Examples of "monotone_likelihood"
respect to the monotone likelihood ratio property.
The monotone likelihood ratio condition in this theorem cannot be weakened, as the next result demonstrates.
If a family of distributions formula_51 has the monotone likelihood ratio property in formula_22,
Theorem 6: Suppose formula_45 (for formula_382) obeys X. Then the family formula_367 obeys X if formula_360 is ordered with respect to the monotone likelihood ratio property.
One-parameter exponential-families have monotone likelihood-functions. In particular, the one-dimensional exponential-family of probability density functions or probability mass functions with
A family of density functions formula_19 indexed by a parameter formula_20 taking values in an ordered set formula_21 is said to have a monotone likelihood ratio (MLR) in the statistic formula_22 if for any formula_23,
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
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).
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_40.
Example: Let formula_9 be an input into a stochastic technology --- worker's effort, for instance --- and formula_31 its output, the likelihood of which is described by a probability density function formula_32 Then the monotone likelihood ratio property (MLRP) of the family formula_7 is expressed as follows: for any formula_34, the fact that formula_35 implies that the ratio formula_36 is increasing in formula_31.
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.
Proposition 2: Let formula_385 and formula_386 be two probability mass functions defined on formula_387 and suppose formula_388 is does not dominate formula_385 with respect to the monotone likelihood ratio property. Then there is a family of functions formula_390, defined on formula_391, that obey single crossing differences, such that formula_392, where formula_393 (for formula_394).
There are methods of construction median-unbiased estimators that are optimal (in a sense analogous to minimum-variance property considered for mean-unbiased estimators). Such constructions exist for probability distributions having monotone likelihood-functions. 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 but for a larger class of loss functions.
There are methods of construction median-unbiased methods for probability distributions that have monotone likelihood-functions, such as one-parameter exponential families, to ensure that they are optimal (in a sense analogous to minimum-variance property considered for mean-unbiased estimators). Such constructions exist for probability distributions having monotone likelihoods. 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.
Let formula_352, and formula_45 be a family of real-valued functions defined on formula_7 that obey single crossing differences or the interval dominance order. Theorem 1 and 3 tell us that formula_355 is increasing in formula_6. Interpreting formula_6 to be the state of the world, this says that the optimal action is increasing in the state if the state is known. Suppose, however, that the action formula_9 is taken before formula_6 is realized; then it seems reasonable that the optimal action should increase with the likelihood of higher states. To capture this notion formally, let formula_360 be a family of density functions parameterized by formula_361 in the poset formula_362, where higher formula_361 is associated with a higher likelihood of higher states, either in the sense of first order stochastic dominance or the monotone likelihood ratio property. Choosing under uncertainty, the agent maximizes