Single or combined location measurements of the same parameter without prior probability. (Altern. title, Parametric inference as useful illusion; Part 1)

Motivation. This version is based solely on the calculus of probability, excluding any statistical principle. "Location measurement" means the pdf of the error is known. When the datum is obtained, intuition suggests something like a pdf for the parameter; here we attempt a critical examination of its meaning. Summary. In default of prior probability the parameter is not defined as a random variable, hence there can be no genuine prior-free parametric inference. Nevertheless prior-free predictive inference regarding any future datum is generated directly from the datum of a location measurement. Such inference turns out as if obtained from a certain pdf ("fiducial") indirectly associated with the parameter. This false pdf can expedite predictive inference, but is inappropriate in the analysis of combined measurements (unless they all are location measurements of the same parameter). Also it has the same distribution as the ostensible Bayesian posterior from a uniform "prior". However, if any of these spurious entities is admitted in the analysis, inconsistent results follow. When we combine measurements, we find that the quantisation errors, inevitable in data recording, must be taken into consideration. These errors cannot be folded into predictive inference in an exact sense; that is, we cannot render a predictive distribution of a future datum except as an approximation. Keywords: location measurement; combination of observations; parametric inference; predictive inference; prior-free inference; quantisation error; digitisation; frequentist interpretation; the fiducial argument; fiducial probability; pivotal inference; intuitive assessment; prior-free assessment