What are the errors we should be seeking to minimise in the estimation process?

Three methods of fitting a gravity model - a triproportional model - to an observed trip matrix are compared. The first is the familiar practical method of choosing row, column and cost factors so that the model has the same row, column and cost sums as the grossed-up data. The second method, a true maximum likelihood estimation, chooses the factors so that the sums of the observed counts (not grossed-up) are matched. This differs from the first method only when the sampling probabilities vary from cell to cell. The third method applies the more modern approach of selecting a loss function which represents the practical effect of differences between the model values and the true values, and then chooses the model factors so that the expected loss, as far as it can be determined fromthe sample data and any prior information, is minimised. Squared error in flow times travel time is proposed as the loss function. It is noted that there is a loss function whose use is equivalent to maximum likelihood. When the sample counts are large and the model fits well, each of the methods reduced to minimising the weighted squared, difference between the model and the saturated value. The variations in these weights show the differences between the three methods.