Monotone likelihood ratio property

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Monotone likelihood ratio property is a set of probability density functions, or PDFs, assumed in theoretical models to characterize risks and uncertanty, which makes more conclusions feasible and often plausible.

[edit] Hypothesis Testing

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

A joint distribution $f θ (x)$ is said to have the monotone likelihood ratio (MLR) in the statistic $T (X)$ if for any two values $θ 1 < θ 2$ , the ratio ${f_{\theta_2} (x)} \over {f_{\theta_1} (x)}$ depends on $X$ only through $T (X)$ , and this ratio is non-decreasing in $T (X)$ . If such a joint distribution has this property, a uniformly most powerful test can easily be determined for the hypotheses $H 0 :θ$ ≤ $θ 0$ versus $H 1 :θ > θ 0$ .

[edit] Example

Example: Let e ("effort") be an input variable into a stochastic production function, and y be the random variable that represent output. Let f(y | e) be the pdf of y for each e. Then the statement that f() has the monotone likelihood ratio property (MLRP) is the same as the statement that: for e2>e1, f(y|e2)/f(y|e1) is increasing in y.

Let y be the random variable that represents the output. E (effort) will be the input variable into a stochastic production function. If F(Y|E) is the pdf of Y for each E, then the statement that F() has the monotone likelihood ratio property (MLRP) is the same as the statement that:

E₂>E₁, F( Y | E₂ ) / F( Y | E₁ ) is increasing in Y