Fisher Snedecor Research Articles

Sojourn measures of Student and Fisher–Snedecor random fields

Limit theorems for the volumes of excursion sets of weakly and strongly dependent heavy-tailed random fields are proved. Some generalizations to sojourn measures above moving levels and for cross-correlated scenarios are presented. Special attention is paid to Student and Fisher-Snedecor random fields. Some simulation results are also presented.

Numerical approximation of high-dimensional Fokker–Planck equations with polynomial coefficients

This paper is concerned with the numerical solution of high-dimensional Fokker–Planck equations related to multi-dimensional diffusion with polynomial coefficients or Pearson diffusions. Classification of multi-dimensional Pearson diffusion follows from the classification of one-dimensional Pearson diffusion. There are six important classes of Pearson diffusion—three of them possess an infinite system of moments (Gaussian, Gamma, Beta) while the other three possess a finite number of moments (inverted Gamma, Student and Fisher–Snedecor). Numerical approximations to the solution of the Fokker–Planck equation are generated using the spectral method. The use of an adaptive reduced basis technique facilitates a significant reduction in the number of degrees of freedom required in the approximation through the determination of an optimal basis using the singular value decomposition (SVD). The basis functions are constructed dynamically so that the numerical approximation is optimal in the current finite-dimensional subspace of the solution space. This is achieved through basis enrichment and projection stages. Numerical results with different boundary conditions are presented to demonstrate the accuracy and efficiency of the numerical scheme.

Kinetics of Jatropha oil methanolysis

The effect of temperature, methanol to oil molar ratio and catalyst concentration on Jatropha oil methanolysis were studied in the range from 40 to 60°C, 3 to 12 and 0.2 to 1wt% (NaOH), respectively. 100% Conversion of Jatropha oil and 97% yield to fatty acid methyl esters (FAME) were obtained after 60min of reaction at 60°C, methanol to oil molar ratio 6:1 and 0.6wt% of NaOH. Catalyst concentration increment between 0.2wt% and 0.6wt% had a positive effect on the reaction rate and on the yield to FAME. Furthermore, catalyst concentrations of 0.2wt% did not show catalytic activity at all the temperatures tested. A second order kinetic model that includes the three consecutive reversible reactions involved in triacylglycerol methanolysis and the effect of temperature and catalyst concentration on reaction rate describes adequately Jatropha oil methanolysis, as validated by the Fisher–Snedecor test of unbiased variances. A set of eighteen parameters: six kinetic constants, six activation energies and six parameters related to the catalyst concentration were determined using a Dual Population Evolutionary Algorithm (DPEA). Finally, the kinetic model parameters of Jatropha oil and palm oil methanolysis were compared. Parameters obtained were similar in both reactions.

Repeatability Estimation for Vibration Measurement Results of Power Transformer Core

The subject matter of this paper refers to the diagnostics of a transformer core, based on measurements and analysis of vibroacoustic signals registered during their normal operation. The paper presents results of the research aimed to evaluate the repeatability of vibration measurements for power transformer core. Recording of vibroacoustic signals was performed under laboratory conditions on dry-insulated transformers. The study concerned evaluation of changes in vibroacoustic indicators of the transformer core during operation with a turned and loosened core. Conclusions from the measurement results were drawn on the basis of the parametric test of signi cance derived from the variance analysis for many means of single classi cation basing on the Fisher Snedecor F-distribution.

Ergodicity and mixing bounds for the Fisher–Snedecor diffusion

We consider the Fisher-Snedecor diffusion; that is, the Kolmogorov-Pearson diffusion with the Fisher-Snedecor invariant distribution. In the nonstationary setting, we give explicit quantitative rates for the convergence rate of respective finite-dimensional distributions to that of the stationary Fisher-Snedecor diffusion, and for the $\beta$-mixing coefficient of this diffusion. As an application, we prove the law of large numbers and the central limit theorem for additive functionals of the Fisher-Snedecor diffusion and construct $P$-consistent and asymptotically normal estimators for the parameters of this diffusion given its nonstationary observation.

Comparison of Parameters of Spinal Curves in the Sagittal Plane Measured by Photogrammetry and Inclinometry

BACKGROUND. The photogrammetric method and inclinometer-based measurements are commonly employed to assess the anteroposterior curvatures of the spine. These methods are used both in clinical trials and for screening purposes. The aim of the study was to compare the parameters used to characterise the anteroposterior spinal curvatures as measured by photogrammetry and inclinometry. MATERIAL AND METHODS. The study enrolled 341 subjects: 169 girls and 172 boys, aged 4 to 9 years, from kindergartens and primary schools in Rzeszów. The anteroposterior spinal curvatures were examined by photogrammetry and with a mechanical inclinometer. RESULTS. There were significant differences in the α angle between the inclinometric and photogrammetric assessment in the Student t test (p=0.017) and the Fisher Snedecor test (p=0.0001), with similar differences in the β angle (Student's t p=0.0001, Fisher Snedecor p=0.007). For the γ angle, significant differences were revealed with Student's t test (p=0.0001), but not with the Fisher Snedecor test (p = 0.22). CONCLUSIONS. 1. Measurements of inclination of particular segments of the spine obtained with the photogrammetric method and the inclinometric method in the same study group revealed statistically significant differences. 2. The results of measurements obtained by photogrammetry and inclinometry are not comparable. 3. Further research on agreement between measurements of the anteroposterior spinal curvatures obtained using the available measurement equipment is recommended.

Spectral representation of transition density of Fisher–Snedecor diffusion

We analyse spectral properties of an ergodic heavy-tailed diffusion with the Fisher–Snedecor invariant distribution and compute spectral representation of its transition density. The spectral representation is given in terms of a sum involving finitely many eigenvalues and eigenfunctions (Fisher–Snedecor orthogonal polynomials) and an integral over the absolutely continuous spectrum of the corresponding Sturm–Liouville operator. This result enables the computation of the two-dimensional density of the Fisher–Snedecor diffusion as well as calculation of moments of the form , where m and n are at most equal to the number of Fisher–Snedecor polynomials. This result is particularly important for explicit calculations associated with this process.

Linear regression models, least-squares problems, normal equations, and stopping criteria for the conjugate gradient method

Minimum-variance unbiased estimates for linear regression models can be obtained by solving least-squares problems. The conjugate gradient method can be successfully used in solving the symmetric and positive definite normal equations obtained from these least-squares problems. Taking into account the results of Golub and Meurant (1997, 2009) [10,11], Hestenes and Stiefel (1952) [17], and Strakoš and Tichý (2002) [16], which make it possible to approximate the energy norm of the error during the conjugate gradient iterative process, we adapt the stopping criterion introduced by Arioli (2005) [18] to the normal equations taking into account the statistical properties of the underpinning linear regression problem. Moreover, we show how the energy norm of the error is linked to the χ2-distribution and to the Fisher–Snedecor distribution. Finally, we present the results of several numerical tests that experimentally validate the effectiveness of our stopping criteria.

Turán Type Inequalities for Tricomi Confluent Hypergeometric Functions

Some sharp two-sided Turan type inequalities for parabolic cylinder functions and Tricomi confluent hypergeometric functions are deduced. The proofs are based on integral representations for quotients of parabolic cylinder functions and Tricomi confluent hypergeometric functions, which arise in the study of the infinite divisibility of the Fisher–Snedecor F distribution. Moreover, some complete monotonicity results are given concerning Turan determinants of Tricomi confluent hypergeometric functions. These complement and improve some of the results of Ismail and Laforgia (in Constr. Approx. 26:1–9, 2007).

Modeling of the Kinetics Degradation of Unstabilized and HALS-Stabilized LDPE Films Under Thermo-Oxidation and Natural Weathering Conditions

Mathematical models are proposed to predict degradation of unstabilized low-density polyethylene (LDPE) and LDPE stabilized with hindered amine light stabilizers (HALS) under both thermo-oxidations at 70, 80 and 90 °C and natural weathering conditions (NWC). The degradation was measured by change in percent elongation at break \({(\varepsilon_{r})}\) with time. The mathematical approach developed was multiple linear regression analysis (MLRA). The reliability of the selected models was analyzed using four statistical criteria: residual variance, coefficient of determination (r2), Student test and Fisher–Snedecor test. The linear systems that resulted from the MLRA were solved using the GAUSS method. The results obtained indicated that the polynomial models developed to predict elongation at break were reliable for both unstabilized and HALS-stabilized samples under thermo-oxidation at 70, 80 and 90 °C and NWC This was also confirmed by the comparison of the half-life time values predicted from the models with those observed experimentally.

Hypothesis testing for Fisher–Snedecor diffusion

We consider the problem of testing the hypothesis on marginal distribution of ergodic diffusion with Fisher–Snedecor invariant distribution, to be called Fisher–Snedecor diffusion. We propose a GMM approach to testing this statistical hypothesis where the moment condition is based on eigenfunctions of the diffusion infinitesimal generator—Fisher–Snedecor polynomials. Statistical test is observed in two different settings: (1) for known values of parameters of the process; (2) for consistent moment based estimators of parameters. Results are illustrated in a short simulation study.

Modeling of the Kinetic Degradation of Unstabilized and HALS-Stabilized LDPE Films in Natural Aging

In this work, we propose new mathematical models describing the kinetic degradation of unstabilized low-density polyethylene (LDPE) and LDPE stabilized with hindered amine light stabilizers (HALS). The samples were exposed to natural weathering conditions, and the degradation was measured by a change in elongation at break with time. The mathematical approach developed was based on multiple linear regression analysis (MLRA). We tested the reliability of the selected models using four statistical criteria: residual variance, coefficient of determination, Student test and Fisher–Snedecor test. The linear systems resulting from the MLRA were solved using the Cholesky method. The results showed that the polynomial models developed to predict elongation at break were reliable for both types of samples under natural weathering conditions. The experimental half-life times are close to those predicted by the models.

Parameter estimation for Fisher–Snedecor diffusion

We consider the problem of parameter estimation for an ergodic diffusion with Fisher–Snedecor invariant distribution, to be called Fisher–Snedecor diffusion. We propose moments-based estimators of unknown parameters, based on both discrete and continuous observations, and prove their consistency and asymptotic normality. The explicit form of the asymptotic covariance matrix is determined by using the properties of eigenfunctions (Fisher–Snedecor polynomials) of the corresponding Sturm–Liouville operator.

Differences of raw cork quality in productive cork oak woods in Sicily in relation to stand density

Differences of raw cork quality, in terms of thickness and porosity, were observed in productive cork oak ( Quercus suber L.) stands in the Nebrodi and Iblei mountains of Sicily in relation to stand density. Five study areas were chosen across a range of different stand density satisfying specific sampling requirements among two managed cork oak woods.Analysis of variance using Fisher–Snedecor‘s F test ( p < 0.05) were used to identify statistically significant differences of cork quality and sylvicultural, dendrometric and cork stripping parameters between study areas within each stand. Relationships between cork quality and sylvicultural and dendrometric parameters were also tested using Pearson's correlation coefficient ( r). Results shown that sylvicultural practices are a key factor influencing cork quality: statistically significant differences in cork thickness and crown area were observed between areas at different stand density; positive correlations were found between cork thickness and crown area in lower density study areas favouring a significant vegetative activity; negative correlations were found between parameters of cork stripping (cork stripping coefficient, cork stripping surface, maximum height of cork stripping) and cork thickness. No correlations were found between cork porosity and dendrometric or sylvicultural parameters, demonstrating the probable genetic origin of cork porosity. Results shown that cork quality parameters and stand attributes may be used in sustainable management models of cork oak forests.