ORDER STATISTICS MEDIANS

ORDER STATISTICS MEDIANS (LET)

Let Subcommand

Generate the N order statistics medians from one of the following distributions:
NORMAL
UNIFORM
HALF NORMAL
EV1 (or GUMBEL)
EV2 (or FRECHET)
WEIBULL
EXPONENTIAL

Given a sample of size N from a distribution, the observations can be ordered from smallest to largest. The smallest value is called the first order statistic and the largest value is called the n-th order statistic. There is an order statistic corresponding to each observation (so there are N order statistics in all). Each of these order statistics has its own sampling distribution with its own statistics (e.g., mean, standard deviation). The order statistics medians command generates the medians of each of the N order statistics.

Order statistics are commonly used in extreme value analysis. In particular, the first (i.e., minimum) and n-th (i.e., maximum) order statistics are of interest.

The order statistic medians (Filliben 1975) can be approximated by:

N_i = G(U_i)

where U_i are the uniform order statistic medians (defined below) and G is the percent point function for the desired distribution.

The uniform order statistic medians are defined as:

m_i = 1 - m_n    for i = 1
m_i = (i - 0.3175)/(n + 0.365)    for i = 2, 3, ..., n-1
m_i = 0.5^(1/n)    for i = n


Dataplot uses order statistic medians in the generation of probability plots. Specifically, the probability plot is formed by:

• Vertical axis: Ordered response values
• Horizontal axis: Order statistic medians for the specified distribution

LET <resp> = <dist> ORDER STATISTICS MEDIANS
                        FOR I = <start> <inc> <stop>
where <dist> identifies the distribution from which the order statistics are derived (NORMAL, HALF NORMAL, UNIFORM, EV1, EV2, WEIBULL, EXPONENTIAL);
            <start> is a number or parameter that identifies the first row of <resp> in which the order statistics medians are saved (typically it has a value of 1);
            <inc> is a number or parameter that identifies the row increment of <resp> in which the order statistics medians are saved (typically it has a value of 1);
            <stop> is a number or parameter that identifies the last row of <resp> in which the order statistics medians are saved;
and where <resp> is a variable where the order statistics medians are saved.

LET Y1 = NORMAL ORDER STATISTICS MEDIANS FOR I = 1 1 100
LET Y2 = NORMAL ORDER STATISTICS MEDIANS FOR I = 1 1 1000
LET Y1 = HALF NORMAL ORDER STATISTICS MEDIANS FOR I = 1 1 100
LET Y1 = UNIFORM ORDER STATISTICS MEDIANS FOR I = 1 1 100
LET Y1 = EV1 ORDER STATISTICS MEDIANS FOR I = 1 1 100


LET GAMMA = 3.2
LET Y1 = WEIBULL ORDER STATISTICS MEDIANS FOR I = 1 1 100


LET GAMMA = 2
LET Y1 = EV2 ORDER STATISTICS MEDIANS FOR I = 1 1 100

For the Weibull and Frechet distributions, you need to specify the value of the shape parameter. To do this, enter the following command before ORDER STATISTIC MEDIAN command
LET GAMMA = <value>

You can use the percent point function to compute order statistic medians for distributions not supported by the ORDER STATISTIC MEDIANS command. For example, to obtain the order statistic medians for the gamma distribution, enter
LET GAMMA = 1.8
LET YTEMP = UNIFORM ORDER STATISTIC MEDIANS FOR I = 1 1 100
LET YOSMED = GAMMPPF(YTEMP,GAMMA)

None

FRECHET for EV1
GUMBEL for EV2

ORDER STATISTIC MEANS	=	Compute order statistic means for a few select distributions.
MEDIAN	=	Compute the median of a variable.
MEAN	=	Compute the mean of a variable.
RANDOM NUMBERS	=	Generate a sequence of random numbers.
NORMAL PLOT	=	Generate a normal plot.
WEIBULL PLOT	=	Generate a Weibull plot.
PROBABILITY PLOT	=	Generates a probability plot.
PPCC PLOT	=	Generates a probability plot correlation coefficient plot.

Extreme value analysis

Filliben (1975), "The Probability Plot Correlation Coefficient Test for Normality," Technometrics, pp. 111-117.

1991/10: Original implementation of uniform, normal and half-normal distributions
1993/05: Added EV1 (Gumbel), EV2 (Frechet) and Weibull distributions
2022/04: Added exponential distribution

SKIP 25
READ ZARR13.DAT Y
LET N = SIZE Y
LET Y = SORT Y
LET X = ORDER STATISTIC MEDIANS FOR I = 1 1 N
.
TITLE CASE ASIS
LABEL CASE ASIS
TITLE Normal Probability Plot for ZARR13.DAT
Y1LABEL Sorted Data
X1LABEL Normal Order Statistic Medians
.
PLOT Y X