Interval estimation of the mean of a normal distribution based on quantized observations

Abstract

ABSTRACT. We consider the problem of making statistical inference about the mean of a normal distribution based on a random sample of quantized (digitized) observations. This problem arises, for example, in a measurement process with errors drawn from a normal distribution and with a measurement device or process with a known resolution, such as the resolution of an analog-to-digital converter or another digital instrument. In this paper we investigate the effect of quantization on subsequent statistical inference about the true mean. If the standard deviation of the measurement error is large with respect to the resolution of the indicating measurement device, the effect of quantization (digitization) diminishes and standard statistical inference is still valid. Hence, in this paper we consider situations where the standard deviation of the measurement error is relatively small. By Monte Carlo simulations we compare small sample properties of the interval estimators of the mean based on standard approach (i.e. by ignoring the fact that the measurements have been quantized) with some recently suggested methods, including the interval estimators based on maximum likelihood approach and the fiducial approach. The paper extends the original study by Hannig et al. (2007).

On the choice of a noninformative prior for Bayesian inference of discretized normal observations

“…We conjecture that the unexpected non-monotonic behavior of the posterior for identical data when using the reference prior is due to this inherent difficulty. Some further aspects pointing towards an inherent difficulty of the discrete problem for x (1) = x (n) follow from the work of Witkovsky and Wimmer (2009), who noted that the maximum likelihood estimator for (μ, σ ) does not exist in this case. Furthermore, they stated that the fiducial solution to the discrete problem requires choosing a 'predetermined random mechanism', as a consequence of which it appears that such solution depends on this choice just as a Bayesian inference depends on the choice of prior.…”

“…This is in contrast to the continuous problem where the fiducial solution is uniquely defined (and is numerically the same as the Bayesian solution with the standard prior). In fact, Witkovsky and Wimmer (2009) proposed a different 'predetermined random mechanism' than that of Hannig et al (2007).…”

“…He derived a procedure for quoting a 'standard uncertainty of measurement', which is a step required in metrology. As an alternative, application of fiducial inference was proposed by Frenkel (2009), Hannig et al (2007 and also, as mentioned above, by Witkovsky and Wimmer (2009). The latter two papers also provided simulation studies which indicate that their fiducial analyses yield similar numerical results to intervals constructed using the results of Willink (2007), i.e.…”

“…This and related problems have received some attention in the statistical and engineering literature (Arnold and Press 1986;Heitjan 1989;Vardeman 2001, 2002;Vardeman and Lee 2005;Witkovsky and Wimmer 2009), but perhaps even more in the metrological literature (Lira and Wöger 1997;Elster 2000;Locci et al 2002;Frenkel and Kirkup 2005;Lira 2006;Taraldsen 2006;Cordero et al 2006;Willink 2007;Hannig et al 2007;Phillips et al 2008;Frenkel 2009). When-as is often the case-the variability of the data is large in comparison to δ, the discrete problem can be treated just as the continuous one.…”

Finite resolution and serial correlations in mass metrology

“…For the theoretical modelling of the problem Monte Carlo simulation [23][24][25][26][27] has been used. Two significant factors contributing to the measurement outcome and uncertainty are always assumed in [13][14][15][16][17][18][19][20][21][22][23][24][25][26] to be present: finite resolution and Gaussian noise. In the majority of these publications, it is generally concluded that if the sample standard deviation s is sufficiently large in comparison to the resolution or quantization step size q of the measurement instrument (about 0.5 ), s q ≥ the effect of the finite resolution will be insignificant, and conventional statistical inference will be valid.…”