Estimating The Quantiles of A Population
There are several methods for estimating the quantiles. The most comprehensive breadth of methods is available in the R programming language, which includes nine sample quantile methods. SAS includes five sample quantile methods, STATA includes two, and Microsoft Excel includes one.
In effect, the methods compute Qp, the estimate for the kth q-quantile, where p = k / q, from a sample of size N by computing a real valued index h. When h is an integer, the hth smallest of the N values, xh, is the quantile estimate. Otherwise a rounding or interpolation scheme is used to compute the quantile estimate from h, x⌊h⌋, and x⌈h⌉. (For notation, see floor and ceiling functions).
Estimate types include:
| Type | h | Qp | Notes |
|---|---|---|---|
| R-1, SAS-3 | Inverse of empirical distribution function. When p = 0, use x1. | ||
| R-2, SAS-5 | The same as R-1, but with averaging at discontinuities. When p = 0, x1. When p = 1, use xN. | ||
| R-3, SAS-2 | The observation numbered closest to Np. Here, ⌊ h ⌉ indicates rounding to the nearest integer, choosing the even integer in the case of a tie. When p ≤ (1/2) / N, use x1. | ||
| R-4, SAS-1 | Linear interpolation of the empirical distribution function. When p < 1 / N, use x1. When p = 1, use xN. | ||
| R-5 | Piecewise linear function where the knots are the values midway through the steps of the empirical distribution function. When p < (1/2) / N, use x1. When p ≥ (N - 1/2) / N, use xN. | ||
| R-6, SAS-4 | Linear interpolation of the expectations for the order statistics for the uniform distribution on . When p < 1 / (N+1), use x1. When p ≥ N / (N + 1), use xN. | ||
| R-7, Excel | Linear interpolation of the modes for the order statistics for the uniform distribution on . When p = 1, use xN. | ||
| R-8 | Linear interpolation of the approximate medians for order statistics. When p < (2/3) / (N + 1/3), use x1. When p ≥ (N - 1/3) / (N + 1/3), use xN. | ||
| R-9 | The resulting quantile estimates are approximately unbiased for the expected order statistics if x is normally distributed. When p < (5/8) / (N + 1/4), use x1. When p ≥ (N - 3/8) / (N + 1/4), use xN. | ||
| If h were rounded, this would give the order statistic with the least expected square deviation relative to p. When p < (3/2) / (N + 2), use x1. When p ≥ (N + 1/2) / (N + 2), use xN. |
Note that R-3 and R-4 do not give h = (N + 1) / 2 when p = 1/2.
The standard error of a quantile estimate can in general be estimated via the bootstrap. The Maritz-Jarrett method can also be used.
Read more about this topic: Quantile
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