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

The type double function is imsl_d_ranks.

A pointer to a vector of length n_observations containing the rank (or optionally, a transformation of the rank) of each observation.

In data without ties, the output values are the ordinary ranks (or a transformation of the ranks) of the data in x. If x[i] has the smallest value among the values in x and there is no other element in x with this value, then ranks[i] = 1. If both x[i] and x[j] have the same smallest value, then the output value depends upon the option used to break ties.

When the ties are resolved randomly, the function imsl_f_random_uniform is used to generate random numbers. Different results may occur from different executions of the program unless the “seed” of the random number generator is set explicitly by use of the function imsl_random_seed_set.

Normal and other functions of the ranks can optionally be returned. Normal scores can be defined as the 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, imsl_f_normal_inverse_cdf, 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, n_observations) 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_value) 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. Then, these normal scores are averaged or selected in the manner specified. (Whether the transformations are made first or ties are resolved first makes no difference except when IMSL_AVERAGE is specified.)

In the Tukey version (see Tukey 1962), the scaling transformation for the rank riis (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 for the Blom normal scores.

In the Van der Waerden version (see Lehmann 1975, p. 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 for the Blom normal scores.

When option IMSL_EXPECTED_NORMAL_SCORES is used, the output values are the expected values of the normal order statistics from a sample of size n_observations. If the value in x[i] is the k-th smallest, then the value output in ranks[i] is E(zk) where E(⋅) is the expectation operator, and zk is the k-th order statistic in a sample of size n_observations from a standard normal distribution. Ties are handled in the same way as for the Blom normal scores.

Savage scores are the expected values of the exponential order statistics from a sample of size n_observations. These values are called Savage scores because of their use in a test discussed by Savage (1956) (see Lehmann 1975). If the value in x[i] is the k-th smallest, then the value output in ranks[i] is E(yk) where yk is the k-th order statistic in a sample of size n_observations from a standard exponential distribution. The expected value of the k-th order statistic from an exponential sample of size n (n_observations) is

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

The data for this example, from Hinkley (1977), contains 30 observations. Note that the fourth and sixth observations are tied, and that the third and twentieth observations are tied.

This example uses all of the score options with the same data set, which contains some ties. Ties are handled in several different ways in this example.