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.
All sample quantiles are defined as weighted averages of
consecutive order statistics. Sample quantiles of type \(i\)
are defined by:
$$Q_{i}(p) = (1 - \gamma)x_{j} + \gamma x_{j+1}$$
where \(1 \le i \le 9\),
\(\frac{j - m}{n} \le p < \frac{j - m + 1}{n}\),
\(x_{j}\) is the \(j\)th order statistic, \(n\) is the
sample size, the value of \(\gamma\) is a function of
\(j = \lfloor np + m\rfloor\) and \(g = np + m - j\),
and \(m\) is a constant determined by the sample quantile type.
Discontinuous sample quantile types 1, 2, and 3
For types 1, 2 and 3, \(Q_i(p)\) is a discontinuous
function of \(p\), with \(m = 0\) when \(i = 1\) and \(i =
2\), and \(m = -1/2\) when \(i = 3\).
- Type 1
Inverse of empirical distribution function.
\(\gamma = 0\) if \(g = 0\), and 1 otherwise.
- Type 2
Similar to type 1 but with averaging at discontinuities.
\(\gamma = 0.5\) if \(g = 0\), and 1 otherwise.
- Type 3
SAS definition: nearest even order statistic.
\(\gamma = 0\) if \(g = 0\) and \(j\) is even,
and 1 otherwise.
Continuous sample quantile types 4 through 9
For types 4 through 9, \(Q_i(p)\) is a continuous function
of \(p\), with \(\gamma = g\) and \(m\) given below. The
sample quantiles can be obtained equivalently by linear interpolation
between the points \((p_k,x_k)\) where \(x_k\)
is the \(k\)th order statistic. Specific expressions for
\(p_k\) are given below.
- Type 4
\(m = 0\). \(p_k = \frac{k}{n}\).
That is, linear interpolation of the empirical cdf.
- Type 5
\(m = 1/2\).
\(p_k = \frac{k - 0.5}{n}\).
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
\(m = p\). \(p_k = \frac{k}{n + 1}\).
Thus \(p_k = \mbox{E}[F(x_{k})]\).
This is used by Minitab and by SPSS.
- Type 7
\(m = 1-p\).
\(p_k = \frac{k - 1}{n - 1}\).
In this case, \(p_k = \mbox{mode}[F(x_{k})]\).
This is used by S.
- Type 8
\(m = (p+1)/3\).
\(p_k = \frac{k - 1/3}{n + 1/3}\).
Then \(p_k \approx \mbox{median}[F(x_{k})]\).
The resulting quantile estimates are approximately median-unbiased
regardless of the distribution of x.
- Type 9
\(m = p/4 + 3/8\).
\(p_k = \frac{k - 3/8}{n + 1/4}\).
The resulting quantile estimates are approximately unbiased for
the expected order statistics if x is normally distributed.
Further details are provided in Hyndman and Fan (1996) who recommended type 8.
The default method is type 7, as used by S and by R < 2.0.0.