scholarly journals Bootstrap With Cluster‐Dependence in Two or More Dimensions

Econometrica ◽  
2021 ◽  
Vol 89 (5) ◽  
pp. 2143-2188
Author(s):  
Konrad Menzel
Keyword(s):  
Asymptotic Distribution ◽  
Limiting Distribution ◽  
Network Data ◽  
Sample Mean ◽  
Uniform Consistency ◽  
Bootstrap Procedure ◽  
Non Gaussian ◽  
Matched Data

We propose a bootstrap procedure for data that may exhibit cluster‐dependence in two or more dimensions. The asymptotic distribution of the sample mean or other statistics may be non‐Gaussian if observations are dependent but uncorrelated within clusters. We show that there exists no procedure for estimating the limiting distribution of the sample mean under two‐way clustering that achieves uniform consistency. However, we propose bootstrap procedures that achieve adaptivity with respect to different uniformity criteria. Important cases and extensions discussed in the paper include regression inference, U‐ and V‐statistics, subgraph counts for network data, and non‐exhaustive samples of matched data.

scholarly journals The Alpha stable distribution in ocean ambient noise modelling

2019 ◽  
Vol 283 ◽  
pp. 08002
Author(s):  
Guoli Song ◽  
Xinyi Guo ◽  
Li Ma
Keyword(s):  
Normal Distribution ◽  
Ambient Noise ◽  
Stable Distribution ◽  
Characteristic Exponent ◽  
Statistical Modelling ◽  
Gaussian State ◽  
Sample Mean ◽  
Noise Interference ◽  
Interference Noise ◽  
Non Gaussian

In view of the non-Gaussian of ocean ambient noise, the  stable distribution is applied to the statistical modelling. Firstly, the one-to-one correspondence between the four parameters of stable distribution and the sample mean, variance, skewness and kurtosis are established according to physical meaning. Then, numerical simulations are conducted to analyze the suitability of stable distribution for non-Gaussian ambient noise. In the case of white noise interference, noise is divided into Gaussian state, leptokurtic, and platykurtic separately. The parameters of stable distribution are estimated by the sample quantile and characteristic function method jointly. The simulation results show that, in the Gaussian state,  stable distribution is equivalent to normal distribution. As for leptokurtic distribution, stable distribution is much better than normal distribution, indicating absolute predominance in impulse-like data modeling. But it is not adaptive for low kurtosis state because its characteristic exponent can’t be bigger than two. Finally, the result is verified by ambient noise collected in three environmental conditions, such as quiet ambient noise, airgun interference noise and ship noise. In all three cases,  stable distribution shows good adaptability and accuracy, especially for the airgun dataset it is far superior to normal distribution.

scholarly journals Fixed‐Effect Regressions on Network Data

Econometrica ◽  
2019 ◽  
Vol 87 (5) ◽  
pp. 1543-1560 ◽  
Author(s):  
Koen Jochmans ◽  
Martin Weidner
Keyword(s):  
Regression Model ◽  
Fixed Effects ◽  
Sufficient Conditions ◽  
Value Added ◽  
Network Data ◽  
Data Sets ◽  
Important Special Case ◽  
Valid Inference ◽  
Value Added Models ◽  
Matched Data

This paper considers inference on fixed effects in a linear regression model estimated from network data. An important special case of our setup is the two‐way regression model. This is a workhorse technique in the analysis of matched data sets, such as employer–employee or student–teacher panel data. We formalize how the structure of the network affects the accuracy with which the fixed effects can be estimated. This allows us to derive sufficient conditions on the network for consistent estimation and asymptotically valid inference to be possible. Estimation of moments is also considered. We allow for general networks and our setup covers both the dense and the sparse case. We provide numerical results for the estimation of teacher value‐added models and regressions with occupational dummies.

scholarly journals Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin

2021 ◽  
Vol 9 (1) ◽  
pp. 424-438
Author(s):  
Guillaume Boglioni Beaulieu ◽  
Pierre Lafaye de Micheaux ◽  
Frédéric Ouimet
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Marginal Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Sample Mean ◽  
Non Gaussian ◽  
Mutually Independent ◽  
Specific Sequences

Abstract We present a general methodology to construct triplewise independent sequences of random variables having a common but arbitrary marginal distribution F (satisfying very mild conditions). For two specific sequences, we obtain in closed form the asymptotic distribution of the sample mean. It is non-Gaussian (and depends on the specific choice of F). This allows us to illustrate the extent of the ‘failure’ of the classical central limit theorem (CLT) under triplewise independence. Our methodology is simple and can also be used to create, for any integer K, new K-tuplewise independent sequences that are not mutually independent. For K [four.tf], it appears that the sequences created using our methodology do verify a CLT, and we explain heuristically why this is the case.

scholarly journals Detecting spatial patterns with the cumulant function – Part 1: The theory

2008 ◽  
Vol 15 (1) ◽  
pp. 159-167 ◽  
Author(s):  
A. Bernacchia ◽  
P. Naveau
Keyword(s):  
Spatial Patterns ◽  
Principal Component ◽  
Second Order ◽  
Empirical Orthogonal Functions ◽  
Dependence Structure ◽  
Orthogonal Functions ◽  
Sample Mean ◽  
Cumulant Function ◽  
Non Gaussian ◽  
Fast Procedure

Abstract. In climate studies, detecting spatial patterns that largely deviate from the sample mean still remains a statistical challenge. Although a Principal Component Analysis (PCA), or equivalently a Empirical Orthogonal Functions (EOF) decomposition, is often applied for this purpose, it provides meaningful results only if the underlying multivariate distribution is Gaussian. Indeed, PCA is based on optimizing second order moments, and the covariance matrix captures the full dependence structure of multivariate Gaussian vectors. Whenever the application at hand can not satisfy this normality hypothesis (e.g. precipitation data), alternatives and/or improvements to PCA have to be developed and studied. To go beyond this second order statistics constraint, that limits the applicability of the PCA, we take advantage of the cumulant function that can produce higher order moments information. The cumulant function, well-known in the statistical literature, allows us to propose a new, simple and fast procedure to identify spatial patterns for non-Gaussian data. Our algorithm consists in maximizing the cumulant function. Three families of multivariate random vectors, for which explicit computations are obtained, are implemented to illustrate our approach. In addition, we show that our algorithm corresponds to selecting the directions along which projected data display the largest spread over the marginal probability density tails.

scholarly journals Linear Programming Estimators and Bootstrapping for Heavy Tailed Phenomena

1997 ◽  
Vol 29 (03) ◽  
pp. 759-805 ◽  
Author(s):  
Paul D. Feigin ◽  
Sidney I. Resnick
Keyword(s):  
Time Series ◽  
Linear Programming ◽  
Asymptotic Distribution ◽  
Rates Of Convergence ◽  
Parameter Estimates ◽  
Autoregressive Time Series ◽  
Bootstrap Procedure ◽  
Heavy Tailed

For autoregressive time series with positive innovations which either have heavy right or left tails, linear programming parameter estimates of the autoregressive coefficients have good rates of convergence. However, the asymptotic distribution of the estimators depends heavily on the distribution of the process and thus cannot be used for inference. A bootstrap procedure circumvents this difficulty. We verify the validity of the bootstrap and also give some general comments on the bootstrapping of heavy tailed phenomena.

scholarly journals Extremal clustering in non-stationary random sequences

Extremes ◽  
2021 ◽  
Author(s):  
Graeme Auld ◽  
Ioannis Papastathopoulos
Keyword(s):  
Asymptotic Distribution ◽  
Long Range ◽  
Extreme Values ◽  
Random Variables ◽  
Limiting Distribution ◽  
Long Range Dependence ◽  
Dependence Structure ◽  
Stationary Sequences ◽  
Random Sequences ◽  
Extremal Clustering

AbstractIt is well known that the distribution of extreme values of strictly stationary sequences differ from those of independent and identically distributed sequences in that extremal clustering may occur. Here we consider non-stationary but identically distributed sequences of random variables subject to suitable long range dependence restrictions. We find that the limiting distribution of appropriately normalized sample maxima depends on a parameter that measures the average extremal clustering of the sequence. Based on this new representation we derive the asymptotic distribution for the time between consecutive extreme observations and construct moment and likelihood based estimators for measures of extremal clustering. We specialize our results to random sequences with periodic dependence structure.

scholarly journals Linear Programming Estimators and Bootstrapping for Heavy Tailed Phenomena

1997 ◽  
Vol 29 (3) ◽  
pp. 759-805 ◽  
Author(s):  
Paul D. Feigin ◽  
Sidney I. Resnick
Keyword(s):  
Time Series ◽  
Linear Programming ◽  
Asymptotic Distribution ◽  
Rates Of Convergence ◽  
Parameter Estimates ◽  
Autoregressive Time Series ◽  
Bootstrap Procedure ◽  
Heavy Tailed

For autoregressive time series with positive innovations which either have heavy right or left tails, linear programming parameter estimates of the autoregressive coefficients have good rates of convergence. However, the asymptotic distribution of the estimators depends heavily on the distribution of the process and thus cannot be used for inference. A bootstrap procedure circumvents this difficulty. We verify the validity of the bootstrap and also give some general comments on the bootstrapping of heavy tailed phenomena.

scholarly journals Two-term Edgeworth expansion for M-estimators of a linear regression parameter without Cramér-type conditions and an application to the bootstrap

1997 ◽  
Vol 62 (3) ◽  
pp. 361-370 ◽  
Author(s):  
I. Karabulut ◽  
S. N. Lahiri
Keyword(s):  
Linear Regression ◽  
Edgeworth Expansion ◽  
Random Variables ◽  
Regression Parameter ◽  
Simple Linear Regression ◽  
Marked Contrast ◽  
Lattice Distribution ◽  
Sample Mean ◽  
Bootstrap Procedure ◽  
M Estimator

AbstractA two-term Edgeworth expansion for the distribution of an M-estimator of a simple linear regression parameter is obtained without assuming any Cramér-type conditions. As an application, it is shown that certain modification of the naive bootstrap procedure is second order correct even when the error variables have a lattice distribution. This is in marked contrast with the results of Singh on the sample mean of independent and identically distributed random variables.

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