The paper is concerned with issues of the estimation of random variable distribution parameters by the Monte Carlo method. Such quantities can correspond to statistical parameters computed based on the data obtained in typical measurement situations. The subject of the research is the mean, the mean square and the variance of random variables with uniform, Gaussian, Student, Simpson, trapezoidal, exponential, gamma and arcsine distributions.

Gravity Recovery and Climate Experiment (GRACE) mission data is widely used in various fields of science. GRACE explored changes of the gravity field regularly from April 2002 to June 2017. In the following research, we examine variance of signal contained in two different formats of GRACE data: standard spherical harmonics and mass concentration blocks (so-called “mascons”) solutions, both provided in the most recent releases. For spherical harmonics-based solution, we use monthly gravity field solutions provided up to degree and order (d/o) 96 by three different computing centers, i.e. the NASA’s Jet Propulsion Laboratory (JPL), the German Research Center for Geosciences (GFZ) and the Center for Space Research (CSR). For the mass concentration blocks, we use values of total water storage provided by the CSR, JPL and the Goddard Space Flight Center (GSFC) computing centers, which we convert to spherical harmonic coefficients up to d/o 96. We show that using the anisotropic DDK3 filter to smooth the north-south stripes present in total wate storage obtained from standard spherical harmonics solution leaves more information than common isotropic Gaussian filter. In the case of mascons, GSFC solution contains much more information than the CSR and JPL releases, relevant for corresponding d/o. Differences in variance of signal arise from different background models as well as various shape and size of mascons used during processing of GRACE observations.

When observations are autocorrelated, standard formulae for the estimators of variance, s2, and variance of the mean, s2 (x), are no longer adequate. They should be replaced by suitably defined estimators, s2a and s2a (x), which are unbiased given that the autocorrelation function is known. The formula for s2a was given by Bayley and Hammersley in 1946, this work provides its simple derivation. The quantity named effective number of observations neff is thoroughly discussed. It replaces the real number of observations n when describing the relationship between the variance and variance of the mean, and can be used to express s2a and s2a (x) in a simple manner. The dispersion of both estimators depends on another effective number called the effective degrees of freedom Veff. Most of the formulae discussed in this paper are scattered throughout the literature and not very well known, this work aims to promote their more widespread use. The presented algorithms represent a natural extension of the GUM formulation of type-A uncertainty for the case of autocorrelated observations.

We discuss the empirical importance of long term cyclical effects in the volatility of financial returns. Following Amado and Teräsvirta (2009), ČiŽek and Spokoiny (2009) and others, we consider a general conditionally heteroscedastic process with stationarity property distorted by a deterministic function that governs the possible time variability of the unconditional variance. The function proposed in this paper can be interpreted as a finite Fourier approximation of an Almost Periodic (AP) function as defined by Corduneanu (1989). The resulting model has a particular form of a GARCH process with time varying parameters, intensively discussed in the recent literature.

In the empirical analyses we apply a generalisation of the Bayesian AR(1)-GARCH model for daily returns of S&P500, covering the period of sixty years of US postwar economy, including the recently observed global financial crisis. The results of a formal Bayesian model comparison clearly indicate the existence of significant long term cyclical patterns in volatility with a strongly supported periodic component corresponding to a 14 year cycle. Our main results are invariant with respect to the changes of the conditional distribution from Normal to Student-tand to the changes of the volatility equation from regular GARCH to the Asymmetric GARCH.

Prior knowledge of the autocorrelation function (ACF) enables an application of analytical formalism for the unbiased estimators of variance s2a and variance of the mean s2a(xmacr;). Both can be expressed with the use of so-called effective number of observations neff. We show how to adopt this formalism if only an estimate {rk} of the ACF derived from a sample is available. A novel method is introduced based on truncation of the {rk} function at the point of its first transit through zero (FTZ). It can be applied to non-negative ACFs with a correlation range smaller than the sample size. Contrary to the other methods described in literature, the FTZ method assures the finite range 1 < neff ≤ n for any data. The effect of replacement of the standard estimator of the ACF by three alternative estimators is also investigated. Monte Carlo simulations, concerning the bias and dispersion of resulting estimators sa and sa(×), suggest that the presented formalism can be effectively used to determine a measurement uncertainty. The described method is illustrated with the exemplary analysis of autocorrelated variations of the intensity of an X-ray beam diffracted from a powder sample, known as the particle statistics effect.

Electroencephalogram (EEG) is one of biomedical signals measured during all-night polysomnography to diagnose sleep disorders, including sleep apnoea. Usually two central EEG channels (C3-A2 and C4- A1) are recorded, but typically only one of them are used. The purpose of this work was to compare discriminative features characterizing normal breathing, as well as obstructive and central sleep apnoeas derived from these central EEG channels. The same methodology of feature extraction and selection was applied separately for the both synchronous signals. The features were extracted by combined discrete wavelet and Hilbert transforms. Afterwards, the statistical indexes were calculated and the features were selected using the analysis of variance and multivariate regression. According to the obtained results, there is a partial difference in information contained in the EEG signals carried by C3-A2 and C4-A1 EEG channels, so data from the both channels should be preferably used together for automatic sleep apnoea detection and differentiation.

The wear behaviour of Cr3C2-25% NiCr laser alloyed nodular cast iron sample were analyzed using a pin-on-disc tribometer. The influence of sliding velocity, temperature and load on laser alloyed sample was focused and the microscopic images were used for metallurgical examination of the worn-out sites. Box-Behnken method was utilised to generate the mathematical model for the condition parameters. The Response Surface Methodology (RSM) based models are varied to analyse the process parameters interaction effects. Analysis of variance was used to analyse the developed model and the results showed that the laser alloyed sample leads to a minimum wear rate (0.6079×10–3 to 1.8570×10–3 mm3/m) and coefficient of friction (CoF) (0.43 to 0.53). From the test results, it was observed that the experimental results correlated well with the predicted results of the developed mathematical model.

The paper presents application of Taguchi method in optimizing the sound transmission loss through sandwich gypsum constructions and those comprising of masonry concrete blocks and gypsum boards in order to investigate the relative influence of the various parameters affecting the sound transmission loss. The application of Taguchi method for optimizing sound transmission loss has been rarely reported. The present work uses the results analytically predicted on “Insul” software for various sandwich material configurations as desired by each experimental run in an L8 orthogonal array. The relative importance of the parameters on single-number rating, Rw (C, Ctr) is evaluated in terms of percentage contribution using Analysis of Variance (ANOVA). The ANOVA method reveals that type of studs, steel stud frame and number of gypsum layers attached are the key factors controlling the sound transmission loss characteristics of sandwich multi-layered constructions.