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 { x 1 , … , x m } {\displaystyle \{x_{1},\ldots ,x_{m}\}} 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 M X = ( g θ , Z ) {\displaystyle M_{X}=(g_{\theta },Z)} , 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 F Θ ( θ ) {\displaystyle F_{\Theta }(\theta )} , 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 M X = ( g θ , Z ) {\displaystyle M_{X}=(g_{\theta },Z)} for the random variable X, we model X = { X 1 , … , X m } {\displaystyle {\boldsymbol {X}}=\{X_{1},\ldots ,X_{m}\}} to be equal to { g θ ( Z 1 ) , … , g θ ( Z m ) } {\displaystyle \{g_{\theta }(Z_{1}),\ldots ,g_{\theta }(Z_{m})\}} . Focusing on a relevant statistic S = h 1 ( X 1 , … , X m ) {\displaystyle S=h_{1}(X_{1},\ldots ,X_{m})} for the parameter θ, the master equation reads
s = h ( g θ ( z 1 ) , … , g θ ( z m ) ) = ρ ( θ ; z 1 , … , z m ) . {\displaystyle s=h(g_{\theta }(z_{1}),\ldots ,g_{\theta }(z_{m}))=\rho (\theta ;z_{1},\ldots ,z_{m}).}
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