should provide (i) parameters, (ii) error estimates on the parameters, and (iii) a statistical measure of goodness-of-fit. When the third item suggests that the model is an unlikely match to the data, then items (i) and (ii) are probably worthless. | Least Squares as a Maximum Likelihood Estimator 657 should provide i parameters ii error estimates on the parameters and iii a statistical measure of goodness-of-fit. When the third item suggests that the model is an unlikely match to the data then items i and ii are probably worthless. Unfortunately many practitioners of parameter estimation never proceed beyond item i . They deem a fit acceptable if a graph of data and model looks good. This approach is known as chi-by-eye. Luckily its practitioners get what they deserve. CITED REFERENCES AND FURTHER READING Bevington . 1969 Data Reduction and Error Analysis for the Physical Sciences New York McGraw-Hill . Brownlee . 1965 Statistical Theory and Methodology 2nd ed. New York Wiley . Martin . 1971 Statistics for Physicists New York Academic Press . von Mises R. 1964 Mathematical Theory of Probability and Statistics New York Academic Press Chapter X. Korn . and Korn . 1968 Mathematical Handbook for Scientists and Engineers 2nd ed. New York McGraw-Hill Chapters 18-19. Least Squares as a Maximum Likelihood Estimator Suppose that we are fitting N data points x yi i 1 . . N to a model that has M adjustable parameters aj j 1 . . M. The model predicts a functional relationship between the measured independent and dependent variables y x y x a1 .aM where the dependence on the parameters is indicated explicitly on the right-hand side. What exactly do we want to minimize to get fitted values for the aj s The first thing that comes to mind is the familiar least-squares fit N minimize over a1 .aM yi y xy a1 .aM 2 i 1 But where does this come from What general principles is it based on The answer to these questions takes us into the subject of maximum likelihood estimators. Given a particular data set of xi s and yi s we have the intuitive feeling that some parameter sets a1. aM are very unlikely those for which the model function y x looks nothing like the data while others may be very .
