Types quantile returns estimates of underlying distribution quantiles based on one or two order statistics from the supplied elements in x at probabilities in probs. One of the nine quantile algorithms discussed in Hyndman and Fan (1996), selected by type, is employed.
Sample quantiles of type i are defined by
Q[i](p) = (1 - gamma) x[j] + gamma x[j+1],
where 1 <= i <= 9, (j-m)/n <= p < (j-m+1)/ n,
x[j] : jth order statistic
n is the sample size
m is a constant determined by the sample quantile type. Here gamma depends on the fractional part of g = np+m-j.
For the continuous sample quantile types (4 through 9), the sample quantiles can be obtained by
linear interpolation between the kth order statistic and p(k):
p(k) = (k - alpha) / (n - alpha - beta + 1),
where α and β are constants determined by the type. Further, m = alpha + p(1 - alpha - beta), and gamma = g.
Discontinuous sample quantile types 1, 2, and 3
Type 1
Inverse of empirical distribution function.
Type 2
Similar to type 1 but with averaging at discontinuities.
Type 3
SAS definition: nearest even order statistic.
Continuous sample quantile types 4 through 9
Type 4
p(k) = k / n. That is, linear interpolation of the empirical cdf.
Type 5
p(k) = (k - 0.5) / n. That is a piecewise linear function where the knots are the values midway through the steps of the empirical cdf. This is popular amongst hydrologists.
Type 6
p(k) = k / (n + 1). Thus p(k) = E[F(x[k])]. This is used by Minitab and by SPSS.
Type 7
p(k) = (k - 1) / (n - 1). In this case, p(k) = mode[F(x[k])]. This is used by S.
Type 8
p(k) = (k - 1/3) / (n + 1/3). Then p(k) =~ median[F(x[k])]. The resulting quantile estimates are approximately median-unbiased regardless of the distribution of x.
Type 9
p(k) = (k - 3/8) / (n + 1/4). The resulting quantile estimates are approximately unbiased for the expected order statistics if x is normally distributed.
Hyndman and Fan (1996) recommend type 8. The default method is type 7, as used by S and by R < 2.0.0.
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