Izotonická regrese v Sobolevových prostorech

We propose a class of nonparametric estimators for the regression models based on least squares over the sets of sufficiently smooth functions. Least squares permit the imposition of additional constraint-isotonia-on nonparametric regression estimation and testing of this constraint. The estimation takes place over the balls of functions which are elements of a suitable Sobolev space-special types of Hilbert spaces that facilitate calculation of the least squares projection. The Hilbertness is allowing us to take projections and hence to decompose spaces into mutually orthogonal complements. Then we transform the problem of searching for the best fitting function in an infinite dimensional space into a finite dimensional optimization problem. We prove that the balls of functions in Sobolev space are bounded and have bounded higher order derivatives. It permits us to estimate over rich set of functions with sufficiently low metric entropy and apply Laws of Large Numbers and Central Limit Theorems.Uvažme třídu neparametrických odhadů pro regresní modely založené na metodě nejmenších čtverců přes množiny dostatečně hladkých funkcí. Nejmenší čtverce dovolují uložení dodatečného omezení, izotonie, na neparametrické regresní odhady a jejich následné testování. Odhady probíhají přes koule funkcí, které jsou prvky vhodných Sobolevových prostorů. Sobolevovy prostory jsou speciální typ Hilbertových prostorů, které umožňují projekci vzhledem k nejmenší čtvercům. Hilbertovskost nám umožňuje dělat projekci a tedy rozložit prostor do navzájem kolmých doplňků. Pak převedeme problém hledání nejlépe aproximující funkce v prostoru nekonečné dimenze na konečně-dimenzionální optimalizační problém. Dokážeme, že koule funkcí Sobolevových prostorech je omezená a má omezené i derivace vyššího řádu. To nám dovoluje odhadovat přes bohatou množinu funkcí s dostatečně malou metrickou entropií a použít zákony velkých čísel a centrální limitní věty.Katedra pravděpodobnosti a matematické statistikyDepartment of Probability and Mathematical StatisticsMatematicko-fyzikální fakultaFaculty of Mathematics and Physic

Metformin treatment for diabetes mellitus correlates with progression and survival in colorectal carcinoma

BACKGROUND: Diabetes mellitus is unfavorably associated with cancer risk. The purpose of this multi-disciplinary project was to evaluate a possible association of diabetes mellitus and other comorbidities and their treatment with progression of colorectal cancer. PATIENTS AND METHODS: We investigated the correlation between pathological characteristics and clinical course, including comorbidities in 1004 Czech patients diagnosed and surgically treated for colorectal adenocarcinoma (CRC) between 1999 and 2016. RESULTS: In our data set, CRC patients treated with metformin due to coexisting diabetes mellitus type 2 (T2DM) developed fewer distant metastases which clinically correlates with slower CRC progression. Survival in metformin subgroup was longer, particularly in men with CRC. Osteoporosis may be a negative factor of survival in CRC patients. CONCLUSIONS: Our findings also indicate that aging, higher tumor grade and TNM stage, coexistence of selected endocrine disorders, and metabolic abnormalities may change the tumor microenvironment and impact survival in colorectal cancer, although mechanism of these observations yet to be explained. Patients with diabetes mellitus type 2 treated with metformin may represent the altered microenvironment with specifically tuned metabolic molecular responses and with various epigenetic characteristics. More awareness and increased understanding of the mechanisms underlying the positive effect of metformin on patients' survival could offer insight into new treatment methods and permit more individualized treatment plans.Web of Science13239238

Stressful Daylight: Differences in Diel Rhythmicity Between Albino and Pigmented Fish

In laboratory experiments, variously colored strains of animals, including those with albino phenotypes, are commonly used. The melanocortin theory suggests, however, that coloration phenotypes alter animal physiology and behavior. Animals with the albino phenotype show photoreceptor degradation associated with lowered visual accuracy, escape reactions, etc., presumably accompanied by prevailing nocturnal activity and lowered aggressiveness. This assumption was tested in small groups of albino and pigmented European catfish, Silurus glanis, during the diel cycle. The frequency of agonistic interactions was observed during mutual contests for shelters, and subsequently, blood plasma, brain, gill, and liver samples were collected to evaluate stress parameters. In an experimental arena with shelters, the light/dark rhythmicity of locomotor activity and aggressiveness of the two phenotypes were comparable; the peak was observed at night, and a lower peak was observed at dawn. In an experimental stream without shelters, the peak of locomotor activity occurred at night for only the pigmented phenotype. In the evaluation of 4 antioxidants and 1 oxidative stress indicator, representing a total of 15 indices, albino fish showed significant rhythmicity for 8 indices, whereas pigmented catfish showed significant rhythmicity for 5 indices. The production of blood stress parameters with the peak during the day occurred only in albino fish. A complex model was fitted with the aim of evaluating the links between behavioral and biochemical indices. Time periodicity was modeled using a sine wave and confirmed parallel courses of agonistic interactions in the catfish groups; the peak at dawn was associated with a 4.08-fold (conf. int. 3.53–4.7) increase in such interactions. The changes in glucose and superoxide dismutase concentrations varied with phenotype, while the effects of cortisol, lactate and catalase did not. In summary, the rhythmicity of locomotor activity and changes in the aggressiveness of catfish were influenced by shelter availability, and the effect of light-induced stress was more apparent in albino fish than in pigmented conspecific fish. The results suggested that laboratory-raised animals with pigmentation patterns naturally occurring in the wild show more reasonable values during experiments than those with an albino phenotype

Constrained General Regression in Pseudo-Sobolev Spaces with Application to Option Pricing

State price density (SPD) contains important information concerning market expectations. In existing literature, a constrained estimator of the SPD is found by nonlinear least squares in a suitable Sobolev space. We improve the behavior of this estimator by implementing a covariance structure taking into account the time of the trade and by considering simultaneously both the observed Put and Call option prices

Modern Asymptotic Perspectives on Errors-in-variables Modeling

Charles University in Prague Faculty of Mathematics and Physics ABSTRACT OF DOCTORAL THESIS Michal Pešta MODERN ASYMPTOTIC PERSPECTIVES ON ERRORS-IN-VARIABLES MODELING A linear regression model, where covariates and a response are subject to errors, is considered in this thesis. For so-called errors-in-variables (EIV) model, suitable error structures are proposed, various unknown parameter estimation techniques are performed, and recent algebraic and statistical results are summarized. An extension of the total least squares (TLS) estimate in the EIV model-the EIV estimate-is in- vented. Its invariant (with respect to scale) and equivariant (with respect to the covariates' rotation, to the change of covariates direction, and to the interchange of covariates) properties are derived. Moreover, it is shown that the EIV estimate coincides with any unitarily invariant penalizing solution to the EIV problem. It is demonstrated that the asymptotic normality of the EIV estimate is computationally useless for a construction of confidence intervals or hypothesis testing. A proper bootstrap procedure is constructed to overcome such an issue. The validity of the bootstrap technique is proved. A simulation study and a real data example assure of its appropriateness. Strong and uniformly strong mixing errors are taken..