Exact Distribution of the Nonlinear Least Squares Estimator

Eugene Demidenko

DHMC / Dartmouth Mathematics Dept


We report on the solution of a 100 years old statistics problem on the exact density distribution of the nonlinear least squares estimator under normal distribution of errors with small sample size. Throughout the years, several authors suggested the formula for the distribution but they all turned to be wrong. We have tested our solution through a result of Fieller (1932), who derived the exact distribution of the ratio of two normally distributed random variables. The exact distribution is tangled with problems of nonexistence and non-uniqueness, frequently encountered when solving nonlinear equations. Applications to confidence intervals and hypothesis testing for multivariate nonlinear regression and estimating equation approaches, as well as simplifying distribution approximations, are discussed.

The proof of the theorem hints to the method of deriving exact (or near-exact) distribution for many nonlinear statistical models with finite sample size - the most formidable problem of the modern statistics.

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