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The traditional bootstrap selects uniform pseudo random numbers between [0, 1], rounds them into a sequence of random integers in [1, T] to de ne the shuf ed selection from the set {xt} It gives each observation an equal chance (1/T) of being included in the resample An advantage of the ME bootstrap is that it uses the true pseudorandom numbers, and not rounded integers The mean-preserving constraint (on order 1 moments of f(x)) is Sxt = S x(t) = Smt, where mt denote the mean of f(x) within intervals It This property is satis ed by the traditional bootstraps Recall from (933) that the traditional bootstrap selects the observations xt with replacement, such that the probability of selection remains (1/T) Formally, the f(x) of the traditional iid bootstrap at each xt is a so-called delta function with the entire mass (1/T) at xt in It.

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similar inaccuracies in its nonlinear solver functions, statistical distribution, and linear models (McCullough and Wilson 1999, 2002) Excel is not alone in its algorithm choice; we entered the numbers in Table 21 into a variety of statistical software programs and found that some, but not all, produced errors similar in magnitude The standard deviation is a simple formula, and the limitations of alternative implementations is well known; Wilkinson and Dallal (1977) pointed out failures in the variance calculations in statistical packages almost three decades ago In our example, the inaccuracy of Excel s standard deviation function is a direct result of the algorithm choice, not a limitation of the precision of its underlying arithmetic operators Excel s fundamental numerical operations are as accurate as those of most other packages that perform the standard deviation calculation correctly.

A C H I E V I N G T R A N S F O R M AT I O N W H I L E M A I N TA I N I N G O W N E R S H I P According to research completed by Cap Gemini Ernst & Young LLC (CGE&Y), management and IT consultants, and research 137

The mean-preserving requirement is too complicated for the ME bootstrap It requires that the mean mt in the interval It be equal to a weighted sum of the order statistic x(t) with weights from the set {075, 050, 025} as explained below Only if the f(x) maximizes our ignorance de ned by the entropy (9C1), we can call it the ME distribution The problem of nding such f(x) was solved long ago in the statistical literature dealing with so-called characterization problems (Kagan et al, 1973) A reference for economists in a different context is Theil and Laitinen (1980) The solution states that the form of the density is exponential in the two open-ended intervals and uniform in all intermediate intervals with a xed range with parameters given as follows: 1 f(x) = (1/q)exp([x - z1]/q), with q = m1 == 075x(1) + 0.

By implementing the textbook equation within Excel, using the average function, we were able to obtain the correct standard deviations for all the cases shown in Table 21 Excel s designers might argue that they made the correct choice in choosing a more time-ef cient calculation over one that is more accurate in some circumstances; and in a program used to analyze massive datasets, serious thought would need to go into these trade-offs However, given the uses to which Excel is normally put and the fact that internal limits in Excel prohibit analysis of truly large datasets, the one-pass algorithm offers no real performance advantage In this case, the textbook formula is more accurate than the algorithm used by Excel However, we do not claim that the textbook formula here is the most robust method to calculate the standard deviation.

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