Input Modeling Using a Computer

Input modeling that involves tting standard univari-ate parametric probability distributions is typically performed using an input modeling package. These packages typically t several distributions to a data set, then determine the distribution with the best t by comparing goodness-of-t statistics. But what if an appropriate input model is not included in one of these packages? The modeler must resort to deriving the appropriate estimators by hand for the appropriate input model. The purpose of this paper is to investigate the use of a prototype Maple-based probability language, known as APPL (A Probability Programming Language), for input modeling. This language allows an analyst to specify a standard or non-standard distribution for an input model, and have the derivations performed automatically. Input modeling serves as an excellent arena for illustrating the applicability and usefulness of APPL. Besides including pre-deened types for over 45 diierent continuous and discrete random variables and over 30 procedures for manipulating random variables (e.g., convolution, transformation), APPL contains input modeling procedures for parameter estimation, plotting empirical and tted CDFs, and performing goodness-of-t tests. Using examples, we illustrate its utility for input modeling. There have been dozens of statistical languages developed over the years to relieve the computations associated with interactive or batch processing of data. APPL's data structures and algorithms were initially developed to accommodate probability problems, but may be used to solve input modeling problems as well. In order to illustrate the syntax and capability of APPL, we begin with some simple examples from probability theory in this section, then address some input modeling problems in the next section. Example 1. Find the probability that the sum of eight independent and identically distributed U(0,1) random variables falls between 7 2 and 11 2. Pr 7 2 < 8 X i=1 X i < 11 2 ! : The two standard methods for approximating the probability are the central limit theorem and Monte Carlo simulation. The central limit theorem approximation gives only one digit of accuracy for this particular problem. Monte Carlo simulation, on the other hand, converges to the exact value if a good random number generator is used, but requires custom coding and requires a 100-fold increase in computing time for each additional digit of accuracy. The APPL statements solve the problem exactly, yielding 3580151 5160960 : ConvolutionIID computes the exact distribution of the sum and stores the result in Y. This may be coded up …