Estimating Probability Distributions by Observing Betting Practices

A bookmaker takes bets on a two-horse race, attempting to minimize expected loss over all possible outcomes of the race. Profits are controlled by manipulation of customer' betting behavior; in order to do this, we need some information about the probability distribution which describes how the customers will bet. We examine what information initial customer' betting behavior provides about this probability distribution, and consider how to use this to estimate the probability distribution for remaining customers.     

Estimating Transition Probabilities for Ignorable Intermittent Missing Data in a Discrete-Time Markov Chain

This article considers a discrete-time Markov chain for modeling transition probabilities when multiple successive observations are missing at random between two observed outcomes using three methods: a na\"?ve analog of complete-case analysis using the observed one-step transitions alone, a non data-augmentation method (NL) by solving nonlinear equations, and a data-augmentation method, the Expectation-Maximization (EM) algorithm. The explicit form of the conditional log-likelihood given the observed information as required by the E step is provided, and the iterative formula in the M step is expressed in a closed form. An empirical study was performed to examine the accuracy and precision of the estimates obtained in the three methods under ignorable missing mechanisms of missing completely at random and missing at random. A dataset from the mental health arena was used for illustration. It was found that both data-augmentation and nonaugmentation methods provide accurate and precise point estimation, and that the na\"?ve method resulted in estimates of the transition probabilities with similar bias but larger MSE. The NL method and the EM algorithm in general provide similar results whereas the latter provides conditional expected row margins leading to smaller standard errors.     

Modeling the Audit Opinions Issued to Bankrupt Companies: A Two‐stage Empirical Analysis*

Many observers are dissatisfied with the accounting profession's ability to warn the public of upcoming bankruptcy filings. Since regulators and users tend to treat an unmodified audit opinion as a “clean bill of health,” they do not expect the business to fail in the near future. Research has shown that more often than not, auditors end up letting users down when it comes to predicting bankruptcy filings with audit opinions. Although auditors assert they are not responsible for predicting future events, it is very clear that their opinion decision is evaluated, at least in part, based on events that occur after the audit report date. The interesting and logical next step is to find out how companies exit bankruptcy. Do they liquidate or reorganize? Successful reorganization may, in the end, exonerate auditors and preserve their role as an early warning device. The opinion prediction model developed in the paper introduces a new bankruptcy resolution variable that proxies for the auditor's prognosis of the ultimate disposition of the soon‐to‐be‐bankrupt company. Using a sample of bankruptcy filings between 1982 and 1992, we find that auditors do not seem to be able to predict filings or resolution. Our tests of bankruptcy resolution support what auditors have been arguing for years: that they are not clairvoyant with respect to a client's future.     

Evaluating and Refining Expert Critiquing Systems: A Methodology*

Expert critiquing systems are a rapidly growing class of intelligent decision support systems that apply artificial intelligence techniques to criticize a user's proposed solution to a problem. Critic programs now exist in the medical, engineering, programming, knowledge acquisition, word processing, and other domains. Critic refinement is a nontrivial activity that, even when done well, consumes a sizable fraction of the complete effort to build and deploy a critiquing system. To ease that effort, it is important to adopt a rigorous approach that allows one to reproducibly measure the degree of success of the current critic version and to predict which refinements will improve the critic further. The current article presents one such approach with actual case studies that illustrate its usage and elaborate selected aspects of the refinement process.