Complete asymptotic expansions for the density function of t-distribution

2018 ◽  
Vol 141 ◽  
pp. 1-6 ◽  
Author(s):  
Chao-Ping Chen
1968 ◽  
Vol 8 (3) ◽  
pp. 405-422
Author(s):  
A. Bikelis

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: А. Бикялис. Асимптотические разложения для плотностей и распределений сумм независимых одинаково распределенных случайных векторов A. Bikelis. Nepriklausomų vienodai pasiskirsčiusių atsitiktinių vektorių sumų tankių ir pasiskirstymo funkcijų asimptotiniai išdėstymai


Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 687
Author(s):  
Fode Zhang ◽  
Xiaolin Shi ◽  
Hon Keung Tony Ng

In geometry and topology, a family of probability distributions can be analyzed as the points on a manifold, known as statistical manifold, with intrinsic coordinates corresponding to the parameters of the distribution. Consider the exponential family of distributions with progressive Type-II censoring as the manifold of a statistical model, we use the information geometry methods to investigate the geometric quantities such as the tangent space, the Fisher metric tensors, the affine connection and the α-connection of the manifold. As an application of the geometric quantities, the asymptotic expansions of the posterior density function and the posterior Bayesian predictive density function of the manifold are discussed. The results show that the asymptotic expansions are related to the coefficients of the α-connections and metric tensors, and the predictive density function is the estimated density function in an asymptotic sense. The main results are illustrated by considering the Rayleigh distribution.


Author(s):  
Vitaly Sobolev

Study of estimation of accuracy of approximations in the Central limit theorem (CLT) is one of the known problems in probability theory. The main result here is the estimate of the theorem of Berry — Esseen. Its low accuracy is well known. So this theorem guarantees accuracy of approximation 103 in the CLT only if the number of summands in the normed sum is greater than 160 000. Therefore, increasing the accuracy of the approximations in the CLT is an actual task. In particular, for this purpose are used asymptotic expansions in the Central limit theorem. As a rule, asymptotic expansions have additive form. Although it is possible to construct expansions in the multiplicative form. So V.M. Kalinin in [3] received the multiplicative form of the asymptotic expansions. However, he constructed asymptotic expansions for probability distributions (multinomial, Poisson, Student’s t-distribution). So very naturally the question arises: how to build multiplicative expansions in CLT? Secondly, what are the forms of decompositions in CLT in terms of accuracy approximations are better: additive or multiplicative? This paper proposes new asymptotic expansions in the central limit theorem which permit us to approximate distributions of normalized sums of independent gamma random variables with explicit estimates of the approximation accuracy and comparing them with expansions in terms of Chebyshev — Hermite polynomials. New asymptotic expansions is presented in the following theorem. Comparing multiplicative asymptotic expansion from theorem 1 with the additive asymptotic expansion from [5], we obtain that multiplicative asymptotic expansion of the density of normalized the sums in the case of gamma distribution give a much greater accuracy numerical calculations are compared with asymptotic additive expansion provided a much smaller number of calculations. The author would like to thank Vladimir Senatov for setting the task and paying attention to this work.


Author(s):  
David Cockayne ◽  
David McKenzie

The technique of Electron Reduced Density Function (RDF) analysis has ben developed into a rapid analytical tool for the analysis of small volumes of amorphous or polycrystalline materials. The energy filtered electron diffraction pattern is collected to high scattering angles (currendy to s = 2 sinθ/λ = 6.5 Å-1) by scanning the selected area electron diffraction pattern across the entrance aperture to a GATAN parallel energy loss spectrometer. The diffraction pattern is then converted to a reduced density function, G(r), using mathematical procedures equivalent to those used in X-ray and neutron diffraction studies.Nearest neighbour distances accurate to 0.01 Å are obtained routinely, and bond distortions of molecules can be determined from the ratio of first to second nearest neighbour distances. The accuracy of coordination number determinations from polycrystalline monatomic materials (eg Pt) is high (5%). In amorphous systems (eg carbon, silicon) it is reasonable (10%), but in multi-element systems there are a number of problems to be overcome; to reduce the diffraction pattern to G(r), the approximation must be made that for all elements i,j in the system, fj(s) = Kji fi,(s) where Kji is independent of s.


Statistics ◽  
2003 ◽  
Vol 37 (4) ◽  
pp. 1-1
Author(s):  
A. K. GUPTA
Keyword(s):  

2015 ◽  
Vol 47 (8) ◽  
pp. 24-40 ◽  
Author(s):  
Telman Abbas ogly Aliev ◽  
Naila F. Musaeva ◽  
Matanat Tair kyzy Suleymanova ◽  
Bahruz Ismail ogly Gazizade

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