Nonasymptotic bounds for the quadratic risk of the Grenander estimator

2020 ◽  
pp. 1-9
Author(s):  
Malkhaz Shashiashvili
Keyword(s):  
Upper Bound ◽  
Finite Sample ◽  
Nonparametric Maximum Likelihood Estimator ◽  
Finite Sample Size ◽  
Kolmogorov Smirnov ◽  
Integrated Squared Error ◽  
Nonasymptotic Bounds ◽  
Grenander Estimator ◽  
First Time ◽  
Straightforward Consequence

Abstract There is an enormous literature on the so-called Grenander estimator, which is merely the nonparametric maximum likelihood estimator of a nonincreasing probability density on [0, 1] (see, for instance, Grenander (1981)), but unfortunately, there is no nonasymptotic (i.e. for arbitrary finite sample size n) explicit upper bound for the quadratic risk of the Grenander estimator readily applicable in practice by statisticians. In this paper, we establish, for the first time, a simple explicit upper bound 2n−1/2 for the latter quadratic risk. It turns out to be a straightforward consequence of an inequality valid with probability one and bounding from above the integrated squared error of the Grenander estimator by the Kolmogorov–Smirnov statistic.

scholarly journals Asymptotic Theory in Model Diagnostic for General Multivariate Spatial Regression

2016 ◽  
Vol 2016 ◽  
pp. 1-16 ◽  
Author(s):  
Wayan Somayasa ◽  
Gusti N. Adhi Wibawa ◽  
La Hamimu ◽  
La Ode Ngkoimani
Keyword(s):  
Asymptotic Theory ◽  
Spatial Regression ◽  
Test Procedure ◽  
Real Data ◽  
Partial Sums ◽  
Finite Sample ◽  
Partial Sums Processes ◽  
Finite Sample Size ◽  
Kolmogorov Smirnov ◽  
Von Mises

We establish an asymptotic approach for checking the appropriateness of an assumed multivariate spatial regression model by considering the set-indexed partial sums process of the least squares residuals of the vector of observations. In this work, we assume that the components of the observation, whose mean is generated by a certain basis, are correlated. By this reason we need more effort in deriving the results. To get the limit process we apply the multivariate analog of the well-known Prohorov’s theorem. To test the hypothesis we define tests which are given by Kolmogorov-Smirnov (KS) and Cramér-von Mises (CvM) functionals of the partial sums processes. The calibration of the probability distribution of the tests is conducted by proposing bootstrap resampling technique based on the residuals. We studied the finite sample size performance of the KS and CvM tests by simulation. The application of the proposed test procedure to real data is also discussed.

scholarly journals New Synthetic Methods of Novel Nanoporous Polycondensates and Excellent Oxygen Permselectivity of Their Composite Membranes

Nanomaterials ◽  
2019 ◽  
Vol 9 (6) ◽  
pp. 859
Author(s):  
Yu Zang ◽  
Toshiki Aoki ◽  
Masahiro Teraguchi ◽  
Takashi Kaneko ◽  
Hongge Jia ◽  
...  
Keyword(s):  
Contact Angle ◽  
Upper Bound ◽  
Oxygen Permeability ◽  
Membrane Surface ◽  
Composite Membranes ◽  
Synthetic Methods ◽  
Preparation Methods ◽  
First Time

Two kinds of novel nanoporous polycondensates (sc(Rf)) have been synthesized by two new preparation methods consisting of polycondensation and highly selective photocyclicaromataization of 1/3 helical cis-cis polyphenylacetylenes with polymerizable groups. By the original methods, new well-defined sheet polymers having nanopores or nanospaces have been synthesized for the first time. Their composite membranes, containing small amounts (1.0 wt%) of sc(Rf), had ultrahigh oxygen permeability (Po2 > 1000 barrer), and their plots were beyond the Robeson’s upper bound line in the graph of oxygen permselectivity (α = Po2/PN2) versus Po2. Both α and Po2 values were enhanced by adding only small amounts (1.0 wt%) of sc(Rf). One of the sc(Rf)s synthesized on the base membrane surface showed the best performance, i.e., Po2 = 5300 barrer and α = 2.5. The membrane surface was effectively covered by sc(Rf), judging from the contact angle values. It is thought that nanopores and nanospaces created in and between sc(Rf) molecules played an important role for the enhancement of both α and Po2/PN2.

Near Observational Equivalence and Theoretical size Problems with Unit Root Tests

1996 ◽  
Vol 12 (4) ◽  
pp. 724-731 ◽  
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.

scholarly journals The string of diamonds is nearly tight for rumour spreading

2019 ◽  
Vol 29 (2) ◽  
pp. 190-199
Author(s):  
Omer Angel ◽  
Abbas Mehrabian ◽  
Yuval Peres
Keyword(s):  
Upper Bound ◽  
Real Numbers ◽  
Vertex Graph ◽  
First Time

AbstarctFor a rumour spreading protocol, the spread time is defined as the first time everyone learns the rumour. We compare the synchronous push&pull rumour spreading protocol with its asynchronous variant, and show that for any n-vertex graph and any starting vertex, the ratio between their expected spread times is bounded by $O({n^{1/3}}{\log ^{2/3}}n)$. This improves the $O(\sqrt n)$ upper bound of Giakkoupis, Nazari and Woelfel (2016). Our bound is tight up to a factor of O(log n), as illustrated by the string of diamonds graph. We also show that if, for a pair α, β of real numbers, there exist infinitely many graphs for which the two spread times are nα and nβ in expectation, then $0 \le \alpha \le 1$ and $\alpha \le \beta \le {1 \over 3} + {2 \over 3} \alpha $; and we show each such pair α, β is achievable.

scholarly journals On the variance parameter estimator in general linear models

Metrika ◽  
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.

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

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.

scholarly journals Statistics of firn closure: a simulation study

1993 ◽  
Vol 39 (131) ◽  
pp. 133-142 ◽  
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 103–104 bubbles, the simulations indicate that the range of variability is small compared to the average duration of the trapping process.

scholarly journals MIDAS: A Benchmarking Multi-Criteria Method for the Identification of Defective Anemometers in Wind Farms

Energies ◽  
2018 ◽  
Vol 12 (1) ◽  
pp. 28 ◽  
Author(s):  
Arkaitz Rabanal ◽  
Alain Ulazia ◽  
Gabriel Ibarra-Berastegi ◽  
Jon Sáenz ◽  
Unai Elosegui
Keyword(s):  
Wind Farm ◽  
Principal Component ◽  
Wind Farms ◽  
General Purpose ◽  
Wind Data ◽  
Kolmogorov Smirnov ◽  
Clear Identification ◽  
Novel Method ◽  
Verification Techniques ◽  
First Time

A novel multi-criteria methodology for the identification of defective anemometers is shown in this paper with a benchmarking approach: it is called MIDAS: multi-technique identification of defective anemometers. The identification of wrong wind data as provided by malfunctioning devices is very important, because the actual power curve of a wind turbine is conditioned by the quality of its anemometer measurements. Here, we present a novel method applied for the first time to anemometers’ data based on the kernel probability density function and the recent reanalysis ERA5. This estimation improves classical unidimensional methods such as the Kolmogorov–Smirnov test, and the use of the global ERA5’s wind data as the first benchmarking reference establishes a general method that can be used anywhere. Therefore, adopting ERA5 as the reference, this method is applied bi-dimensionally for the zonal and meridional components of wind, thus checking both components at the same time. This technique allows the identification of defective anemometers, as well as clear identification of the group of anemometers that works properly. After that, other verification techniques were used versus the faultless anemometers (Taylor diagrams, running correlation and R M S E , and principal component analysis), and coherent results were obtained for all statistical techniques with respect to the multidimensional method. The developed methodology combines the use of this set of techniques and was able to identify the defective anemometers in a wind farm with 10 anemometers located in Northern Europe in a terrain with forests and woodlands. Nevertheless, this methodology is general-purpose and not site-dependent, and in the future, its performance will be studied in other types of terrain and wind farms.

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