This paper deals with variability issues connected with the proton transfer reaction-mass spectrometry (PTR-MS) measurements of isoprene concentration. We focus on isoprene as an abundant and widely studied compound in human breath. The variability caused by the measurement process is described by the within-sample distribution. Thus, based on the formula for computing isoprene concentration that reflects the principle of the PTR-MS, a theoretical model for the within-sample distribution of isoprene concentration is suggested. This model, which assumes that the distribution is proportional to a quotient of two independent Poisson-distributed random variables, is then confronted with empirical distributions obtained from 17 breath samples collected from a healthy individual within a month. (In each sample, isoprene concentration was determined 97 times.) The empirical within-sample distributions are also compared to normal and log-normal distributions. While those seem to be satisfactory approximations, the theoretical model is found suitable only in 10 out of 17 breath samples. We also comment on the stability of samples during the measurement process in the PTR-MS instrument and, for the sake of comparison, determine the within-sample and the within-subject variability of isoprene concentrations in our data. The respective geometric standard deviations are 1.01 and 1.29.

In this paper, we deal with the comparative calibration problem, i.e. with the situation when one instrument or measurement technique is calibrated against another, each of which is subject to the measurement error. We propose an approximate, small sample, calibration confidence interval of the unknown true value of the measured substance in units of the more precise instrument, given measurement in units of the less precise instrument. Here we deal with the simplest case—single-use linear univariate calibration, i.e. the case in which we assume linear relationship between the two measurement techniques (instruments), and, further, that the calibration procedure is conducted in order to obtain one value for an unknown, reported together with an interval estimate. The method for deriving the approximate confidence interval is based on estimation of the calibration line via the replicated errors-in-variables model. The model is locally linearized and the Wald-type F-statistic is constructed. An essential point in this approach is the use of the F-approximation of the distribution of the F-statistic suggested by Kenward and Roger [Kenward, M.G. and Roger, J.H., 1997, Small sample inference for fixed effects from restricted maximum likelihood. Biometrics, 53, 983–997]. The statistical properties of the proposed interval estimator are verified by simulations for wide spectrum of experimental designs and compared with two standard univariate interval estimates—the classical approach for deriving the calibration interval was proposed by Eisenhart [Eisenhart, C., 1939, The interpretation of certain methods and their use in biological and industrial research. Annals of Mathematical Statistics, 10, 162–186] and the inverse method was proposed by Krutchkoff [Krutchkoff, R.G., 1967, Classical and inverse methods of calibration. Technometrics, 9, 425–439].

In linguistic modelling, a number of probability distributions must be modified because of different individual influences on the data and gradual shifts to new attractors. Several kinds of modifications, estimations and tests of already published models are presented for fitting purposes.

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).