Disturbance parameters
In the statistic is a perturbation parameter ( English nuisance parameter , not to be confused with noise factor or disturbance ) is an arbitrary parameter that is not of direct interest, however, that in the analysis parameters which are primarily of interest, must be considered. In general, any parameter that affects the analysis of another can be viewed as a disturbance parameter.
Disturbance parameters are often, but not always, variances, e.g. B. the variance of a normal distribution if the expected value is of primary interest; or the unknown “ true ” position of every observation in error-in-the-variable models .
A parameter can cease to be a "troublemaker" by becoming the primary subject of investigation.
Statistical theory
The general treatment of disturbance parameters can be largely similar in terms of theoretical statistics between Frequentist and Bayesian approaches . It is based on the attempt to partition the likelihood function (plausibility function ) into components that provide information about the parameters of interest and information about the other (interfering) parameters. This can include approaches using exhaustive statistics and non-distribution statistics . If this partitioning can be achieved, it may be possible to perform a Bayesian analysis on the parameters of interest by algebraically determining their common a posteriori distribution . Partitioning enables frequentist theory to develop general estimation approaches in the presence of interfering parameters. If partitioning cannot be achieved, approximate partitioning may still be used.
In some special cases it is possible to formulate methods that circumvent the presence of interfering parameters. The t-test provides a useful test in practice because the test statistic does not depend on the unknown variance. It is a case where use can be made of a pivot statistic . In other cases, however, such a workaround is not known.
Statistical Practice
Practical approaches to statistical analysis treat the disturbance parameters somewhat differently in the frequentistic and Bayesian methodology.
A general approach in a frequentist analysis can be based on plausibility quotient tests. These provide both significance tests and confidence intervals for the parameters of interest, which are approximately valid for medium to large sample sizes and take into account the presence of interfering parameters. See Basu (1977) for a general discussion and Spall and Garner (1990) for a discussion of identifying parameters in linear dynamic models (i.e., state space representation ).
In Bayesian statistics, a generally applicable approach generates random samples from the common a posteriori distribution of all parameters (see Markov chain Monte Carlo method ). Under these circumstances, the common distribution of only the parameters of interest can be determined by marginalizing the interference parameters. However, this approach may not always be computationally efficient if some or all of the spurious parameters can be eliminated on a theoretical basis.
literature
- Basu, D. (1977): On the Elimination of Nuisance Parameters. , Journal of the American Statistical Association , Vol. 77, pp. 355-366. doi : 10.1080 / 01621459.1977.10481002
- Bernardo, JM, Smith, AFM (2000): Bayesian Theory. Wiley., ISBN 0-471-49464-X
- Cox, DR, Hinkley, DV (1974) Theoretical Statistics. Chapman and Hall., ISBN 0-412-12420-3
- Spall, JC and Garner, JP (1990): Parameter Identification for State-Space Models with Nuisance Parameters. , IEEE Transactions on Aerospace and Electronic Systems , Vol. 26 (6), pp. 992-998.
- Young, GA, Smith, RL (2005): Essentials of Statistical Inference , CUP., ISBN 0-521-83971-8