INTRODUCTION
The Monte Carlo method provides approximate solutions to a variety of mathematical problems (Bauer, 1958). As is well known, the MonteCarlo method is composed from three composite parts. Firstly, this is a simulation of random variables with the known distributions, secondly, construction of probability models for real processes and at last, problems of theory of the statistical estimation (Rubenstein, 1981). Certainly, the basic ideas of this method are the law of large numbers and the central limit theorem (Ermakov, 1971; Bevrani, 2003; Gentle, 2004). In both cases the sample size is unknown. Frequently there is a question, whether enough the available statistical data that the inference made on their basis, were exact and reliable, in other words, whether available sampling is representative. Also it is rather general problem. Therefore the purpose of the given article is the estimation of sample’s value for the MonteCarlo method.
THE MONTECARLO METHOD
Let it is required to calculate approximately model I with the help of a MonteCarlo method. Then it is necessary to find an random variable U, such, that its mathematical expectation is equal I: EU = I.
Let’s consider (n+1) independent identically distributed random variables U_{1},
U_{2},...,U_{n} with the finite second moments. Then from the
central limiting theorem it follows that;
where, Φ (x) is a standard normal distribution function.
This relation means, that if we have sufficiently big amount of observations
U_{1}, U_{2},..., U_{n}, the required model can be approximately
calculated as follows:
Thus, with the probability near to 0.998, we mistake on value, not exceeding.
Easy to see, that EI^{*}_{n} = I.
ESTIMATION OF SAMPLE SIZE
Let’s consider the problem on the accuracy of the approximation I^{*}_{n}
≈. I. Unfortunately, unlike the determined (nonrandom) schemes, analysis
of random data requires more then one parameter describing the accuracy, as
event is
random, for any (0,
1), that is, for one sampling this event may happen and for any anothermay
not. Therefore alongside with the parameter ε describing the accuracy,
we’ll set one more parameter γ (0,
1) confidence of a statistical inference. We’ll require, that the probability
of the indicated event was not less then γ, that is,
Thus it is clear, that ε should be close to zero and γ should be
close to unit, characterizing our confidence of the regularity of the inference.
Now we are passing to the estimation of a sample size. We’ll start with traditional
approaches, using the Chebyshev’s inequality and the central limiting theorem.
Then we’ll consider more accurate estimates which take into account an error
of normal approximation. These estimates will be based of the BerryEsseen’s
inequality and it’s more exact analogue for the case of smooth distributions.
Solution based of the Chebyshev’s inequality: On the Chebyshev’s inequality:
Hence, condition (3) is satisfied, if Denote
DU = σ^{2}, then and
a low bound for the number of observations will look like:
Solution based on the central limit theorem: As is well known, the Chebyshev’s
inequality is rather rough, therefore, using the Central Limiting Theorem (CLT)
instead of it permits to hope, that estimates for the necessary sample size
and appropriate accuracy would be more optimistically. CLT implies, that for
the sufficiently big n
Taking into account requirements on the confidence of our inference, we obtain,
that probability (6) should be no more than 1–γ:
when in view of definition αquantiles z_{α} of the standard
normal law we obtain
As it is easy to see, estimates (5) and (7)
differ only in the factors (1–γ)^{1} and .For
example, let us assign γ = 0.95, then the condition (5)
requires that the relation was not
less than 20, while the (7) oneonly 3.85 (z_{0.975}
= 1.96), that is more, than five times better. Such in the image, the CLT allows
to receive more optimistically estimates, however optimism from apparent advantage
of the solution based on the CLT, doesn’t owe us to weaken. The matter is that
the Chebyshev’s inequality gives though rough, but absolutely correct, guaranteed
estimates for the sample’s value and for the accuracy. At the same time, attracting
the CLT, we use approximate equality (6), which brings itself
an error into the inference. In the following section we’ll correct this lack.
Solutions which take into account the accuracy of the normal approximation: The BerryEsseen inequality as an estimate of the rate of convergence in the CLT is well known in the probability theory. This estimate holds for an arbitrary distribution with the finite third moment.
Assume, that the random variable U has the finite third moment and denote β^{3}
= MU–I^{3}. Then, applying the BerryEsseen’s inequality to
the accuracy estimation of relation (6), we obtain:
where,
and C_{0} is an absolute constant with the upper bound C_{0}<0.7655
(Shiganov, 1986; Korolev and Shevtsova, 2006). Thus, more accurate estimate
for the sample’s value is as follows:
Results analysis: Let σ^{2} = 1. Then the required sample’s
value can be easily computed with the help of relations Eq. 5,
7, 9 and 11. The outcomes of these computations
are shown in the Table 1 and 2 (Appendix).
The first Table 1 is constructed for ε = 0.001 and the
second one for ε = 0.01. The upper rows contain values of confidence level
γ, the second and the third onesvalues of the samples sizes, obtained
by using the Chebyshev’s inequality (Eq. 5) and the CLT (Eq.
7), accordingly. A marginal left column contains the values of L_{3}
(from 0.7655 till 2.1655 with the step 0.1). We consider so lower bound for
L_{3}, because as it follows from the Lyapunov’s inequality and
therefore L_{3}>=C_{0}. The sample’s value can be found in the
intersection of the row with appropriate value L_{3} and the column
with required confidence level γ.
APPENDIX
Table 1: 
Estimations for the sample’s value when ε = 0.001 

Table 2: 
Estimations for the sample’s value when ε = 0.01 

ACKNOWLEDGMENT
This research has been supported by the Research Institute for Fundamental Sciences, Tabriz, Iran. The authors would like to thank this support.