Outlier Detection

2018 ◽  
Author(s):  
Bent Nielsen
Keyword(s):  
Asymptotic Theory ◽  
Empirical Process ◽  
Robust Statistics ◽  
Formal Theory ◽  
Simulation Studies ◽  
Asymptotic Results ◽  
Forthcoming Article ◽  
Empirical Process Theory ◽  
Impulse Indicator Saturation ◽  
Detection Of Outliers

This is an advance summary of a forthcoming article in the Oxford Research Encyclopedia of Economics and Finance. Please check back later for the full article. Detection of outliers is an important explorative step in empirical analysis. Once detected, the investigator will have to decide how to model the outliers depending on the context. Indeed, the outliers may represent noisy observations that are best left out of the analysis or they may be very informative observations that would have a particularly important role in the analysis. For regression analysis in time series a number of outlier algorithms are available, including impulse indicator saturation and methods from robust statistics. The algorithms are complex and their statistical properties are not fully understood. Extensive simulation studies have been made, but the formal theory is lacking. Some progress has been made toward an asymptotic theory of the algorithms. A number of asymptotic results are already available building on empirical process theory.

Standard Asymptotic Theory

2017 ◽  
Author(s):  
Russell Cheng
Keyword(s):  
Asymptotic Theory ◽  
Asymptotic Variance ◽  
Statistical Significance ◽  
Regularity Conditions ◽  
Parameter Estimates ◽  
Asymptotic Results ◽  
Ml Estimation ◽  
True Parameter ◽  
Log Likelihood ◽  
Ml Estimator

This book relies on maximum likelihood (ML) estimation of parameters. Asymptotic theory assumes regularity conditions hold when the ML estimator is consistent. Typically an additional third derivative condition is assumed to ensure that the ML estimator is also asymptotically normally distributed. Standard asymptotic results that then hold are summarized in this chapter; for example, the asymptotic variance of the ML estimator is then given by the Fisher information formula, and the log-likelihood ratio, the Wald and the score statistics for testing the statistical significance of parameter estimates are all asymptotically equivalent. Also, the useful profile log-likelihood then behaves exactly as a standard log-likelihood only in a parameter space of just one dimension. Further, the model can be reparametrized to make it locally orthogonal in the neighbourhood of the true parameter value. The large exponential family of models is briefly reviewed where a unified set of regular conditions can be obtained.

scholarly journals A Diagnostic for Seasonality Based Upon Polynomial Roots of ARMA Models

2021 ◽  
Vol 37 (2) ◽  
pp. 367-394
Author(s):  
Tucker McElroy
Keyword(s):  
Moving Average ◽  
Type I ◽  
Simulation Studies ◽  
Arma Models ◽  
Asymptotic Results ◽  
Global Test ◽  
Raw Data ◽  
Polynomial Roots ◽  
Oscillatory Character ◽  
Statistical Agencies

Abstract Methodology for seasonality diagnostics is extremely important for statistical agencies, because such tools are necessary for making decisions whether to seasonally adjust a given series, and whether such an adjustment is adequate. This methodology must be statistical, in order to furnish quantification of Type I and II errors, and also to provide understanding about the requisite assumptions. We connect the concept of seasonality to a mathematical definition regarding the oscillatory character of the moving average (MA) representation coefficients, and define a new seasonality diagnostic based on autoregressive (AR) roots. The diagnostic is able to assess different forms of seasonality: dynamic versus stable, of arbitrary seasonal periods, for both raw data and seasonally adjusted data. An extension of the AR diagnostic to an MA diagnostic allows for the detection of over-adjustment. Joint asymptotic results are provided for the diagnostics as they are applied to multiple seasonal frequencies, allowing for a global test of seasonality. We illustrate the method through simulation studies and several empirical examples.

Statistical spatial series modelling II: Some further results on unilateral lattice processes

1983 ◽  
Vol 15 (3) ◽  
pp. 562-584 ◽  
Author(s):  
Dag Tjøstheim
Keyword(s):  
Time Series ◽  
Half Space ◽  
Special Interest ◽  
Asymptotic Theory ◽  
Innovation Process ◽  
Asymptotic Results ◽  
Practical Applications ◽  
Spatial Series ◽  
Lattice Processes ◽  
Series Criteria

An asymptotic theory of estimation is developed for classes of spatial series F(x1, · ··, xn), where (x1, · ··, xn) varies over a regular cartesian lattice. Two classes of unilateral models are studied, namely half-space models and causal (quadrant-type) models. It is shown that a number of asymptotic results are common for these models. Of special interest for practical applications is the problem of determining how many parameters should be included to describe the degree of dependence in each direction. Here we are able to obtain weakly consistent generalizations of familiar time-series criteria under the assumption that the generating variables of the model are independently and identically distributed. For causal models we introduce the concepts of spatial innovation process and lattice martingale and use these to extend some of the asymptotic theory to the case where a certain type of dependence is permitted in the generating variables.

scholarly journals Detection of Outliers in the Volatility of Malaysia Shariah Compliant Index Return: The Impulse Indicator Saturation Approach

2020 ◽  
pp. 1-7
Author(s):  
Ida Normaya Mohd Nasir ◽  
Mohd Tahir Ismail
Keyword(s):  
Time Series Data ◽  
Global Financial Crisis ◽  
Financial Time Series ◽  
The United States ◽  
Series Data ◽  
Crude Oil Prices ◽  
Global Events ◽  
Using Data ◽  
Impulse Indicator Saturation ◽  
Detection Of Outliers

Financial time series data often affected by various unexpected events which known as the outliers. The aim of this study is to detect the outliers in high frequency data using Impulse Indicator Saturation approach (IIS).Monte Carlo simulations illustrate the ability of IIS to detect outliers by using data with various simulation settings. For empirical application, we have chosen the Malaysia Shariah compliant index which is the FBM EMAS Shariah (FBMS) index. The result of this study discovered the presence of 47 outliers which related to several global events such as global financial crisis (2008 & 2009), the falling of stock market (2011), the United States debt-ceiling crisis (2013) and the declination of international crude oil prices (2014). Keywords: outliers; volatility; stock indices; IIS

On resampling schemes for polytopes

2019 ◽  
Vol 56 (4) ◽  
pp. 959-980
Author(s):  
Weinan Qi ◽  
Mahmoud Zarepour
Keyword(s):  
Convex Hull ◽  
Asymptotic Theory ◽  
Convex Set ◽  
Real Life ◽  
Convex Hulls ◽  
Asymptotic Results ◽  
Underlying Distribution ◽  
Asymptotic Consistency ◽  
Computational Intractability ◽  
Practical Implications

AbstractThe convex hull of a sample is used to approximate the support of the underlying distribution. This approximation has many practical implications in real life. To approximate the distribution of the functionals of convex hulls, asymptotic theory plays a crucial role. Unfortunately most of the asymptotic results are computationally intractable. To address this computational intractability, we consider consistent bootstrapping schemes for certain cases. Let $S_n=\{X_i\}_{i=1}^{n}$ be a sequence of independent and identically distributed random points uniformly distributed on an unknown convex set in $\mathbb{R}^{d}$ ($d\ge 2$ ). We suggest a bootstrapping scheme that relies on resampling uniformly from the convex hull of $S_n$ . Moreover, the resampling asymptotic consistency of certain functionals of convex hulls is derived under this bootstrapping scheme. In particular, we apply our bootstrapping technique to the Hausdorff distance between the actual convex set and its estimator. For $d=2$ , we investigate the asymptotic consistency of the suggested bootstrapping scheme for the area of the symmetric difference and the perimeter difference between the actual convex set and its estimate. In all cases the consistency allows us to rely on the suggested resampling scheme to study the actual distributions, which are not computationally tractable.

scholarly journals The linear spin-up of a stratified, rotating fluid in a square cylinder

2012 ◽  
Vol 712 ◽  
pp. 7-40 ◽  
Author(s):  
M. R. Foster ◽  
R. J. Munro
Keyword(s):  
Time Scale ◽  
Asymptotic Theory ◽  
Horizontal Velocity ◽  
Rotating Fluid ◽  
Asymptotic Results ◽  
Ekman Layers ◽  
Starting Flow ◽  
Particle Imaging ◽  
Diffusive Time ◽  
Theoretical Results

AbstractHere we present experimental and theoretical results for how a stratified fluid, initially rotating as a solid body with constant angular velocity, $\Omega $, within a closed cylinder of square cross-section, is spun up when subject to a small, impulsive increase, $ \mrm{\Delta} \Omega $, in the cylinder’s rotation rate. The fluid’s adjustment to the new state of solid rotation can be characterized by: (a) an inviscid, horizontal starting flow which conserves the vorticity of the initial condition; (b) the eruption of Ekman layer fluid from the perimeter region of the cylinder’s base and lid; (c) horizontal-velocity Rayleigh layers that grow into the interior from the container’s sidewalls; and (d) the formation and decay of columnar vortices in the vertical corner regions. Asymptotic results describe the inviscid starting flow, and the subsequent interior spin-up that occurs due to the combined effects of Ekman suction through the base and lid Ekman layers, and the growth of the sidewall Rayleigh layers. Attention is focused on the flow development over the spin-up time scale ${T}_{s} = {E}^{\ensuremath{-} 1/ 2} {\Omega }^{\ensuremath{-} 1} $, where $E$ is the Ekman number. (The spin-up process over the much longer diffusive time scale, ${T}_{d} = {E}^{\ensuremath{-} 1} {\Omega }^{\ensuremath{-} 1} $, is not considered here.) Experiments were performed using particle imaging velocimetry (PIV) to measure horizontal velocity components at fixed heights within the flow interior and at regular stages during the spin-up period. The velocity data obtained are shown to be in excellent agreement with the asymptotic theory.

scholarly journals Asymptotic results for the empirical process of stationary sequences

2009 ◽  
Vol 119 (4) ◽  
pp. 1298-1324 ◽  
Author(s):  
István Berkes ◽  
Siegfried Hörmann ◽  
Johannes Schauer
Keyword(s):  
Empirical Process ◽  
Stationary Sequences ◽  
Asymptotic Results

scholarly journals General Bootstrap for Dual ϕ-Divergence Estimates

2012 ◽  
Vol 2012 ◽  
pp. 1-33 ◽  
Author(s):  
Salim Bouzebda ◽  
Mohamed Cherfi
Keyword(s):  
Coverage Probability ◽  
Empirical Process ◽  
Asymptotic Properties ◽  
Process Theory ◽  
General Notion ◽  
Finite Sample ◽  
Confidence Set ◽  
Empirical Process Theory ◽  
Divergence Estimates ◽  
Simulation Results

A general notion of bootstrappedϕ-divergence estimates constructed by exchangeably weighting sample is introduced. Asymptotic properties of these generalized bootstrappedϕ-divergence estimates are obtained, by means of the empirical process theory, which are applied to construct the bootstrap confidence set with asymptotically correct coverage probability. Some of practical problems are discussed, including, in particular, the choice of escort parameter, and several examples of divergences are investigated. Simulation results are provided to illustrate the finite sample performance of the proposed estimators.

scholarly journals Threshold behaviour and final outcome of an epidemic on a random network with household structure

2009 ◽  
Vol 41 (3) ◽  
pp. 765-796 ◽  
Author(s):  
Frank Ball ◽  
David Sirl ◽  
Pieter Trapman
Keyword(s):  
Limit Theorems ◽  
Branching Process ◽  
Random Network ◽  
Household Structure ◽  
Simulation Studies ◽  
Threshold Parameter ◽  
Threshold Behaviour ◽  
Asymptotic Results ◽  
Social Contacts ◽  
Major Outbreak

In this paper we consider a stochastic SIR (susceptible→infective→removed) epidemic model in which individuals may make infectious contacts in two ways, both within ‘households’ (which for ease of exposition are assumed to have equal size) and along the edges of a random graph describing additional social contacts. Heuristically motivated branching process approximations are described, which lead to a threshold parameter for the model and methods for calculating the probability of a major outbreak, given few initial infectives, and the expected proportion of the population who are ultimately infected by such a major outbreak. These approximate results are shown to be exact as the number of households tends to infinity by proving associated limit theorems. Moreover, simulation studies indicate that these asymptotic results provide good approximations for modestly sized finite populations. The extension to unequal-sized households is discussed briefly.

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