Confidence Distribution Xie and Singh (2013): Confidence distribution, the frequentist distribution estimator of a parameter: A Review Céline Cunen, 15/09/2014
Outline of Article Introduction The concept of Confidence Distribution (CD) A classical Definition and the History of the CD Concept A modern definition and interpretation Illustrative examples Basic parametric examples Significant (p-value) functions Bootstrap distributions Likelihood functions Asymptotically third-order accurate confidence distributions CD, Bootstrap, Fiducial and Bayesian approaches CD-random variable, Bootstrap estimator and fiducial-less interpretation CD, fiducial distribution and Belief function CD and Bayesian inference Inferences using a CD Confidence Interval Point estimation Hypothesis testing Optimality (comparison) of CDs Combining CDs from independent sources Combination of CDs and a unified framework for Meta-Analysis Incorporation of Expert opinions in clinical trials CD-based new methodologies, examples and applications CD-based likelihood caluculations Confidence curve CD-based simulation methods Additional examples and applications of CD-developments Summary
CD: a sample-dependent distribution that can represent confidence intervals of all levels for a parameter of interest
CD: a broad concept = covers all approaches that can build confidence intervals at all levels
Interpretation A distribution on the parameter space A Distribution estimator = contains information for many types of inference An estimator FOR the parameter of interest, instead of an inherent distribution OF the parameter Purely frequentist: The parameter is a fixed, non-random quantity 95% CI: the true parameter value will be covered by the CIs 95% of the time
Fiducial Inference Fisher's biggest blunder? Fiducial Faith A fiducial distribution: describes the level of faith attached to different values of the unknown parameter Sometimes normalized likelihood function are interpreted as distributions of the parameter ~ the parameters are both fixed and random quantities Interpretation of 95% CI: there is a 95% probability that the parameter lies inside the CI.
2.2 Modern definition A function Hn () is called a confidence distribution for a parameter θ if: R1: H n () is a cummulative distribution function on the parameter space R2: at the true parameter value θ = θ 0 H n (θ 0 ) as a function of the sample x follows the uniform distribution U[0,1] R2 is important!
2.3.1: Basic parametric examples A sample: With σ² known The function satisfies the requirements in the CD-definition R1: it clearly is a cdf R2:
2.3.2 p-value functions One-sided test H0 : θ b vs H 1 : θ > b P-value function: Usually CDs or (acds): Because cdf And because when b=θ 0, H 0 is true and p-values are uniformly distributed when H 0 is true.
2.3.3 Bootstrap distributions True (unknown) parameter Original sample, with estimator Bootstrap sample: a sample of equal size as the original sample, sampled with replacement from the original sample Compute the estimator on each bootstrap samples (get many ) --> the Bootstrap Distribution The Bootstrap Distribution is an acd!
2.3.4 Likelihood functions Under some mild conditions: normalized likelihood functions are density functions of asymptotic normal CDs Method for obtaining CDs from likelihoods CD-based inference likelihood inference
3.1 CD-random Variable,... The CD-random variable: The CD is not a distribution of θ! CD-random variable = Bootstrap estimator, and this is useful because it: Help understanding CD-inference and develop new methods Clarifies the interpretation of CDs: CDs are not distributions of θ, so therefore it is not a problem that a transformation g(θ) of θ does not generally lead to a CD for g(θ)
3.2 CD, Fiducial Distribution... Methods from fiducial reasoning (and from CDs) are supposed to have good statistical performance in the frequentist sense CIs should have the exact coverage property For tests: the actual rate of type I error is equal to the specified level of the test Many fiducial distributions are CDs Fiducial reasoning: provides a procedure for finding CDs
3.3 CD and Bayesian Inference Bayesian credible intervals to not possess the exact coverage property But asymptotically they can obtain it Then: posterior distributions are acds Bayes methods can produce CDs! Benefits with CBs compared to bayesian methods: Nuisance parameters
4 Inferences using a CD CDs contain information for any type of frequentist inference
4.1 Confidence Intervals CDs allows us to construct CIs for all levels of α CI constructed from CDs have the exact coverage property:
4.2 Point Estimation Median: Mean: Mode: Under some conditions, these are consistent estimators (= converging in probability to the true value)
4.3 Hypothesis Testing One-sided test: The support on C: Reject H 0 if the support on C is less than α The rejection region corresponds to a level α test Support = p-value (often) Two-sided test Same story, but with a more complicated rejection region
5 Optimality of CDs Can have multiple CDs for the same parameter A better CD = a CD more concentrated around the true parameter value
6.1 Combination of CDs... k independent studies, estimate the same parameter of interest θ Study i with sample xi and CD H i (.) Propose a general recipe for combining k independent CDs:
Combining k p-values - Fisher's method k p-values from k independent studies Under H0 : Remember that So that the teststatistic Will be distributed under H0
Example 2 studies aim to estimate the μ parameter from a normal model with known sigmas Study 1: n1 =30, Study 2: n2 =40, With CDs: We choose And then
6.2 Incorporation of Expert Opinions How can one incorporate existing knowledge in an analysis? Bayesian approaches: prior = existing knowledge In a frequentist setting: CD! CD-approach: CD e : summarizes the existing information/opinions CD d : from the data Combines these two by the methods in the last section Advantages: Easy to implement/ computationally cheap No need for priors on nuisance parameter! Avoids the discrepant posterior phenomenon
7 New methodologies, Examples... Different CD-related methods: Obtaining likelihood functions from CDs Presenting CDs: the confidence curve CD-based simulation methods
8 Summary CDs is a broad concept which contains many well-known notions and results Most types of frequentist inference can be derived from CDs Advantages of the CD-approach: Handles nuisance parameters well Easy to combine information Problems (need of further study): Multivariate CDs Cases where it difficult to obtain CDs Model uncertainty/ diagnosis/ selection