# quantile¶

## Purpose¶

Computes quantiles from data in a matrix, given specified probabilities.

## Format¶

y = quantile(x, e[, tp])
Parameters:
• x (NxK matrix) – data

• e (Lx1 vector) – quantile levels or probabilities.

• tp (scalar) – 1, 2, …, 9. Sample quantile type. Default is 4.

Returns:

y (LxK matrix) – quantiles.

## Examples¶

// Set the rng seed for repeatable random numbers
rndseed 345567;

// Create a 1000x4 random normal matrix
x = rndn(1000, 4);

// Quantile levels
e = { .025, .5, .975 };

y = quantile(x, e);

print "medians";
print y[2, .];
print;
print "95 percentiles";
print y[1, .];
print y[3, .];


Produces the following output:

medians
-0.037801917   0.029923972  -0.010477829  -0.023937160

95 percentiles
-2.0074122    -2.0798579    -1.9982702    -1.9605009
2.0437573     2.0271770     1.9025695     1.9228044


## Remarks¶

Let { $$x_{(1)},...,x_{(n)}\,$$ } denote the order statistics, and let $${\overset{\hat{}}{Q}}_{i}\left( p \right)\, = \,\left( 1 - \gamma \right)x_{(j)} + \gamma x_{(j + 1)}$$ denotes the sample quantiles, where $$\frac{j - m}{n} \leq p < \frac{j - m - 1}{n},\, m \in {\mathbb{R}},\, 0 \leq \gamma \leq 1.$$ The value of $$\gamma$$ is a function of integer part $$j = \, floor\left( pn + m \right)$$ and fractional part $$g = \, pn + m - j$$. The $$m$$ is a constant determined by sample quantile type.

Type

Definition

1

Discrete sample quantile type 1. Inverse of empirical distribution function.

2

Discrete sample quantile type 2. Similar to type 1 except that averaging at discontinuities.

3

Discrete sample quantile type 3. SAS definition, choose the nearest even order statistics.

4

Continuous sample quantile type 4. Interpolating the step function of definition 1.

5

Continuous sample quantile type 5. This is the value midway through each step of definition 1.

6

Continuous sample quantile type 6. The vertices divide the sample space into n+1 regions, each with probability 1/(n+1).

7

Continuous sample quantile type 7. The vertices divide the range into n-1 regions, and 100p% of the intervals lie to the left and 100(1-p)% lie to the right.

8

Continuous sample quantile type 8. The resulting sample quantile is median unbiased regardless the distribution.

9

Continuous sample quantile type 9. The resulting sample quantile is median unbiased if normal distribution.

quantile.src