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A general method for debiasing a Monte Carlo estimator
Don McLeish
TL;DR
Numerical and Monte Carlo procedures can produce biased estimates of limiting values, despite the appeal of unbiased Monte Carlo estimators with error estimates. The paper randomizes consistent sequences to estimate their limits without bias and applies the approach to numerical integration, root-finding, and finance. It reports successful implementations and emphasizes creating unbiased estimators where their construction is not obvious.
Problem
Numerical methods can yield biased estimates, while the common argument for Monte Carlo methods emphasizes unbiasedness and readily estimated error.
Source: §1 · p. 2Method
Results
The procedure was successfully implemented in root-finding and finance problems, and the paper reports an unbiased estimator with an easily estimated standard error.
Source: §2 · p. 3, §4 · p. 10Takeaways & Limitations
The procedure is presented as a way to create unbiased estimators when construction is not obvious, especially for nonlinear functions of variables that can be estimated unbiasedly.
Source: §3 · p. 8The variance optimization assumes sequence-generation time is O(n), but some applications, including the numerical integral example, require O(2^n) computation.
Source: §2 · p. 5
Abstract
from arXiv · showhide
Consider a process, stochastic or deterministic, obtained by using a numerical integration scheme, or from Monte-Carlo methods involving an approximation to an integral, or a Newton-Raphson iteration to approximate the root of an equation. We will assume that we can sample from the distribution of the process from time 0 to finite time n. We propose a scheme for unbiased estimation of the limiting value of the process, together with estimates of standard error and apply this to examples including numerical integrals, root-finding and option pricing in a Heston Stochastic Volatility model. This results in unbiased estimators in place of biased ones i nmany potential applications.
1 Introduction.
The introduction challenges the view that Monte Carlo methods are preferable because they provide unbiased estimates and readily estimated error. It proposes that consistent estimator sequences can generally be randomized to eliminate bias and estimate error, at the cost of additional randomization and potentially higher MSE.
1 Introduction.
- Monte Carlo methods are often favored over numerical methods because their estimators are usually unbiased and have estimable variance.The cited argument also points to consistency as sample size increases and standard errors available at any sample size.Source: §1 · p. 2
- The note argues that this advantage is not decisive: consistent estimator sequences can generally be made unbiased and given an error estimate.The authors say the aim is to eliminate bias completely, rather than merely reduce it.Source: §1 · p. 2
- The proposed debiasing adds randomization to the sequence and may increase mean squared error.The additional randomization is the stated cost of obtaining unbiasedness.Source: §1 · p. 2
2 The debiased sequence and its variance
The section develops an unbiased estimator for a sequence’s limit using random truncation, then examines its variance and how to choose the truncation distribution. The MSE cost of debiasing and the computation cost of generating the sequence both shape that choice.
2 The debiased sequence and its variance
- Y is unbiased for x∞ with an easily estimated standard error when it is integrable and sum and expectation can be interchanged.The construction uses terms F_n with conditional mean one and a finite cutoff.Source: §2 · pp. 2–3
- Variance analysis assumes E(X_n−x∞)^2 tends to zero as n tends to infinity.Source: §2 · p. 3
- A convergence-rate estimate can guide Q_n: if x∞−X_n ∼ ar^n, the section suggests Q_n ∼ cr^(n/2) or geometric tails.The ideal design would require knowing x∞, so the rate-based choice is an asymptotic alternative.Source: §2 · p. 4
- About 3 times the sample size is needed near p=.4 for the debiased estimator to match the conventional iteration’s MSE; near |r|=1, the relative MSE increase is very small.The comparison is with stopping after µN iterations.Source: §2 · p. 5, Figure 1 · p. 6
- For reasonably fast convergence, minimum variance favors a large guaranteed N, with the remaining budget used to choose q.The optimization assumes O(n) sequence-generation time; for the numerical integral with O(2^n) work, the budget should instead reflect that cost.Source: §2 · pp. 5–6
3 Examples
The examples apply randomized truncation debiasing to root finding and quadrature, then to Heston option pricing. The root-finding results motivate adapting the truncation when iterates have not converged.
Example 1 Unbiased Estimation of a root
- The randomized root estimator had sample mean 1.0023 and sample variance 0.011 over 100,000 simulations, but worked well when starting sufficiently close to the root.The example argues for allowing larger N when the sequence suggests it has not converged.Source: §3 · p. 7
- When the initial value is far from the root, N can be a stopping time; conditional tail probabilities given the iterates suffice for unbiasedness, while the paper focuses on the simplest shift rule.The authors note that more powerful rules for choosing the shift are possible.Source: §3 · p. 7
Example 2 Simpson’s rule
- The reported X was approximately 6.41 × 10−6, corresponding to more than a 2,000-fold efficiency gain over crude Monte Carlo in a comparison using the same expected seven function evaluations.For Simpson’s rule, with grid spacing ∆x = 2−n, a bounded fourth derivative gives error O(16−n).Source: §3 · pp. 7–8
Example 3 Heston Stochastic Volatility model
- For the Heston example, the option payoff is a highly nonlinear function of terminal volatility and integrated variance, so the integral is approximated by trapezoidal rules while volatility is simulated at finite times.For sufficiently smooth drift and diffusion coefficients, the cited Euler approximation has expected-value error of order 2^-n.Source: §3 · p. 9
- With 1,000,000 simulations, the Heston option estimate was 34.9846 (SE 0.0107) versus a reported true price around 34.9998; another setting gave 6.8115 (SE 0.0048998), in agreement with 6.801.The reported parameter settings fail the Feller condition, so volatility hits zero with probability one and may do so frequently for some values.Source: §3 · p. 10
4 Conclusion
The conclusion presents a procedure for debiasing numerical estimates and obtaining statistical error estimates. It reports applications to root finding and finance, and points to potential Monte Carlo extensions of numerical procedures.
4 Conclusion
- The proposed procedure removes bias from estimates produced by numerical quadrature or numerical equation-solving methods and provides statistical error estimates.Source: §4 · p. 10
- The procedure is reported as successfully implemented in simple root-finding problems and more complicated financial problems.Source: §4 · p. 10
- The authors identify potential for Monte Carlo extensions of numerical procedures that provide unbiased estimates and error estimates.Source: §4 · p. 10