Theory of best integer equivariant estimation for contaminated normal and multivariate t-distribution with applications

Integer Least Squares ◽

Wide Range ◽

Best Linear Unbiased ◽

Best Linear Unbiased Estimators ◽

T Distribution ◽

Best integer equivariant (BIE) estimators provide minimum mean squared error (MMSE) solutions to the problem of GNSS carrier-phase ambiguity resolution for a wide range of distributions. The associated BIE estimators are universally optimal in the sense that they have an accuracy which is never poorer than that of any integer estimator and any linear unbiased estimator. Their accuracy is therefore always better or the same as that of Integer Least-Squares (ILS) estimators and Best Linear Unbiased Estimators (BLUEs).Current theory is based on using BIE for the multivariate normal distribution. In this contribution this will be generalized to the contaminated normal distribution and the multivariate t-distribution, both of which have heavier tails than the normal. Their computational formulae are presented and discussed in relation to that of the normal distribution. In addition a GNSS real-data based analysis is carried out to demonstrate the universal MMSE properties of the BIE estimators for GNSS-baselines and associated parameters.&#160;Keywords: Integer equivariant (IE) estimation &#183; Best integer equivariant (BIE) &#183; Integer Least-Squares (ILS) . Best linear unbiased estimation (BLUE) &#183; Multivariate contaminated normal &#183; Multivariate t-distribution . Global Navigation Satellite Systems (GNSSs)

Some Extensions of the Multivariate t-Distribution and the Multivariate Generalization of the Distribution of the Regression Coefficient

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100034885 ◽

1961 ◽

Vol 57 (1) ◽

pp. 80-85 ◽

Cited By ~ 31

Author(s):

A. M. Kshirsagar

Keyword(s):

Normal Distribution ◽

Degrees Of Freedom ◽

Regression Coefficient ◽

Bivariate Normal Distribution ◽

Practical Applications ◽

The Matrix ◽

Pre Treatment ◽

T Distribution ◽

If the components x1, x2,…, xk of a vector X have a non-singular multivariate normal distribution having a null vector of means and variance-covariance matrix Σ= σ2, the matrix R=[ρij] (where ρii = 1) is known in certain cases but σ2 is unknown. If s2 is an estimate of σ2 based on ƒ degrees of freedom and is distributed independently of X, the distribution of the vector t=x/s is known as the multivariate t-distribution. This distribution was first obtained by Dunnett and Sobel (6) and independently by Cornish (3). Dunnett, Sobel and Bechhofer(2) have discussed some practical applications of this distribution. Cornish (3) obtained this distribution while considering the pre-treatment to be given to certain types of replicated experiments. This distribution possesses some useful properties and makes it suitable as a basis for exact tests of significance in various problems, and Dunnett and Sobel (6), by providing tables of the probability integral, have taken the first step towards its use in practice. Cornish, in a later paper (4) considered the sampling distribution of statistics derived from the multivariate t-distribution and using this he obtained the well-known ((7), (8)) distribution of the sample regression coefficient of one variate with respect to another, when both have a bivariate normal distribution.

BAYESIAN ANALYSIS OF DYNAMIC FACTOR MODELS USING MULTIVARIATE T DISTRIBUTION

REVISTA BRASILEIRA DE BIOMETRIA ◽

10.28951/rbb.v36i1.155 ◽

2018 ◽

Vol 36 (1) ◽

pp. 140

Author(s):

Larissa Ribeiro de ANDRADE ◽

Daniel Furtado FERREIRA ◽

Thelma SÁFADI ◽

Lúcia Pereira BARROSO

Keyword(s):

Normal Distribution ◽

Factor Model ◽

Graphical Analysis ◽

Dynamic Factor ◽

Multivariate Normal ◽

Chi Square ◽

Convergence Diagnostics ◽

T Distribution ◽

The multivariate t models are symmetric and have heavier tail than the normal distribution and produce robust inference procedures for applications. In this paper, the Bayesian estimation of a dynamic factor model is presented, where the factors follow a multivariate autoregressive model, using the multivariate t distribution. Since the multivariate t distribution is complex, it was represented in this work as a mix of the multivariate normal distribution and a square root of a chi-square distribution. This method allowed the complete dene of all the posterior distributions. The inference on the parameters was made taking a sample of the posterior distribution through a Gibbs Sampler. The convergence was veried through graphical analysis and the convergence diagnostics of Geweke (1992) and Raftery and Lewis (1992).

Calculations Involving the Multivariate Normal and Multivariate t Distributions with and without Truncation

The Stata Journal Promoting communications on statistics and Stata ◽

10.1177/1536867x1801800405 ◽

2018 ◽

Vol 18 (4) ◽

pp. 826-843

Author(s):

Michael J. Grayling ◽

Adrian P. Mander

Keyword(s):

Normal Distribution ◽

Multivariate Normal Distribution ◽

Distribution Functions ◽

Multivariate Normal ◽

Random Vectors ◽

T Distribution ◽

In this article, we present a set of commands and Mata functions to evaluate different distributional quantities of the multivariate normal distribution and a particular type of noncentral multivariate t distribution. Specifically, their densities, distribution functions, equicoordinate quantiles, and pseudo–random vectors can be computed efficiently, in either the absence or the presence of variable truncation.

Adjustment models for multivariate geodetic time series with vector-autoregressive errors

Journal of Applied Geodesy ◽

10.1515/jag-2021-0013 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Boris Kargoll ◽

Alexander Dorndorf ◽

Mohammad Omidalizarandi ◽

Jens-André Paffenholz ◽

Hamza Alkhatib

Keyword(s):

Least Squares Method ◽

Functional Model ◽

Expectation Maximization Algorithm ◽

Two Dimensions ◽

Vector Autoregressive ◽

Wide Range ◽

Engineering Geodesy ◽

Cross Correlations ◽

T Distribution ◽

Abstract In this contribution, a vector-autoregressive (VAR) process with multivariate t-distributed random deviations is incorporated into the Gauss-Helmert model (GHM), resulting in an innovative adjustment model. This model is versatile since it allows for a wide range of functional models, unknown forms of auto- and cross-correlations, and outlier patterns. Subsequently, a computationally convenient iteratively reweighted least squares method based on an expectation maximization algorithm is derived in order to estimate the parameters of the functional model, the unknown coefficients of the VAR process, the cofactor matrix, and the degree of freedom of the t-distribution. The proposed method is validated in terms of its estimation bias and convergence behavior by means of a Monte Carlo simulation based on a GHM of a circle in two dimensions. The methodology is applied in two different fields of application within engineering geodesy: In the first scenario, the offset and linear drift of a noisy accelerometer are estimated based on a Gauss-Markov model with VAR and multivariate t-distributed errors, as a special case of the proposed GHM. In the second scenario real laser tracker measurements with outliers are adjusted to estimate the parameters of a sphere employing the proposed GHM with VAR and multivariate t-distributed errors. For both scenarios the estimated parameters of the fitted VAR model and multivariate t-distribution are analyzed for evidence of auto- or cross-correlations and deviation from a normal distribution regarding the measurement noise.

Robust Curve Clustering Based on a Multivariate $t$-Distribution Model

IEEE Transactions on Neural Networks ◽

10.1109/tnn.2010.2079946 ◽

2010 ◽

Vol 21 (12) ◽

pp. 1976-1984 ◽

Cited By ~ 2

Author(s):

Zhi Min Wang ◽

Qing Song ◽

Yeng Chai Soh ◽

Kang Sim

Keyword(s):

Distribution Model ◽

Curve Clustering ◽

T Distribution ◽

Cumulants of Multivariate Symmetric and Skew Symmetric Distributions

Symmetry ◽

10.3390/sym13081383 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1383

Author(s):

Sreenivasa Rao Jammalamadaka ◽

Emanuele Taufer ◽

Gyorgy H. Terdik

Keyword(s):

Comprehensive Treatment ◽

Multivariate Distributions ◽

Symmetric Distributions ◽

T Distribution ◽

This paper provides a systematic and comprehensive treatment for obtaining general expressions of any order, for the moments and cumulants of spherically and elliptically symmetric multivariate distributions; results for the case of multivariate t-distribution and related skew-t distribution are discussed in some detail.