Predictive Inference for Linear and Circular Concomitants with Biomedical Applications

Abstract

Let (X_i, Y_i), for i=1,...,n, be a random sample from a bivariate distribution. If the sample is ordered with respect to one of the variables, say X, then the rth ordered X-value is called the rth order statistic and is denoted X_{r:n}. The Y-value corresponding to this value is called the concomitant of the rth order statistic and is denoted Y_{[r:n]}. In biomedical research, there is an interest in predicting the concomitant variable corresponding to the rth order statistic of the other variable. For example, one may be interested in predicting the time at which a patient has the peak blood pressure or the mercury level in fish where the water is most polluted. One such distribution of interest is the bivariate exponential conditionals distribution (BEC), whose conditional distributions are both exponential. The asymptotic predictive distribution of the concomitants of order statistics from the BEC is derived. The results are used in a prediction problem involving the mercury concentration in largemouth bass sampled from Florida lakes, as a function of surface water pollution level. Clinicians are often confronted with data such that one variable is linear and the other variable is circular, i.e., measured as an angle. A particular linear-circular distribution of interest is the exponential circular normal distribution. The predictive distribution of concomitants of order statistics from the exponential circular normal distribution is derived. The results are applied to predicting the future value of time at maximum heart rate in subjects from the Augusta Heart Study, a longitudinal study of normotensive children with verified family histories of cardiovascular diseases (e.g., hypertension and premature myocardial infarction).