denoting the bracketed term by Yj, we now compute the 
following partial derivatives as if j were a continuous variable: 
. J + 2 dYi_ 
d3 
1 
'—"
i +
^y 
•In 1 
d2Y 
= 1 
(23) 
M 
J + 2 
In2 1 
' (24) 
The right-hand side of (24) is positive, and using the inequality 
ln( l + 1 ) >t-12/2 we can bound (23) for j = 2 as 
4 dYi_ 
d3 J = 2 
i - - U | i + 
1 
Mj 
1 
1 1 
2 V M 
(25) 
M 
> 0 . 
Therefore the right-hand side of (23) is positive for j > 2. 
Consequently, in order to establish that yj > 0, it suffices to 
show that I2 > 0. Defining m = 1 / A / M , I2 can be expressed 
as —m2(m3 + 3m 2 + 2m — 2). According to Descartes' sign 
rule, this polynomial has only one positive root. For m —> 00 
the polynomial takes negative values, and for m = 1/2 it is 
1764 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 11, NOVEMBER 2008 
positive. Therefore, it is positive for m < 1/2, i.e. for M > 4, 
or N > 5. 
For N = 4, j > 2, we have 
_ (j + 2)(1 + 2J+1) + (j + l)i/J+2 - 3(j + 2 ) ^ + 1 
Ü + 2)ü + 1)^+1 
Let Zj denote the numerator in (26). Since 
§ ^ = Ü + 2)Ü + l V > - 3 ) < 0 , (27) 
xó is seen to be nonnegative for v' < M ' — \JM' — 1/4, 
and thus for v' < M' - A/M7 , i.e. for n < (N - 1/2 -
^/N — l /2)/p +1 /2 . We now prove that the remaining coef-
ficients x'j, j > 1 are also nonnegative for v' < M ' — A /M' . 
We begin with the case N > 5, j > 2. Using the inequality 
(26) 
N - l 
E 
J + l 
> 
N - l 
1/2 
J + l 
(¿Í 
it suffices to consider v = 3 — A/3. Bounding 
( * • 
3\J+2 
2 / ( M ' - l ) J + 2 
(35) 
Zo as 
^ > Ü + 2)(2^'+1 - 3^'+1) + Ü + 1 V + 2 , (28) 
i IÍ/=3-\/3 *S S e e n t 0 ^ e Positive f° r 3 ^ 2. 
For N = 3, j > 2, 
j + 2 + (j + i y + 2 - 2(j + 2 )^ ' + 1 
J + 2 j + 2 
in (34), we can bound 
(M' - 1)J'+2 + (j + 1)^ '+2 - (j + 2)M>is>i+1 
XA > 
u + w + zyj+l 
(36) 
(29) (¿ + 2)07 + 1)^+1 
and similar arguments to those for AT = 4, j > 2 show that 
Xj > 0. 
For AT > 3, j = 1, using the identity 
The denominator in (36) is positive, and the numerator is as 
that in (20) with M replaced by M' and v replaced by v'. It 
stems that x'j > 0 for M' > 4, i.e. N > 5, and j > 2. 
For N = 4, j > 2, (34) gives 
N - l 
E ( * - D 
¿=1 
( N - 2 ) 3 ( W - 2 ) 2 AT-2 
3 + 2 + 6 
( M - l ) 3 ( M - l ) 2 M - l 
x i 
(30) 
3 2 6 
we obtain from (18) 
_ 4z/3 - QMv2 + 2(M - l ) 3 + 3(M - l ) 2 + M - 1 
Xj
' ~ 12V2 • 
(31) 
From Descartes' sign rule, the numerator of (31) considered 
as a polynomial in v has two positive roots at most. This 
polynomial is positive for v = 0 and for v —> oo, and negative 
for v = M. Thus, it will be positive for v < M — \ /M if it is 
for v = M — A/M. Substituting this v and defining m = A/M, 
the numerator is expressed as m2(3m2 — 4m + 1), which is 
positive for m > 1, or equivalently M > 1, and thus for 
AT > 3 . 
(ii) With x' defined as 
ü + 2)((i)J+1 + (|)J+1 + (f) J + 1)+ü + l ) ^ + 2 
- | Ü + 2)^+1]/[Ü + 2)Ü + l)^+1], 
(37) 
which is shown to be positive with analogous arguments as 
for XJ. 
For N = 3, j > 2, 
^ = [ ü + 2)((é)J+1 + (§)J+1)+Ü + l)^+ 2 
-lü'+2y í+ l]/[ü'+2)ü' + iy j '+1], 
(38) 
and similarly it is shown to be positive. 
For N > 3, j = 1, using the identity 
v (»-*) "-VC»-*)* 
N - l ,
 x 2 
• i . , _ 2 ' = 
E (AT ! ) 3 (AT I)
2 
( M ' - l ) 3 ( M ' - l ) 2 
AT-I 
M ' - l 
(39) 
p (n-l)(-N-1)(l-p)n-N 
N - l / . 1 
6 
1 
n-N 
(32) 
p 
ln(l - p ) 1 n
" 2 
and taking into account (14), in order to fulfil condition (ii) it is 
sufficient that x' > Oforn < (N-í/2-y/N - l/2)/p+l/2. 
Defining M' = N - 1/2 and v' = (n - l/2)p, 
pü v v' ) 
_I(^_M 'V(l-r i - -
P \P j p 
(33) 
Proceeding as with x, we can express x' = ^27^0 x'i^ w^ 
1 ^ 
X
^ (j + i y j + i Z ^ ^
 2 
j + i 
J=0 *"j-f 
M' 
3 + 2 j + 1 (34) 
we obtain an expression for #'• which coincides with (31) 
replacing M by M' and z/ by z/'. Therefore, x'- is positive for 
M' > 1, and thus for N > 3. 
According to the foregoing, conditions (i) and (ii) hold for 
N > 3. This establishes the stated result (15). • 
ACKNOWLEDGMENT 
The authors would like to thank the anonymous reviewers 
and the Editor for Wireless Systems Performance, F. Santucci, 
for their helpful comments. 
REFERENCES 
[1] L. Meiido and J. M. Hernando, "A simple sequential stopping rule for 
Monte Carlo simulation," IEEE Trans. Commun., vol. 54, no. 2, pp. 231— 
241, Feb. 2006. 


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