Twisting properties
Twisting properties in general terms are associated with the properties of samples that identify with statistics that are suitable for exchange. Description. Starting with a sample formula_1 observed from a random variable "X" having a given distribution law with a non-set parameter, a parametric inference problem consists of computing suitable values – call them estimates – of this parameter precisely on the basis of the sample. An estimate is suitable if replacing it with the unknown parameter does not cause major damage in next computations. In algorithmic inference, suitability of an estimate reads in terms of compatibility with the  observed sample. In turn, parameter compatibility is a probability measure that we derive from the probability distribution of the random variable to which the parameter refers. In this way we identify a random parameter Θ compatible with an observed sample. Given a sampling mechanism formula_2, the rationale of this operation lies in using the "Z" seed distribution law to determine both the "X" distribution law for the given θ, and the Θ distribution law given an "X" sample. Hence, we may derive the latter distribution directly from the former if we are able to relate domains of the sample space  to subsets of Θ support. In more abstract terms, we speak about twisting properties of samples with properties of parameters and identify the former with statistics that are suitable for this exchange, so denoting a well behavior w.r.t. the unknown parameters. The operational goal is to write the analytic expression of the cumulative distribution function formula_3, in light of the observed value "s" of a statistic "S", as a function of the "S" distribution law when the "X" parameter is exactly θ. Method. Given a sampling mechanism formula_2 for the random variable "X", we model formula_5 to be equal to  formula_6. Focusing on a relevant statistic formula_7 for the parameter "θ", the master equation reads When "s" is a well-behaved statistic w.r.t the parameter, we are sure that a monotone relation exists for each formula_9 between "s" and θ. We are also assured that Θ, as a function of formula_10 for given "s", is a random variable since the master equation provides solutions that are feasible and independent of other (hidden) parameters. The direction of the monotony determines for any formula_11 a relation between events of the type formula_12 or "vice versa" formula_13, where formula_14 is computed by the  master equation with formula_15. In the case that "s" assumes discrete values the first relation changes into formula_16 where formula_17 is the size of the "s" discretization grain, idem with the opposite monotony trend. Resuming these relations on all seeds, for "s" continuous we have either or For "s" discrete we have an interval where formula_20 lies, because of formula_17. The whole logical contrivance is called a twisting argument. A procedure implementing it is as follows. Remark. The rationale behind twisting arguments does not change when parameters are vectors, though some complication arises from the management of joint inequalities. Instead , the difficulty of dealing with a vector of parameters proved to be the Achilles heel of Fisher's approach to the fiducial distribution of parameters . Also Fraser’s constructive probabilities devised for the same purpose do not treat this point completely. Example. For formula_22 drawn from a gamma distribution, whose specification requires values for the parameters λ and "k", a twisting argument may be stated by following the below procedure. Given the meaning of these parameters we know that where formula_25 and formula_26. This leads to a joint cumulative distribution function Using the first factorization and replacing formula_28 with formula_29  in order to have a distribution of formula_30 that is independent of formula_31, we have with "m" denoting the sample size, formula_34 and formula_35 are the observed statistics (hence with indices denoted by capital letters), formula_36 the incomplete gamma function and formula_37 the Fox's H function that can be approximated with a gamma distribution again with proper parameters (for instance estimated through the method of moments) as a function of "k" and "m". With a sample size formula_38 and formula_39 formula_40, you may find the joint p.d.f. of the Gamma parameters "K" and formula_31 on the left. The marginal  distribution of "K" is reported in the picture on the right.