 Monotone likelihood ratio

A monotonic likelihood ratio in distributions f(x) and g(x)
The ratio of the density functions above is increasing in the parameter x, so f(x)/g(x) 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
 for any x_{1} > x_{0},
that is, if the ratio is nondecreasing in the argument x.
If the functions are firstdifferentiable, 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)."
Contents
Intuition
The MLRP is used to represent a datagenerating process that enjoys a straightforward relationship between the magnitude of some observed variable and the distribution it draws from. If f(x) satisfies the MLRP with respect to g(x), the higher the observed value x, the more likely it was drawn from distribution f rather than g. As usual for monotonic relationships, the likelihood ratio's monotonicity comes in handy in statistics, particularly when using maximumlikelihood estimation. Also, distribution families with MLR have a number of wellbehaved stochastic properties, such as firstorder 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 e and the quality of the resulting project q. If the MLRP holds for the distribution of q conditional on your effort e, the higher the quality the more likely you worked hard. Conversely, the lower the quality the more likely you slacked off.
 Choose effort where H means high, L means low
 Observe q drawn from f(q  e). By Bayes' law with a uniform prior,
 Suppose f(q  e) satisfies the MLRP. Rearranging, the probability the worker worked hard is

 which, thanks to the MLRP, is monotonically increasing in q. Hence if some employer is doing a "performance review" he can infer his employee's behavior from the merits of his work.
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 indexed by a parameter θ taking values in an ordered set Θ is said to have a monotone likelihood ratio (MLR) in the statistic T(X) if for any θ_{1} < θ_{2},
 is a nondecreasing function of T(X).
Then we say the family of distributions "has MLR in T(X)".
List of families
Family T(X) in which f_{θ}(X) has the MLR Exponential[λ] observations Binomial[n,p] observations Poisson[λ] observations Normal[μ,σ] if σ known, observations Hypothesis testing
If the family of random variables has the MLRP in T(X), a uniformly most powerful test can easily be determined for the hypotheses versus H_{1}:θ > θ_{0}.
Example:Effort and output
Example: Let e be an input into a stochastic technology  worker's effort, for instance  and y its output, the likelihood of which is described by a probability density function f(y;e). Then the monotone likelihood ratio property (MLRP) of the family 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.
Relation to other statistical properties
If a family of distributions f_{θ}(x) has the monotone likelihood ratio property in T(X),
 the family has monotone decreasing hazard rates in θ (but not necessarily in T(X))
 the family exhibits the firstorder (and hence secondorder) stochastic dominance in x, and the best Bayesian update of θ is increasing in T(X).
But not conversely: neither monotone hazard rates nor stochastic dominance imply the MLRP.
Proofs
Let distribution family f_{θ} satisfy MLR in x, so that for θ_{1} > θ_{0} and x_{1} > x_{0}:
or equivalently:
Integrating this epression twice, we obtain:
1. To x_{1} with respect to x_{0} integrate and rearrange to obtain
2. From x_{0} with respect to x_{1} integrate and rearrange to obtain
Firstorder stochastic dominance
Combine the two inequalities above to get firstorder dominance:
Monotone hazard rate
Use only the second inequality above to get a monotone hazard rate:
Example
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.
Theory of probability distributions probability mass function (pmf) · probability density function (pdf) · cumulative distribution function (cdf) · quantile functionCategories: Theory of probability distributions
 Hypothesis testing
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