Minimum distance estimation

Minimum distance estimation (MDE) is a statistical method for fitting a mathematical model to data, usually the empirical distribution. Minimum distance estimation (MDE) is a statistical method for fitting a mathematical model to data, usually the empirical distribution. Let X 1 , … , X n {displaystyle displaystyle X_{1},ldots ,X_{n}} be an independent and identically distributed (iid) random sample from a population with distribution F ( x ; θ ) : θ ∈ Θ {displaystyle F(x; heta )colon heta in Theta } and Θ ⊆ R k ( k ≥ 1 ) {displaystyle Theta subseteq mathbb {R} ^{k}(kgeq 1)} . Let F n ( x ) {displaystyle displaystyle F_{n}(x)} be the empirical distribution function based on the sample. Let θ ^ {displaystyle {hat { heta }}} be an estimator for θ {displaystyle displaystyle heta } . Then F ( x ; θ ^ ) {displaystyle F(x;{hat { heta }})} is an estimator for F ( x ; θ ) {displaystyle displaystyle F(x; heta )} . Let d [ ⋅ , ⋅ ] {displaystyle d} be a functional returning some measure of 'distance' between the two arguments. The functional d {displaystyle displaystyle d} is also called the criterion function. If there exists a θ ^ ∈ Θ {displaystyle {hat { heta }}in Theta } such that d [ F ( x ; θ ^ ) , F n ( x ) ] = inf { d [ F ( x ; θ ) , F n ( x ) ] ; θ ∈ Θ } {displaystyle d=inf{d; heta in Theta }} , then θ ^ {displaystyle {hat { heta }}} is called the minimum distance estimate of θ {displaystyle displaystyle heta } . (Drossos & Philippou 1980, p. 121) Most theoretical studies of minimum distance estimation, and most applications, make use of 'distance' measures which underlie already-established goodness of fit tests: the test statistic used in one of these tests is used as the distance measure to be minimised. Below are some examples of statistical tests that have been used for minimum distance estimation. The chi-square test uses as its criterion the sum, over predefined groups, of the squared difference between the increases of the empirical distribution and the estimated distribution, weighted by the increase in the estimate for that group.