Computes the ranks, normal scores, or exponential scores for a vector of observations.

The routine RANKS determines the ranks, or various transformations of the ranks of the data in X. Ties in the data can be resolved in four different ways, as specified in ITIE.

For this option, the values output in SCORE are the ordinary ranks of the data in X. If X(I) has the smallest value among those in X and there is no other element in X with this value, then
SCORE(I) = 1. If both X(I) and X(J) have the same smallest value, then

When the ties are resolved by use of routine RNUNF (see Chapter 18, “Random Number Generation”) to generate random numbers, different results may occur when running the same program at different times unless the “seed” of the random number generator is set explicitly by use of the routine RNSET (see Chapter 18, “Random Number Generation”). Ordinarily, there is no need to call the routine to set the seed, even if there are ties in the data.

Normal scores are expected values, or approximations to the expected values, of order statistics from a normal distribution. The simplest approximations are obtained by evaluating the inverse cumulative normal distribution function (routine ANORIN, see Chapter 18, “Random Number Generation”) at the ranks scaled into the open interval (0, 1). In the Blom version (see Blom 1958), the scaling transformation for the rank ri(1 ≤ ri ≤ n, where n is the sample size, NOBS) is (ri ‑ 3/8)/(n + 1/4). The Blom normal score corresponding to the observation with rank ri is

where Φ( ) is the normal cumulative distribution function.

Adjustments for ties are made after the normal score transformation. That is, if X(I) equals X(J) (within FUZZ) and their value is the k-th smallest in the data set, the Blom normal scores are determined for ranks of k and k + 1, and then these normal scores are averaged or selected in the manner specified by ITIE. (Whether the transformations are made first or ties are resolved first makes no difference except when averaging is done.)

In the Tukey version (see Tukey 1962), the scaling transformation for the rank ri is (ri ‑ 1/3)/(n + 1/3). The Tukey normal score corresponding to the observation with rank ri is

Ties are handled in the same way as discussed above for the Blom normal scores.

In the Van der Waerden version (see Lehmann 1975, page 97), the scaling transformation for the rank ri is ri / (n + 1). The Van der Waerden normal score corresponding to the observation with rank ri is

Ties are handled in the same way as discussed above for the Blom normal scores.

For this option, the values output in SCORE are the expected values of the normal order statistics from a sample of size NOBS. If the value in X(I) is the k-th smallest, then the value output in SCORE(I) is E(Zk), where E( ) is the expectation operator and Zk is the k-th order statistic in a sample of size NOBS from a standard normal distribution. Such expected values are computed by the routine ENOS (see Chapter 20, “Mathematical Support”). Ties are handled in the same way as discussed above for the Blom normal scores.

For this option, the values output in SCORE are the expected values of the exponential order statistics from a sample of size NOBS. These values are called Savage scores because of their use in a test discussed by Savage (1956) (see Lehman 1975). If the value in X(I) is the k-th smallest, then the value output in SCORE(I) is E(Yk), where Yk is the k-th order statistic in a sample of size NOBS from a standard exponential distribution. The expected value of the k-th order statistic from an exponential sample of size n (NOBS) is

Ties are handled in the same way as discussed above for the Blom normal scores.

The example uses all of these options with the same data set, which contains some ties. The ties are handled different ways in this example.

The data for this example, from Hinkley (1977), are the same used in several examples in this chapter. There are 30 observations. Note that the fourth and sixth observations are tied and that the third and twentieth are tied.