Learning Block Structures in U-statistic Based Matrices

Biometrika ◽  
2020 ◽  
Author(s):  
Weiping Zhang ◽  
Baisuo Jin ◽  
Zhidong Bai
Keyword(s):  
Matrix Theory ◽  
Block Structure ◽  
Group Structure ◽  
Asymptotic Properties ◽  
Finite Sample ◽  
Eigenvalues And Eigenvectors ◽  
Sample Matrix ◽  
U Statistics ◽  
Large Matrix ◽  
Block Structures

Abstract We introduce a conceptually simple, efficient and easily implemented approach for learning the block structure in a large matrix. Using the properties of U-statistics and large dimensional random matrix theory, the group structure of many variables can be directly identified based on the eigenvalues and eigenvectors of the scaled sample matrix. We also established the asymptotic properties of the proposed approach under mild conditions. The finite-sample performance of the approach is examined by extensive simulations and data examples.

Estimation of error variance via ridge regression

Biometrika ◽  
2020 ◽  
Author(s):  
X Liu ◽  
S Zheng ◽  
X Feng
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Ridge Regression ◽  
Matrix Theory ◽  
Numerical Study ◽  
Asymptotic Properties ◽  
Error Variance ◽  
High Dimensional ◽  
Finite Sample ◽  
Estimation Of Error

Summary We propose a novel estimator of error variance and establish its asymptotic properties based on ridge regression and random matrix theory. The proposed estimator is valid under both low- and high-dimensional models, and performs well not only in nonsparse cases, but also in sparse ones. The finite-sample performance of the proposed method is assessed through an intensive numerical study, which indicates that the method is promising compared with its competitors in many interesting scenarios.

scholarly journals Stratigraphy of amethyst geode-bearing lavas and fault-block structures of the Entre Rios mining district, Paraná volcanic province, southern Brazil

2013 ◽  
Vol 86 (1) ◽  
pp. 187-198 ◽  
Author(s):  
LÉO A. HARTMANN ◽  
LUCAS M. ANTUNES ◽  
LEONARDO M. ROSENSTENGEL
Keyword(s):  
Block Structure ◽  
Underground Mines ◽  
Mining District ◽  
Volcanic Province ◽  
Fault Block ◽  
River Bed ◽  
World Class ◽  
Entre Rios ◽  
Block Structures

The Entre Rios mining district produces a large volume of amethyst geodes in underground mines and is part of the world class deposits in the Paraná volcanic province of South America. Two producing basalt flows are numbered 4 and 5 in the lava stratigraphy. A total of seven basalt flows and one rhyodacite flow are present in the district. At the base of the stratigraphy, beginning at the Chapecó river bed, two basalt flows are Esmeralda, low-Ti type. The third flow in the sequence is a rhyodacite, Chapecó type, Guarapuava subtype. Above the rhyodacite flow, four basalt flows are Pitanga, high-Ti type including the two mineralized flows; only the topmost basalt in the stratigraphy is a Paranapanema, intermediate-Ti type. Each individual flow is uniquely identified from its geochemical and gamma-spectrometric properties. The study of several sections in the district allowed for the identification of a fault-block structure. Blocks are elongated NW and the block on the west side of the fault was downthrown. This important structural characterization of the mining district will have significant consequences in the search for new amethyst geode deposits and in the understanding of the evolution of the Paraná volcanic province.

scholarly journals ALGORITMA POLINOMIAL MINIMUM UNTUK MEMBENTUK MATRIKS DIAGONAL DARI MATRIKS PERSEGI

2017 ◽  
Vol 6 (2) ◽  
pp. 282
Author(s):  
Himmatul Mursyidah
Keyword(s):  
Polynomial Algorithm ◽  
Matrix Theory ◽  
Equation System ◽  
Diagonal Matrix ◽  
Eigenvalues And Eigenvectors ◽  
State Classification ◽  
Linear Equation System ◽  
Special Matrix ◽  
Minimum Polynomial ◽  
Square Matrix

In mathematics, matrices have many uses, they are finding solutions of a linear equation system, looking for specific solutions of differential equations, determining state classification on Markov chains, and so on. There is a special matrix in matrix theory, that is a diagonal matrix. The diagonal matrix is a matrix whose all non-diagonal entries are primarily zero so that the product of the diagonal matrix can be computed by considering only the components along the main diagonal. A square matrix can sometimes be formed into a diagonal matrix. If a non-diagonal square matrix A can be conjugated with a diagonal matrix, then there is an invertible matrix P so PAP-1=D, where D is a diagonal matrix and P is said to diagonalize A. To find a square matrix diagonalizable or not, many researchers usually use eigenvalues and eigenvectors evaluation. In this study, we discuss that the other way to form a diagonal matrix by using Minimum Polynomial Algorithm.

scholarly journals Gradient and Likelihood Ratio Tests in Cure Rate Models

2016 ◽  
Vol 5 (4) ◽  
pp. 9 ◽  
Author(s):  
Hérica P. A. Carneiro ◽  
Dione M. Valença
Keyword(s):  
Likelihood Ratio ◽  
Asymptotic Properties ◽  
Information Matrix ◽  
Survival Models ◽  
Finite Sample ◽  
Cure Rate ◽  
Time Model ◽  
Cure Rate Models ◽  
Large Sample ◽  
Finite Samples

In some survival studies part of the population may be no longer subject to the event of interest. The called cure rate models take this fact into account. They have been extensively studied for several authors who have proposed extensions and applications in real lifetime data. Classic large sample tests are usually considered in these applications, especially the likelihood ratio. Recently  a new test called \textit{gradient test} has been proposed. The gradient statistic shares the same asymptotic properties with the classic likelihood ratio and does not involve knowledge of the information matrix, which can be an advantage in survival models. Some simulation studies have been carried out to explore the behavior of the gradient test in finite samples and compare it with the classic tests in different models. However little is known about the properties of these large sample tests in finite sample for cure rate models. In this work we  performed a simulation study based on the promotion time model with Weibull distribution, to assess the performance of likelihood ratio and gradient tests in finite samples. An application is presented to illustrate the results.

scholarly journals Heterogeneous individual risk modelling of recurrent events

Biometrika ◽  
2020 ◽  
Author(s):  
Huijuan Ma ◽  
Limin Peng ◽  
Chiung-Yu Huang ◽  
Haoda Fu
Keyword(s):  
Recurrent Events ◽  
Asymptotic Properties ◽  
Dynamic Modelling ◽  
Individual Risk ◽  
Simulation Studies ◽  
Finite Sample ◽  
Risk Modelling ◽  
Modelling Framework ◽  
Proposed Model

Summary Progression of chronic disease is often manifested by repeated occurrences of disease-related events over time. Delineating the heterogeneity in the risk of such recurrent events can provide valuable scientific insight for guiding customized disease management. We propose a new sensible measure of individual risk of recurrent events and present a dynamic modelling framework thereof, which accounts for both observed covariates and unobservable frailty. The proposed modelling requires no distributional specification of the unobservable frailty, while permitting exploration of the dynamic effects of the observed covariates. We develop estimation and inference procedures for the proposed model through a novel adaptation of the principle of conditional score. The asymptotic properties of the proposed estimator, including the uniform consistency and weak convergence, are established. Extensive simulation studies demonstrate satisfactory finite-sample performance of the proposed method. We illustrate the practical utility of the new method via an application to a diabetes clinical trial that explores the risk patterns of hypoglycemia in type 2 diabetes patients.

scholarly journals Stability Analysis and Monitoring Method for the Key Block Structure of the Basic Roof of Noncoal Pillar Mining with Automatically Formed Gob-Side Entry

2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Jianning Liu ◽  
Manchao He ◽  
Yajun Wang ◽  
Ruifeng Huang ◽  
Jun Yang ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Coal Mine ◽  
Block Structure ◽  
Monitoring Method ◽  
Roof Caving ◽  
Pillar Mining ◽  
Key Block ◽  
The Stability ◽  
Block Structures ◽  
Basic Roof

The key block of the basic roof is the main contributor to the structural stability of a roadway. Research on the stability of the key block structure is of great significance for the promotion of noncoal pillar mining with automatically formed gob-side entry (GEFANM) technology. This paper is set in the engineering context of the GEFANM experiment at the Ningtiaota Coal Mine. The study fully considered the differences in the gob roof caving on the roof-cutting-line side, and the range of rotation angles to maintain a stable key block was determined. Based on this range of rotation angles, the range of safe bulking coefficients of gangue was calculated. The bulking coefficient of the gangue on the gravel side of the roadway was used as the metric in a new monitoring method and in the calculation of the field parameters. The range of safe bulking coefficients was determined to be 1.40–1.37. Field monitoring was conducted to obtain the gangue bulking coefficient on the gravel side. Combining the roof and floor convergence data, when the bulking coefficient fell within the safe range, the convergence was 95–113 mm. In this stage, the key block structure was stable. When the gangue bulking coefficient fell outside the safe range, the convergence was larger, and cracks were observed. The key block may be vulnerable to instability. The results affirmed that the gangue bulking coefficient can be used as a monitoring metric to study the stability of key block structures.

scholarly journals Estimation of the tail-index in a conditional location-scale family of heavy-tailed distributions

2019 ◽  
Vol 7 (1) ◽  
pp. 394-417
Author(s):  
Aboubacrène Ag Ahmad ◽  
El Hadji Deme ◽  
Aliou Diop ◽  
Stéphane Girard
Keyword(s):  
Asymptotic Properties ◽  
Real Data ◽  
Scale Model ◽  
Extreme Value Index ◽  
Finite Sample ◽  
Scale Functions ◽  
Nonparametric Estimators ◽  
Heavy Tailed Distributions ◽  
Finite Sample Properties ◽  
Heavy Tailed

AbstractWe introduce a location-scale model for conditional heavy-tailed distributions when the covariate is deterministic. First, nonparametric estimators of the location and scale functions are introduced. Second, an estimator of the conditional extreme-value index is derived. The asymptotic properties of the estimators are established under mild assumptions and their finite sample properties are illustrated both on simulated and real data.

QUASI-INDIRECT INFERENCE FOR DIFFUSION PROCESSES

1998 ◽  
Vol 14 (2) ◽  
pp. 161-186 ◽  
Author(s):  
Laurence Broze ◽  
Olivier Scaillet ◽  
Jean-Michel Zakoïan
Keyword(s):  
Diffusion Processes ◽  
Asymptotic Properties ◽  
Estimation Procedure ◽  
Indirect Inference ◽  
Sampled Data ◽  
Finite Sample ◽  
Inference Procedure ◽  
Monte Carlo Experiments ◽  
Simulation Step ◽  
Finite Sample Properties

We discuss an estimation procedure for continuous-time models based on discrete sampled data with a fixed unit of time between two consecutive observations. Because in general the conditional likelihood of the model cannot be derived, an indirect inference procedure following Gouriéroux, Monfort, and Renault (1993, Journal of Applied Econometrics 8, 85–118) is developed. It is based on simulations of a discretized model. We study the asymptotic properties of this “quasi”-indirect estimator and examine some particular cases. Because this method critically depends on simulations, we pay particular attention to the appropriate choice of the simulation step. Finally, finite-sample properties are studied through Monte Carlo experiments.

Eigenvalues and Eigenvectors of a Large Matrix

SIAM Review ◽  
1985 ◽  
Vol 27 (2) ◽  
pp. 250-250
Author(s):  
N. Oden
Keyword(s):  
Eigenvalues And Eigenvectors ◽  
Large Matrix

THE NONSTATIONARY FRACTIONAL UNIT ROOT

1999 ◽  
Vol 15 (4) ◽  
pp. 549-582 ◽  
Author(s):  
Katsuto Tanaka
Keyword(s):  
Unit Root ◽  
Asymptotic Properties ◽  
Test Statistics ◽  
Finite Sample ◽  
Asymptotic Results ◽  
Simulation Experiments ◽  
Normality Assumption ◽  
Asymptotically Efficient ◽  
Uniformly Most Powerful ◽  
Fractional Unit

This paper deals with a scalar I(d) process {yj}, where the integration order d is any real number. Under this setting, we first explore asymptotic properties of various statistics associated with {yj}, assuming that d is known and is greater than or equal to ½. Note that {yj} becomes stationary when d < ½, whose case is not our concern here. It turns out that the case of d = ½ needs a separate treatment from d > ½. We then consider, under the normality assumption, testing and estimation for d, allowing for any value of d. The tests suggested here are asymptotically uniformly most powerful invariant, whereas the maximum likelihood estimator is asymptotically efficient. The asymptotic theory for these results will not assume normality. Unlike in the usual unit root problem based on autoregressive models, standard asymptotic results hold for test statistics and estimators, where d need not be restricted to d ≥ ½. Simulation experiments are conducted to examine the finite sample performance of both the tests and estimators.

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