
Monotone likelihood ratio

A monotonic likelihood ratio in distributions formula_1 and formula_2
The ratio of the density functions above is increasing in the parameter formula_3, so formula_4 satisfies the monotone likelihood ratio property.
In statistics, the monotone likelihood ratio property is a property of the ratio of two probability density functions (PDFs). Formally, distributions "ƒ"("x") and "g"("x") bear the property if
that is, if the ratio is nondecreasing in the argument formula_3.
If the functions are first-differentiable, the property may sometimes be stated
For two distributions that satisfy the definition with respect to some argument x, we say they "have the MLRP in "x"." For a family of distributions that all satisfy the definition with respect to some statistic "T"("X"), we say they "have the MLR in "T"("X")."
Intuition.
The MLRP is used to represent a data-generating process that enjoys a straightforward relationship between the magnitude of some observed variable and the distribution it draws from. If formula_1 satisfies the MLRP with respect to formula_2, the higher the observed value formula_3, the more likely it was drawn from distribution formula_11 rather than formula_12. As usual for monotonic relationships, the likelihood ratio's monotonicity comes in handy in statistics, particularly when using maximum-likelihood estimation. Also, distribution families with MLR have a number of well-behaved stochastic properties, such as first-order stochastic dominance and increasing hazard ratios. Unfortunately, as is also usual, the strength of this assumption comes at the price of realism. Many processes in the world do not exhibit a monotonic correspondence between input and output.
Example: Working hard or slacking off.
Suppose you are working on a project, and you can either work hard or slack off. Call your choice of effort formula_13 and the quality of the resulting project formula_14. If the MLRP holds for the distribution of "q" conditional on your effort formula_13, the higher the quality the more likely you worked hard. Conversely, the lower the quality the more likely you slacked off.
Families of distributions satisfying MLR.
Statistical models often assume that data are generated by a distribution from some family of distributions and seek to determine that distribution. This task is simplified if the family has the monotone likelihood ratio property (MLRP).
A family of density functions formula_25 indexed by a parameter formula_26 taking values in an ordered set formula_27 is said to have a monotone likelihood ratio (MLR) in the statistic formula_28 if for any formula_29,
Then we say the family of distributions "has MLR in formula_28".
Hypothesis testing.
If the family of random variables has the MLRP in formula_28, a uniformly most powerful test can easily be determined for the hypothesis formula_34 versus formula_35.
Example: Effort and output.
Example: Let formula_13 be an input into a stochastic technology – worker's effort, for instance – and formula_37 its output, the likelihood of which is described by a probability density function formula_38 Then the monotone likelihood ratio property (MLRP) of the family formula_11 is expressed as follows: for any formula_40, the fact that formula_41 implies that the ratio formula_42 is increasing in formula_37.
Relation to other statistical properties.
Monotone likelihoods are used in several areas of statistical theory, including point estimation and hypothesis testing, as well as in probability models. 
Exponential families.
One-parameter exponential families have monotone likelihood-functions. In particular, the one-dimensional exponential family of probability density functions or probability mass functions with
has a monotone non-decreasing likelihood ratio in the sufficient statistic "T"("x"), provided that formula_45 is non-decreasing.
Most powerful tests: The Karlin–Rubin theorem.
Monotone likelihood functions are used to construct uniformly most powerful tests, according to the Karlin–Rubin theorem. Consider a scalar measurement having a probability density function parameterized by a scalar parameter "θ", and define the likelihood ratio formula_46.
If formula_47 is monotone non-decreasing, in formula_3, for any pair formula_49 (meaning that the greater formula_3 is, the more likely formula_51 is), then the threshold test:
is the UMP test of size "α" for testing formula_55
Note that exactly the same test is also UMP for testing formula_56
Median unbiased estimation.
Monotone likelihood-functions are used to construct median-unbiased estimators, using methods specified by Johann Pfanzagl and others. One such procedure is an analogue of the Rao–Blackwell procedure for mean-unbiased estimators: The procedure holds for a smaller class of probability distributions than does the Rao–Blackwell procedure for mean-unbiased estimation but for a larger class of loss functions.
Lifetime analysis: Survival analysis and reliability.
If a family of distributions formula_57 has the monotone likelihood ratio property in formula_28,
But not conversely: neither monotone hazard rates nor stochastic dominance imply the MLRP.
Proofs.
Let distribution family formula_64 satisfy MLR in "x", so that for formula_65 and formula_66:
or equivalently:
Integrating this expression twice, we obtain:
First-order stochastic dominance.
Combine the two inequalities above to get first-order dominance:
Monotone hazard rate.
Use only the second inequality above to get a monotone hazard rate:
Uses.
Economics.
The MLR is an important condition on the type distribution of agents in mechanism design. Most solutions to mechanism design models assume a type distribution to satisfy the MLR to take advantage of a common solution method.