Monotone likelihood ratio property

Monotone likelihood ratio property is a property of a family of probability distributions described by their probability density functions (PDFs).

A family of density functions $\{\; f\_\; heta\; (x)\}\_\{\; hetain\; Theta\}$ indexed by a parameter $heta$ taking values in a set $Theta$ is said to have the monotone likelihood ratio (MLR) in a statistic $T(X)$ if for any two parameter values , the ratio $\{f\_\{\; heta\_2\}\; (x)\}/\{f\_\{\; heta\_1\}\; (x)\}$ is a non-decreasing function of $T(X)$.

Hypothesis Testing

The monotone likelihood ratio property can be used in hypothesis tests, primarily when dealing with composite null hypotheses.

If the family of random variables has the MLRP, a uniformly most powerful test can easily be determined for the hypotheses $H\_0\; :\; heta$ ≤ $heta\_0$ versus $H\_1\; :\; heta\; >\; heta\_0$.

Example

Example: Let e ("effort") be an input variable into a stochastic production function, and y be the random variable that represents output. Let f(y | e) be the pdf of y for each e. Then the monotone likelihood ratio property (MLRP) of $f$ is expressed as follows: for any $e_1,e_2,$ the fact that $e_2 > e_1$ implies that the ratio $f(y|e_2)/f(y|e_1)$ is increasing in y.

References

*cite web
last = Econoterms
first =
authorlink =
coauthors =
title = Glossary
work =
publisher = Experimental Economics Center
date = 2006
url = http://econport.gsu.edu/econport/request?page=web_glossary&glossaryLetter=M
format =
doi =
accessdate = 2006-10-26