ABSTRACT
In Chapter 4, we stressed that it is essential to assess the sampling variability of a point estimate, θ̃, of a characterization θ of a simulated random variable. To assess variability, our approach has been to first calculate the theoretical variance of our estimator, var(θ̃), and then to use our simulated data to estimate this quantity. For a sample average, θ̃ = X̃, we showed in Chapter 4 that var ( θ ˜ ) = σ X 2 / m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203710319/bea86b0b-3659-4f28-8359-acd56fa761ad/content/inq9_251_1.tif"/> , where m is the sample size, and we then estimated var(θ̃) by S X 2 / m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203710319/bea86b0b-3659-4f28-8359-acd56fa761ad/content/inq9_251_2.tif"/> , where S X 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203710319/bea86b0b-3659-4f28-8359-acd56fa761ad/content/inq9_251_3.tif"/> is the sample variance. For more complicated estimators such as sample quantiles or ratio estimators, say of the coefficient of variation, the results in Chapter 6 and particularly Sections 6.1.2 and 6.3.2 allow calculation of var(θ̃). However, the estimates of variability that follow from these calculations rely on unstable estimates of the probability density function (Equation 6.3.14) or on estimates of higher distributional moments. Hence, in this chapter, we present three alternative methods for estimating var(θ̃) that can be used for any estimate θ̃, regardless of its complexity and regardless of whether var(θ̃) can be evaluated exactly or approximately, in closed form. We also show how each of these three methods can be used to produce confidence intervals around the true value θ. Since confidence intervals are sometimes preferred to point estimates, we conclude this chapter with a brief, general discussion of the way simulations can be used to evaluate and compare competing methods for computing confidence intervals.
