Randomized statistical inference: A unified statistical inference frame of frequentist, fiducial, and Bayesian inference

We propose randomized inference (RI), a new statistical inference approach. RI may be realized through a randomized estimate (RE) of a parameter vector, which is a random vector that takes values in the parameter space with a probability density function (PDF) that depends on the sample or sufficient statistics, such as the posterior distributions in Bayesian inference. Based on the PDF of an RE of an unknown parameter, we propose a framework for both the vertical density representation (VDR) test and the construction of a confidence region. This approach is explained with the aid of examples. For the equality hypothesis of multiple normal means without the condition of variance homogeneity, we present an exact VDR test, which is shown as an extension of one-way analysis of variance (ANOVA). In the case of two populations, the PDF of the Welch statistics is given using RE. Furthermore, through simulations, we show that the empirical distribution function, the approximated t, and the RE distribution function of Welch statistics are almost equal. The VDR test of the homogeneity of variance is shown to be more efficient than both the Bartlett test and the revised Bartlett test. Finally, we discuss the prospects of RI.