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lecturenotes09 - MS&E 223 Simulation Brad Null Lecture...

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MS&E 223 Lecture Notes #9 Simulation Quantile Estimation Brad Null Spring Quarter 2008-09 Page 1 of 9 Quantile Estimation 1. Quantiles (Definition and Simple Point Estimate) Quantiles form an important class of performance measures. The following examples show the utility of quantile estimation: Example 1 : Let X be the return on an investment. One possible measure of the downside risk (also called Value at Risk) of the investment is the determination of that value q such that the likelihood that X takes on a smaller value than q is some prescribed value, say 0.01. This is one simple way of quantifying a “worst-case scenario” when assessing the riskiness of an investment. 0 f X (x) 99% q 1% The value q is known as the 0.01-quantile of X. In general, we may be interested in computing the pth quantile of X (0<p<1). Such a quantile p q is defined by 1 p X q F (p) - = When X F is continuous and increasing, then p q is simply the unique root of the equation F X (q p ) = p. When X F has jumps or piecewise-constant sections, we define p q as 1 X F (p) - , where we use our general definition of the inverse of a distribution (see our earlier discussion of the inversion method for random variate generation). This leads to the general definition { } p X q min q :F (q) p = . Example 2 : A measure of spread (or dispersion) in a distribution that is sometimes used in preference to the variance is the inter-quartile range , given by 0.75 0.25 q q α = - , where 0.75 q and 0.25 q are the 0.25th and 0.75th quantiles, respectively, of the distribution under consideration. The inter-quartile range is much less sensitive than the variance to “outlier” observations.
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MS&E 223 Lecture Notes #9 Simulation Quantile Estimation Brad Null Spring Quarter 2008-09 Page 2 of 9 However, the estimation of quantiles is very different from the estimation of functions (both linear and nonlinear) of means. In other words, the above estimation problems involve methodologies that have not previously been addressed in this course. These procedures are discussed below. The natural estimator for p q is, of course, the pth sample quantile. Specifically, set 1 n n Q F (p) - = where n F is the empirical distribution function, defined by n n j j 1 1 F (x) I(X x) n = = . Here, 1 n X , ,X K are i.i.d. replicates of the random variable X. (The function n F is piecewise constant, with jumps at the values 1 n X , ,X K . The height of a jump at the point i x X = is k / n , where k is the number of data points with a value equal to i X .) An equivalent way of defining n Q is to let (i) X be the “ith order statistic” of the sample 1 n X , ,X K , so that (1) (2) (n) X X X L . Then ( 29 n np Q X = That is, n Q is the   np th smallest observation in the sample 1 n X , ,X K . So, developing a point estimator for p q is easy. (There are many other quantile estimators that are used in practice, typically involving some sort of interpolation between ( 29
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lecturenotes09 - MS&amp;E 223 Simulation Brad Null Lecture...

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