Statistics of firn closure: a simulation study

AbstractThe trapping of bubbles of air in polar ice has provided a unique record of past atmospheric composition. However, the interpretation of measured concentrations depends on the statistics of the trapping process. Measurements of trace atmospheric constituents whose concentrations are changing steadily can be interpreted in terms of an “effective age” of the gas which differs from the age of the ice by a delay which corresponds to the mean trapping time. The statistics of bubble trapping can be modelled as a percolation model which is one of a class of models whose transitions are characterized by large critical fluctuations. These critical fluctuations cause an intrinsic sample-to-sample variability in the delay time and thus in the effective age. Monte Carlo simulations using a lattice model of the firn are presented, showing the effect of finite sample size on the age distribution of trapped gas. For samples containing more than about 103–104 bubbles, the simulations indicate that the range of variability is small compared to the average duration of the trapping process.

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Statistics of firn closure: a simulation study

Journal of Glaciology ◽

10.1017/s0022143000015781 ◽

1993 ◽

Vol 39 (131) ◽

pp. 133-142 ◽

Cited By ~ 6

Author(s):

I. G. Enting

Keyword(s):

Age Distribution ◽

Atmospheric Composition ◽

Finite Sample ◽

Critical Fluctuations ◽

Effective Age ◽

Finite Sample Size ◽

Trapped Gas ◽

Process Measurements ◽

Bubble Trapping ◽

Trapping Process

AbstractThe trapping of bubbles of air in polar ice has provided a unique record of past atmospheric composition. However, the interpretation of measured concentrations depends on the statistics of the trapping process. Measurements of trace atmospheric constituents whose concentrations are changing steadily can be interpreted in terms of an “effective age” of the gas which differs from the age of the ice by a delay which corresponds to the mean trapping time. The statistics of bubble trapping can be modelled as a percolation model which is one of a class of models whose transitions are characterized by large critical fluctuations. These critical fluctuations cause an intrinsic sample-to-sample variability in the delay time and thus in the effective age. Monte Carlo simulations using a lattice model of the firn are presented, showing the effect of finite sample size on the age distribution of trapped gas. For samples containing more than about 103–104bubbles, the simulations indicate that the range of variability is small compared to the average duration of the trapping process.

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A SMOOTH TEST FOR THE EQUALITY OF DISTRIBUTIONS

Econometric Theory ◽

10.1017/s0266466612000370 ◽

2012 ◽

Vol 29 (2) ◽

pp. 419-446 ◽

Cited By ~ 7

Author(s):

Anil K. Bera ◽

Aurobindo Ghosh ◽

Zhijie Xiao

Keyword(s):

New York ◽

Goodness Of Fit ◽

Age Distribution ◽

Distribution Functions ◽

Type I ◽

Finite Sample ◽

Community Rating ◽

Smooth Test ◽

Two Samples ◽

Group Insurance

The two-sample version of the celebrated Pearson goodness-of-fit problem has been a topic of extensive research, and several tests like the Kolmogorov-Smirnov and Cramér-von Mises have been suggested. Although these tests perform fairly well as omnibus tests for comparing two probability density functions (PDFs), they may have poor power against specific departures such as in location, scale, skewness, and kurtosis. We propose a new test for the equality of two PDFs based on a modified version of the Neyman smooth test using empirical distribution functions minimizing size distortion in finite samples. The suggested test can detect the specific directions of departure from the null hypothesis. Specifically, it can identify deviations in the directions of mean, variance, skewness, or tail behavior. In a finite sample, the actual probability of type-I error depends on the relative sizes of the two samples. We propose two different approaches to deal with this problem and show that, under appropriate conditions, the proposed tests are asymptotically distributed as chi-squared. We also study the finite sample size and power properties of our proposed test. As an application of our procedure, we compare the age distributions of employees with small employers in New York and Pennsylvania with group insurance before and after the enactment of the “community rating” legislation in New York. It has been conventional wisdom that if community rating is enforced (where the group health insurance premium does not depend on age or any other physical characteristics of the insured), then the insurance market will collapse, since only older or less healthy patients would prefer group insurance. We find that there are significant changes in the age distribution in the population in New York owing mainly to a shift in location and scale.

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Near Observational Equivalence and Theoretical size Problems with Unit Root Tests

Econometric Theory ◽

10.1017/s0266466600007003 ◽

1996 ◽

Vol 12 (4) ◽

pp. 724-731 ◽

Cited By ~ 30

Author(s):

Jon Faust

Keyword(s):

Unit Root ◽

Sufficient Conditions ◽

Unit Root Test ◽

Asymptotic Distributions ◽

Unit Root Tests ◽

Finite Sample ◽

Test Procedures ◽

Asymptotic Size ◽

Finite Sample Size ◽

Topological Sense

Said and Dickey (1984,Biometrika71, 599–608) and Phillips and Perron (1988,Biometrika75, 335–346) have derived unit root tests that have asymptotic distributions free of nuisance parameters under very general maintained models. Under models as general as those assumed by these authors, the size of the unit root test procedures will converge to one, not the size under the asymptotic distribution. Solving this problem requires restricting attention to a model that is small, in a topological sense, relative to the original. Sufficient conditions for solving the asymptotic size problem yield some suggestions for improving finite-sample size performance of standard tests.

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On the variance parameter estimator in general linear models

Metrika ◽

10.1007/s00184-019-00751-4 ◽

2019 ◽

Vol 83 (2) ◽

pp. 243-254

Author(s):

Mathias Lindholm ◽

Felix Wahl

Keyword(s):

Sample Size ◽

Linear Models ◽

General Linear ◽

Finite Sample ◽

Parameter Estimator ◽

General Linear Models ◽

Vector Autoregressive ◽

Finite Sample Size ◽

Error Terms ◽

Variance Parameter

Abstract In the present note we consider general linear models where the covariates may be both random and non-random, and where the only restrictions on the error terms are that they are independent and have finite fourth moments. For this class of models we analyse the variance parameter estimator. In particular we obtain finite sample size bounds for the variance of the variance parameter estimator which are independent of covariate information regardless of whether the covariates are random or not. For the case with random covariates this immediately yields bounds on the unconditional variance of the variance estimator—a situation which in general is analytically intractable. The situation with random covariates is illustrated in an example where a certain vector autoregressive model which appears naturally within the area of insurance mathematics is analysed. Further, the obtained bounds are sharp in the sense that both the lower and upper bound will converge to the same asymptotic limit when scaled with the sample size. By using the derived bounds it is simple to show convergence in mean square of the variance parameter estimator for both random and non-random covariates. Moreover, the derivation of the bounds for the above general linear model is based on a lemma which applies in greater generality. This is illustrated by applying the used techniques to a class of mixed effects models.

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FINITE-SAMPLE SIZE CONTROL OF IVX-BASED TESTS IN PREDICTIVE REGRESSIONS

Econometric Theory ◽

10.1017/s0266466620000298 ◽

2020 ◽

pp. 1-25

Author(s):

Mehdi Hosseinkouchack ◽

Matei Demetrescu

Keyword(s):

Size Control ◽

Finite Sample ◽

Limiting Distributions ◽

Valid Inference ◽

Tuning Parameters ◽

Predictive Regressions ◽

Chi Squared ◽

Standard Normal ◽

Finite Sample Size

Abstract In predictive regressions with variables of unknown persistence, the use of extended IV (IVX) instruments leads to asymptotically valid inference. Under highly persistent regressors, the standard normal or chi-squared limiting distributions for the usual t and Wald statistics may, however, differ markedly from the actual finite-sample distributions which exhibit in particular noncentrality. Convergence to the limiting distributions is shown to occur at a rate depending on the choice of the IVX tuning parameters and can be very slow in practice. A characterization of the leading higher-order terms of the t statistic is provided for the simple regression case, which motivates finite-sample corrections. Monte Carlo simulations confirm the usefulness of the proposed methods.

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