scholarly journals ESTIMATES OF DERIVATIVES OF (LOG) DENSITIES AND RELATED OBJECTS

2021 ◽  
pp. 1-36
Author(s):  
Joris Pinkse ◽  
Karl Schurter
Keyword(s):  
Likelihood Estimation ◽  
Alternative Methods ◽  
Asymptotic Performance ◽  
Density Estimates ◽  
Optimal Rate Of Convergence ◽  
Local Polynomial ◽  
Finite Samples ◽  
Unknown Density ◽  
Optimal Inputs ◽  
Derivatives Of

We estimate the density and its derivatives using a local polynomial approximation to the logarithm of an unknown density function f. The estimator is guaranteed to be non-negative and achieves the same optimal rate of convergence in the interior as on the boundary of the support of f. The estimator is therefore well-suited to applications in which non-negative density estimates are required, such as in semiparametric maximum likelihood estimation. In addition, we show that our estimator compares favorably with other kernel-based methods, both in terms of asymptotic performance and computational ease. Simulation results confirm that our method can perform similarly or better in finite samples compared to these alternative methods when they are used with optimal inputs, that is, an Epanechnikov kernel and optimally chosen bandwidth sequence. We provide code in several languages.

scholarly journals Numerical differentiation on scattered data through multivariate polynomial interpolation

2021 ◽  
Author(s):  
Francesco Dell’Accio ◽  
Filomena Di Tommaso ◽  
Najoua Siar ◽  
Marco Vianello
Keyword(s):  
Polynomial Interpolation ◽  
Numerical Differentiation ◽  
Scattered Data ◽  
Monomial Basis ◽  
Multivariate Polynomial Interpolation ◽  
Differentiation Formula ◽  
Local Polynomial ◽  
Leja Points ◽  
Derivatives Of ◽  
Numerical Differentiation Formula

AbstractWe discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor’s formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.

scholarly journals The Kumaraswamy Exponentiated U-Quadratic Distribution: Properties and Application

2018 ◽  
pp. 1-17
Author(s):  
Mustapha Muhammad ◽  
Isyaku Muhammad ◽  
Aisha Muhammad Yaya
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Likelihood Estimation ◽  
Quantile Function ◽  
Moment Generating Function ◽  
Maximum Likelihood Estimates ◽  
Alternative Methods ◽  
Finite Sample ◽  
Modified Maximum Likelihood ◽  
Probability Weighted Moments

In this paper, a new lifetime model called Kumaraswamy exponentiated U-quadratic (KwEUq) distribution is proposed. Several mathematical and statistical properties are derived and studied such as the explicit form of the quantile function, moments, moment generating function, order statistics, probability weighted moments, Shannon entropy and Renyi entropy. We also found that the usual maximum likelihood estimates (MLEs) fail to hold for the KwEUq distribution. Two alternative methods are suggested for the parameter estimation of the KwEUq, the alternative maximum likelihood estimation (AMLE) and modified maximum likelihood estimation (MMLE). Simulation studies were conducted to assess the finite sample behavior of the AMLEs and MMLEs. Finally, we provide application of the KwEUq for illustration purposes.

scholarly journals Application of Non-polynomial Splines to Solving Differential Equations

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Heat Conduction Problem ◽  
Numerical Differentiation ◽  
Exponential Polynomial ◽  
Order Of Approximation ◽  
Polynomial Splines ◽  
Implicit Difference Scheme ◽  
Local Polynomial ◽  
Second Derivatives ◽  
Implicit Difference Schemes ◽  
Derivatives Of

The application of the local polynomial and non-polynomial to the construction of methods for numerically solving the heat conduction problem is discussed. The non-polynomial splines are used here to approximate the partial derivatives. Formulas for numerical differentiation based on the application of the nonpolynomial splines of the fourth order of approximation are constructed. Particular attention is paid to polynomial, trigonometric, exponential, polynomial-trigonometric and polynomial-exponential splines. This approach allows us to construct explicit and implicit difference schemes. The main focus of the paper is on implicit difference scheme. New approximations with splines of the Lagrange and Hermite type with new properties are obtained. These approximations take into account the first and second derivatives of the function being approximated. Numerical examples are given.

Estimation of the integral of the square of derivatives of symmetric probability densities of one-dimensional random variables

Metrologiya ◽  
2020 ◽  
pp. 15-27
Author(s):  
Aleksandr V. Lapko ◽  
Vasiliy A. Lapko
Keyword(s):  
Standard Deviation ◽  
Probability Density ◽  
Random Variables ◽  
Random Variable ◽  
Density Estimates ◽  
One Dimensional ◽  
Probability Densities ◽  
The One ◽  
Derivatives Of ◽  
Selection Of

When substantiating the method of fast selection of the bandwidth of kernel probability density estimates, a constant was found that is a functional of the second density derivative. In this paper, the obtained result is generalized to derivatives of symmetric probability densities of different orders. The functional dependences of the constants under study on the coeffi cient of antikurtosis of a random variable are established. The regularities peculiar to them are investigated. Based on the results obtained, a method for estimating functionals from derived probability densities has been developed, which involves the following actions. In the original sample estimated standard deviation of the one-dimensional random variables and the coeffi cient of antikurtosis. Using the reconstructed functional dependences on the antikurtosis coeffi cient, the constants are estimated, which are functionals of the derivatives of the probability density. With known estimates of the standard deviation of the investigated random variable and the considered constant, the values of the functional from the derivative of the probability density of the selected order are calculated. The obtained results are confi rmed by the analysis of the data of computational experiments. It is established that with increasing order of the derivative, the values of the estimates of the studied functionals increase. This fact is explained by the complication of the integrand function in the considered functionals. The proposed method provides objective results for the fi rst three derivatives of the probability density of a random variable. The obtained conclusions are confi rmed by the results of the confi dence estimation of the investigated functionals.

11,13-Methylenation of sesquiterpene exomethylene-γ-lactones: Synthesis of sesquiterpene cyclopropyl-γ-lactones

1991 ◽  
Vol 56 (5) ◽  
pp. 1052-1063 ◽  
Author(s):  
Giovanni Appendino ◽  
Elisabetta Ghilardi ◽  
Giancarlo Cravotto ◽  
Pierluigi Gariboldi
Keyword(s):  
Comparative Study ◽  
Silica Gel ◽  
Good Yield ◽  
Alternative Methods ◽  
Basic Alumina ◽  
Alkylation Reactions ◽  
Derivatives Of

11,13-Methylene derivatives of sesquiterpene exomethylene-γ-lactones were prepared by reaction of diazomethane and photolysis of the resulting pyrazoline adduct(s). Variable amounts of exomethylene- and ethylidene-γ-lactones were also formed in the photolysis reaction. These compounds could bee removed after reaction with diethylamine and chromatography on a bilayer (basic alumina/silica gel) column. Also 11,13-methylene derivatives of acid-and thermolabile germacranolides could be prepared in good yield with this procedure. Results obtained using alternative methods of methylenation are reported. A comparative study of the reactivity towards nucleophiles of exomethylene- and cyclopropyl sesquiterpene-γ-lactones provided little support for the involvement of compounds of the latter class in in vivo alkylation reactions.

Monocyclic and cascade rearrangements of furoxans

2004 ◽  
Vol 76 (9) ◽  
pp. 1691-1703 ◽  
Author(s):  
N. N. Makhova ◽  
I. V. Ovchinnikov ◽  
A. S. Kulikov ◽  
S. I. Molotov ◽  
E. L. Baryshnikova
Keyword(s):  
Triazole Ring ◽  
Alternative Methods ◽  
Alkyl Aryl ◽  
Thermally Induced ◽  
Triazole Derivatives ◽  
The Third ◽  
Azo Group ◽  
Derivatives Of ◽  
Cascade Type

Monocyclic rearrangements of azoles are extensively studied as alternative methods for the preparation of new heterocyclic systems. The present work is devoted to investigation of monocyclic and cascade rearrangements of 1,2,5-oxadiazole 2-oxide (furoxan) derivatives. It was found during investigations that rearrangements of furoxan ring had some peculiarities in comparison with analogous rearrangements of other azoles. Therefore, three different kinds of rearrangements were found. The first of them occurred through a dinitroso-ethylene intermediate and resulted in the synthesis of 1,2,3-triazole 1-oxides [oximes of 5-acetyl-4-phenyl(methyl)-2-phenyl-2H -1,2,3-triazole 1-oxides and 2-(furoxan-4-yl)-4-nitro-5-R-2H -1,2,3-triazole 1-oxides ]by thermal recyclization accordingly of 3-methyl-4-acetyl(benzoyl)furoxans phenylhydrazones and 3,3'-(R)-disubstituted-4,4'-azofuroxans. The latter reaction was performed in an oxidizing medium. The second kind of rearrangement (classical variant) was the synthesis of new azoles containing the 1-nitroalkyl substituent. These rearrangements were performed using three examples: base-induced interconversion of furoxanyl ketone phenylhydrazones into 5-(1-nitroalkyl)-2H-1,2,3-triazole derivatives and of 1-alkyl(aryl)-3-(furoxan-4-yl)amidines into 1-substituted 3-(1-nitroalkyl)-1,2,4-triazoles as well as a thermally induced rearrangement of 4-thioureido-3-R-furoxans into derivatives of 5-amino-3-(1-nitroalkyl)-1,2,4-thiadiazole including (5-amino-1,2,4-thiadiazol-3-yl)nitro-formaldehyde arylhydrazones (where R =N=N –Ar). Rearrangements of the third kind were those of the cascade type. Three new cascade rearrangements of azofuroxan derivatives [3,3'-azo-4,4'-bis(acetylamino)furoxans, 3-arylazo-4-acetylaminofuroxans, and 3-arylazo-4-(3- ethoxycarbonylureido)furoxans] into 4-amino-5-nitro-2H-1,2,3-triazole derivatives were discovered. These three reactions were assumed to include two consecutive (cascade) rearrangements: a 1,2,4-oxadiazole ring was formed at the first step and then transformed into a 1,2,3-triazole ring with the participation of an azo group.

scholarly journals A Fourier inversion method for the estimation of a density and its derivatives

1977 ◽  
Vol 23 (2) ◽  
pp. 166-171 ◽  
Author(s):  
J. Blum ◽  
V. Susarla
Keyword(s):  
Almost Sure Convergence ◽  
Characteristic Functions ◽  
Inversion Method ◽  
Fourier Inversion ◽  
Bivariate Normal ◽  
Unknown Density ◽  
The Mean ◽  
Image Position ◽  
Almost All ◽  
Derivatives Of

During the last two decades a variety of methods have been developed for the problem of estimation of unknown density f wrt Lebesgue measure and its vth dervative g(= f(v)) using i.i.d random variables X1, …, Xn when X1 ∼ f. For example, see Wegman (1972). In almost all the papers on the estimation of f(x) or g(x), various authors assumed the existence of derivatives of f of order r(>v) at x to obtain rates for the mean-square convergences and other desirable properties for their estimators. Here it is shown that ifthen estimators gˇ(x) can be constructed for which E[(gˇ(x)-g(x))2] = O(n−(2a-δ)/(2a+2v+1)) for anygiveng δ > 0. Simiar statements hold for almost sure convergence of gˇ(x). It can also be shown that (gˇ(x1), gˇ(x2)) is asymptotically bivariate normal under certain conditions for x1, ≠ x2. If (Al) is satisfied with a ≦ l, then our estimators have all the desirable properties while other methods are not applicable in this situation since they require differentiability conditions on g. (For example, see Susarla and Kumar (1975) and its references.) Our estimators are defined by using the inversion theorem for some absolutely integrable characteristic functions. The motivation for our estimators is given in O'Bryan and Susarla (1975, 76) and Susarla and O'Bryan (1975).

scholarly journals ALTERNATIVE METHODS OF SYNTHESIS OF NOVEL HETEROSYNTHONES – FUNCTIONALIZED HYPOXANTHINE PYRIMIDINES

2021 ◽  
Vol 26 (2(78)) ◽  
pp. 32-39
Author(s):  
V. M. Povstyanoy ◽  
T. A. Yuyrova ◽  
A. N. Retchitskiy ◽  
A. A. Krysko
Keyword(s):  
Wide Spectrum ◽  
Physiological Activity ◽  
Biginelli Reaction ◽  
Aromatic Aldehydes ◽  
13C Nmr Spectroscopy ◽  
Alternative Methods ◽  
Reaction Products ◽  
Optimal Method ◽  
Methylene Bridge ◽  
Derivatives Of

It has been known that derivatives of dihydropyrimidine and xanthine possess the physiological activity of the wide spectrum of action. The combination of the specified heterocyclic fragments within one molecule can lead to the increase of its known types of biological activity as well as to the discovery of novel types of activity. We have previously reported the synthesis of intermediates, which consist of functionalized dihydropyrimidines, connected via a methylene bridge with the halogen substituted derivatives of the ophylline, 3-methylxanthine and imidazole. It was also observed that these compounds would react with N‑nucleophiles with the formation of various branched and cyclic products. The aim of this work was to determine the optimal conditions for obtaining heterocyclic products as a result of conjugation of bromomethyldihydropyrimidine and hypoxanthine at the positions C6 and C1 via a methylene bridge. It is important to note, that the latter can be widely modified by using structurally diverse aromatic aldehydes during the synthesis of dihydropyrimidine core by using Biginelli reaction, which explains structural diversity of the reaction products. After having tried various reaction conditions, we have concluded that the optimal method for obtaining the products entailed keeping equimolar ratios of bromomethyl substituted dihydropyrimidines and 1-potassium‑2-chloro‑7-methylhypoxanthine in dry ethanol for 4 hours. The ester group at the C5 of dihydropyrimidine ring and a chlorine leaving group at C2 of hypoxanthine fragments of the molecule allow to consider these structures as pro missing synthons for farther synthesis of condensed pyrimidine and xanthine systems. The structures of novel compounds have been confirmed with the methods such as HPLC/MS, 1H, and 13C NMR spectroscopy.

scholarly journals cRacle: R Tools for Estimating Climate from Vegetation

2019 ◽  
Author(s):  
Robert S. Harbert ◽  
Alex A. Baryiames
Keyword(s):  
Plant Community ◽  
Best Practice ◽  
Likelihood Estimation ◽  
Data Access ◽  
Alternative Methods ◽  
Model Parameters ◽  
Mean Annual Temperature ◽  
Modeling Tools ◽  
Model Correction ◽  
Community Data

ABSTRACTPremise of the studyThe Climate Reconstruction Analysis using Coexistence Likelihood Estimation (CRACLE) method for the estimation of climate from vegetation is a robust set of modeling tools for estimating climate and paleoclimate that makes use of large repositories of biodiversity data and open-access R software.MethodsHere we implement a new R library for the estimation of climate from vegetation. The ‘cRacle’ library implements functions for data access, aggregation, and models to estimate climate from plant community composition. ‘cRacle’ is modular and features many best-practice features.ResultsPerformance tests using modern vegetation survey data from North and South America shows that CRACLE outperforms alternative methods. CRACLE estimates of mean annual temperature (MAT) are usually within 1°C of the actual when optimal model parameters are used. Generalized Boosted Regression (GBR) model correction is also shown here to improve on CRACLE models by reducing bias.DiscussionCRACLE provides accurate estimates of climate from modern plant communities. Non-parametric CRACLE modeling coupled to GBR model correction produces the most accurate results to date. The ‘cRacle’ R library streamlines the estimation of climate from plant community data, and will make this modeling more accessible to a wider range of users.

Sign in / Sign up

Export Citation Format

Share Document