Question

Discuss how sampling distributions are used for inference

Answer 1

=>A sampling distribution or finite-sample distribution is the probability distribution of a given random-sample-based statistic

=> If an arbitrarily large number of samples, each involving multiple observations (data points), were separately used in order to compute one value of a statistic

=> In many contexts, only one sample is observed, but the sampling distribution can be found theoretically.

=> Sampling distributions are important in statistics because they provide a major simplification en route to statistical inference

=> More specifically, they allow analytical considerations to be based on the probability distribution of a statistic, rather than on the joint probability distribution of all the individual sample values

=> In the theory of statistical inference, the idea of a sufficient statistic provides the basis of choosing a statistic (as a function of the sample data points) in such a way that no information is lost by replacing the full probabilistic description of the sample with the sampling distribution of the selected statistic.

=> In frequentist inference, for example in the development of a statistical hypothesis test or a confidence interval, the availability of the sampling distribution of a statistic (or an approximation to this in the form of an asymptotic distribution) can allow the ready formulation of such procedures, whereas the development of procedures starting from the joint distribution of the sample would be less straightforward.

=> In Bayesian inference, when the sampling distribution of a statistic is available, one can consider replacing the final outcome of such procedures, specifically the conditional distributions of any unknown quantities given the sample data, by the conditional distributions of any unknown quantities given selected sample statistics. Such a procedure would involve the sampling distribution of the statistics. The results would be identical provided the statistics chosen are jointly sufficient statistics.