Computing and graphing probability values of pearson distributions: a SAS/IML macro

Background Any empirical data can be approximated to one of Pearson distributions using the first four moments of the data (Elderton WP, Johnson NL. Systems of Frequency Curves. 1969; Pearson K. Philos Trans R Soc Lond Ser A. 186:343–414 1895; Solomon H, Stephens MA. J Am Stat Assoc. 73(361):153–60 1978). Thus, Pearson distributions made statistical analysis possible for data with unknown distributions. There are both extant, old-fashioned in-print tables (Pearson ES, Hartley HO. Biometrika Tables for Statisticians, vol. II. 1972) and contemporary computer programs (Amos DE, Daniel SL. Tables of percentage points of standardized pearson distributions. 1971; Bouver H, Bargmann RE. Tables of the standardized percentage points of the pearson system of curves in terms of β1 and β2. 1974; Bowman KO, Shenton LR. Biometrika. 66(1):147–51 1979; Davis CS, Stephens MA. Appl Stat. 32(3):322–7 1983; Pan W. J Stat Softw. 31(Code Snippet 2):1–6 2009) available for obtaining percentage points of Pearson distributions corresponding to certain pre-specified percentages (or probability values; e.g., 1.0%, 2.5%, 5.0%, etc.), but they are little useful in statistical analysis because we have to rely on unwieldy second difference interpolation to calculate a probability value of a Pearson distribution corresponding to a given percentage point, such as an observed test statistic in hypothesis testing. Results The present study develops a macro program to identify the appropriate type of Pearson distribution based on either input of dataset or the values of four moments and then compute and graph probability values of Pearson distributions for any given percentage points. Conclusions The SAS macro program returns accurate approximations to Pearson distributions and can efficiently facilitate researchers to conduct statistical analysis on data with unknown distributions.

Probability Distributions of Means of IA and IF for Gaussian Noise and Its Application to an Anomaly Detection

The Hilbert–Huang transform (HHT) extracts the intrinsic oscillation modes of input data, and estimates instantaneous amplitude (IA) and frequency (IF) for each mode. The HHT is applied to detection of some anomaly structures of signals as well as to analysis of signals. However, only qualitative discussions have been conducted on the applications to the detections. To make more statistically-based arguments on the application of the HHT, we investigated the probability distribution of the means of IA and IF for white Gaussian noise and found that it fits the Pearson distribution rather than the normal distribution. We defined a feature value for an anomaly detection by using the probability density function estimated on the basis of the Pearson distribution. Our method does not require different models for different lengths of the segment over which the mean is calculated, and therefore it is useful especially for the case that the length cannot be fixed.