Synonyms for monotone_likelihood or Related words with monotone_likelihood
σx youden wilks_lambda_distribution maximum_likelihood_estimators pearson_correlation_coefficient shannon_entropy σb studentized_range kurtosis σy mlrp eudysmic matthews_correlation_coefficient variance_covariance differintegral mean_squared_error regression_coefficient skewness exponentiated additivity weighted_sum poisson_distributions hellinger_distance superpartient hypergeometric_distribution bivariate memorylessness kullback_leibler_divergence noncentrality covariance_matrices weibull_distribution excess_kurtosis hotelling_squared_distribution radon_nikodym minimum_variance_unbiased σa pareto_distribution lyapunov_exponents weierstrass_transform scalar_valued scalar_curvature logarithmic_derivative bose_einstein hölder_condition nonparametric_skew point_biserial_correlation unnormalized softmax_function sortino_ratio tsallis_entropyExamples of "monotone_likelihood" |
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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 |