No abstract

The stochastic error distance (SED) introduced by Diebold and Shin (2017) ranks forecast models by divergence between distributions of the errors of the actual and perfect forecast models. The basic SED is defined by the variation distance and provides a representation of the mean absolute error, but by basing ranking on the entire error distribution and divergence, the SED moves beyond the traditional forecast evaluations. First, we establish connections between ranking forecast models by the SED, error entropy and some partial orderings of distributions. Then, we introduce the notion of excess error for forecast errors of magnitudes larger than a tolerance threshold and give the SED representation of the mean excess error (MEE). As a function of the threshold, the MEE is a local risk measure. With the distribution of the absolute error as a prior for the threshold, its Bayes risk is the entropy functional of the survival function, which is a known measure in the information theory and reliability. Notions and results are illustrated using various distributions for the error. The empirical versions of SED, MEE and its Bayes risk are compared with the mean squared error in ranking regression and autoregressive integrated moving average models for forecasting bond risk premia.

No abstract

Big data enables reliable estimation of continuous probability density, cumulative distribution, survival, hazard rate, and mean residual functions (MRFs). We illustrate that plot of the MRF provides the best resolution for distinguishing between distributions. At each point, the MRF gives the mean excess of the data beyond the threshold. Graph of the empirical MRF, called here the MR plot, provides an effective visualization tool. A variety of theoretical and data driven examples illustrate that MR plots of big data preserve the shape of the MRF and complex models require bigger data. The MRF is an optimal predictor of the excess of the random variable. With a suitable prior, the expected MRF gives the Bayes risk in the form of the entropy functional of the survival function, called here the survival entropy. We show that the survival entropy is dominated by the standard deviation (SD) and the equality between the two measures characterizes the exponential distribution. The empirical survival entropy provides a data concentration statistic which is strongly consistent, easy to compute, and less sensitive than the SD to heavy tailed data. An application uses the New York City Taxi database with millions of trip times to illustrate the MR plot as a powerful tool for distinguishing distributions.

No abstract