Tilted Edgeworth expansions for asymptotically normal vectors

2008 ◽  
Vol 62 (6) ◽  
pp. 1113-1142 ◽  
Author(s):  
Christopher S. Withers ◽  
Saralees Nadarajah
Keyword(s):  
Edgeworth Expansions ◽  
Asymptotically Normal ◽  
Normal Vectors

scholarly journals The Asymptotic Local Structure of the Cox Modified Likelihood-Ratio Statistic for Testing Non-Nested Hypotheses

1992 ◽  
Vol 8 (4) ◽  
pp. 553-569 ◽  
Author(s):  
Jerzy Szroeter
Keyword(s):  
Likelihood Ratio ◽  
Local Structure ◽  
Recent Literature ◽  
Likelihood Ratio Statistic ◽  
Close Relative ◽  
Asymptotically Normal ◽  
Data Generating Process ◽  
Normal Vectors ◽  
Modified Likelihood ◽  
Made In

It is shown that the Cox modified likelihood-ratio statistic for testing partially non-nested hypotheses H0 and H1 is asymptotically equivalent to a bilinear form in nondegenerate asymptotically normal random vectors for sequences of data-generating processes converging to the intersection of H0 and H1 but not necessarily belonging to either H0 or H1. One of the asymptotically normal vectors is the complete parametric encompassing vector of Mizon and Richard, while the other is a close relative. The results are valid regardless of whether or not the data-generating process is exponential and imply that the Cox statistic is not generally asymptotically locally normal. This corrects an assumption made in recent literature.

scholarly journals Chebyshev–Edgeworth-Type Approximations for Statistics Based on Samples with Random Sizes

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 775
Author(s):  
Gerd Christoph ◽  
Vladimir V. Ulyanov
Keyword(s):  
Random Sample ◽  
Weighted Sums ◽  
Sample Sizes ◽  
Edgeworth Expansions ◽  
Chi Square ◽  
Limit Laws ◽  
Asymptotically Normal ◽  
Generalized Gamma ◽  
Standard Normal ◽  
T Distribution

Second-order Chebyshev–Edgeworth expansions are derived for various statistics from samples with random sample sizes, where the asymptotic laws are scale mixtures of the standard normal or chi-square distributions with scale mixing gamma or inverse exponential distributions. A formal construction of asymptotic expansions is developed. Therefore, the results can be applied to a whole family of asymptotically normal or chi-square statistics. The random mean, the normalized Student t-distribution and the Student t-statistic under non-normality with the normal limit law are considered. With the chi-square limit distribution, Hotelling’s generalized T02 statistics and scale mixture of chi-square distributions are used. We present the first Chebyshev–Edgeworth expansions for asymptotically chi-square statistics based on samples with random sample sizes. The statistics allow non-random, random, and mixed normalization factors. Depending on the type of normalization, we can find three different limit distributions for each of the statistics considered. Limit laws are Student t-, standard normal, inverse Pareto, generalized gamma, Laplace and generalized Laplace as well as weighted sums of generalized gamma distributions. The paper continues the authors’ studies on the approximation of statistics for randomly sized samples.

scholarly journals A New Class of Estimators Based on a General Relative Loss Function

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1138
Author(s):  
Tao Hu ◽  
Baosheng Liang
Keyword(s):  
Loss Function ◽  
Asymptotically Normal ◽  
New Class ◽  
Statistical Inferences ◽  
Relative Loss ◽  
Special Cases ◽  
Class Of Estimators ◽  
Box Cox Transformation ◽  
Loss Estimator ◽  
Prostate Cancer Study

Motivated by the relative loss estimator of the median, we propose a new class of estimators for linear quantile models using a general relative loss function defined by the Box–Cox transformation function. The proposed method is very flexible. It includes a traditional quantile regression and median regression under the relative loss as special cases. Compared to the traditional linear quantile estimator, the proposed estimator has smaller variance and hence is more efficient in making statistical inferences. We show that, in theory, the proposed estimator is consistent and asymptotically normal under appropriate conditions. Extensive simulation studies were conducted, demonstrating good performance of the proposed method. An application of the proposed method in a prostate cancer study is provided.

scholarly journals Elastic deformation of the mandibular jaw revisited—a clinical comparison between digital and conventional impressions using a reference

2021 ◽  
Author(s):  
Alexander Schmidt ◽  
Leona Klussmann ◽  
Maximiliane A. Schlenz ◽  
Bernd Wöstmann
Keyword(s):  
Statistical Analysis ◽  
Elastic Deformation ◽  
Clinical Relevance ◽  
Mouth Opening ◽  
Reference Structure ◽  
Intraoral Scanner ◽  
Digital Impressions ◽  
Normal Vectors ◽  
Mandibular Deformation ◽  
Conventional Impression

Abstract Objectives Due to the partly strongly differing results in the literature, the aim of the present study was to investigate a possible deformation of the mandible during mouth opening using an intraoral scanner (IOS) and a conventional impression for comparison with a reference aid. Materials and methods Four steel spheres were reversibly luted in the mandibular (n = 50) with a metallic reference aid at maximum mouth opening (MMO). Two digital impressions (Trios3), at MMO and at slightly mouth opening SMO and a conventional impression (Impregum), were taken as the measuring accuracy of the reference structure was already known. Difference between MMO-SMO for digital impressions and deviations between digital and conventional (SMO) were calculated. Furthermore, the angle between the normal vectors of two constructed planes was measured. Statistical analysis was performed with SPSS25. Results Deviations for linear distances ranged from −1 ± 3 μm up to 17 ± 78 μm (digital impressions, MMO-SMO), from 19 ± 16 μm up to 132 ± 90 μm (digital impressions, SMO), and from 28 ± 17 μm up to 60 ± 52 μm (conventional impressions, SMO). There were no significant differences for digital impressions (MMO-SMO), and there were significant differences between the conventional and digital impressions at SMO. Conclusions Based on the results of the present study, no mandibular deformation could be detected during mouth opening with regard to the digital impressions. The results were rather within the measuring tolerance of the intraoral scanner. Clinical relevance Based on the present study, no deformation of the mandibular during mouth opening could be observed at the level previously assumed. Therewith related, dental techniques related to a possible mandibular deformation therefore should be reconsidered.

A Fast and Rigorously Parallel Surface Voxelization Technique for GPU-Accelerated CFD Simulations

2015 ◽  
Vol 17 (5) ◽  
pp. 1246-1270 ◽  
Author(s):  
C. F. Janßen ◽  
N. Koliha ◽  
T. Rung
Keyword(s):  
Graphics Processing Units ◽  
Parallel Implementation ◽  
Parallel Execution ◽  
Time Step ◽  
Cfd Simulations ◽  
Performance Loss ◽  
Neighbor Search ◽  
Body Shell ◽  
Normal Vectors ◽  
Grid Nodes

AbstractThis paper presents a fast surface voxelization technique for the mapping of tessellated triangular surface meshes to uniform and structured grids that provide a basis for CFD simulations with the lattice Boltzmann method (LBM). The core algorithm is optimized for massively parallel execution on graphics processing units (GPUs) and is based on a unique dissection of the inner body shell. This unique definition necessitates a topology based neighbor search as a preprocessing step, but also enables parallel implementation. More specifically, normal vectors of adjacent triangular tessellations are used to construct half-angles that clearly separate the per-triangle regions. For each triangle, the grid nodes inside the axis-aligned bounding box (AABB) are tested for their distance to the triangle in question and for certain well-defined relative angles. The performance of the presented grid generation procedure is superior to the performance of the GPU-accelerated flow field computations per time step which allows efficient fluid-structure interaction simulations, without noticeable performance loss due to the dynamic grid update.

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