Scale-invariant Estimates Research Articles

Transference of scale-invariant estimates from Lipschitz to nontangentially accessible to uniformly rectifiable domains

Transference of scale-invariant estimates from Lipschitz to nontangentially accessible to uniformly rectifiable domains

Estimation of the scale parameter of a selected gamma population with unequal shape parameters under stein loss function

The problem of estimation of the parameter of a selected population arises when we encounter with several populations and would like to estimate the parameter of the best (worst) selected population. Suppose be independent random samples drawn from populations respectively, where observations from Π i have a -distribution with unequal known shape parameters In this paper, we use a selection rule to select the best (worst) population, and estimate the best (worst) scale parameter of the selected population under the Stein loss function. The uniformly minimum risk unbiased (UMRU) estimators of θS and θJ are obtained. A sufficient condition for inadmissibility of scale-invariant estimators of is obtained and it is shown that the UMRU estimator of is inadmissible. For k = 2, a sufficient condition for minimaxity of a given estimator of is obtained, and the generalized Bayes estimator of θS is shown to be minimax. Finally, the risk functions of the various competing estimators are compared numerically, and a real data is provided to compute the proposed estimates and their expected risks.

Scale invariant regularity estimates for second order elliptic equations with lower order coefficients in optimal spaces

We show local and global scale invariant regularity estimates for subsolutions and supersolutions to the equation −div(A∇u+bu)+c∇u+du=−divf+g, assuming that A is elliptic and bounded. In the setting of Lorentz spaces, under the assumptions b,f∈Ln,1, d,g∈Ln2,1 and c∈Ln,q for q≤∞, we show that, with the surprising exception of the reverse Moser estimate, scale invariant estimates with “good” constants (that is, depending only on the norms of the coefficients) do not hold in general. On the other hand, assuming a necessary smallness condition on b,d or c,d, we show a maximum principle and Moser's estimate for subsolutions with “good” constants. We also show the reverse Moser estimate for nonnegative supersolutions with “good” constants, under no smallness assumptions when q<∞, leading to the Harnack inequality for nonnegative solutions and local continuity of solutions. Finally, we show that, in the setting of Lorentz spaces, our assumptions are the sharp ones to guarantee these estimates.

Application of Liar Remote Sensing Forest Leaf Area Index Extraction Method Based on Big Data Network

In the past, liar was mainly used to study forest ecosystem flight time. So far, the ecological application of SLR is mainly to use one-dimensional index to characterize the canopy height. A new 3D analysis method of liar waveform is proposed. Many models developed before are scale dependent and need to be fitted and then applied with the same scale or pixel size. A method of scale invariant estimation of forest biomass based on liar data is proposed. Two equivalent biomass models were proposed: linear biomass model and linear biomass model. The development of sensors and algorithms is evaluated. The specific areas discussed include the selection of analysis units (grid units, segments or individual trees), canopy height model and point based analysis, the role of multiple benefits and benefit intensity, and the comparison of common algorithms. The results showed that the highest biomass of bark was 0.2645, the lowest biomass of above ground was 0.0267, and the biomass of branch was 0.021 higher than that of trunk.

Scale-Invariant Estimates and Vorticity Alignment for Navier–Stokes in the Half-Space with No-Slip Boundary Conditions

This paper is concerned with geometric regularity criteria for the Navier–Stokes equations in \({\mathbb {R}}^3_{+}\times (0,T)\) with a no-slip boundary condition, with the assumption that the solution satisfies the ‘ODE blow-up rate’ Type I condition. More precisely, we prove that if the vorticity direction is uniformly continuous on subsets of $$\begin{aligned} \bigcup _{t\in (T-1,T)} \big (B(0,R)\cap {\mathbb {R}}^3_{+}\big )\times {\{t\}},\,\,\,\,\,\, R=O(\sqrt{T-t}), \end{aligned}$$where the vorticity has large magnitude, then (0, T) is a regular point. This result is inspired by and improves the regularity criteria given by Giga et al. [20]. We also obtain new local versions for suitable weak solutions near the flat boundary. Our method hinges on new scaled Morrey estimates, blow-up and compactness arguments and ‘persistence of singularites’ on the flat boundary. The scaled Morrey estimates seem to be of independent interest.

Fundamental matrices and Green matrices for non-homogeneous elliptic systems

In this paper, we establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems with bounded measurable coefficients in Rn and for the corresponding Green functions in arbitrary open sets. We impose certain non-homogeneous versions of de Giorgi– Nash–Moser bounds on the weak solutions and investigate in detail the assumptions on the lower order terms sufficient to guarantee such conditions. Our results, in particular, establish the existence and fundamental estimates for the Green functions associated to the Schr¨odinger (−∆ + V ) and generalized Schrodinger (− div A∇ + V ) operators with real and complex coefficients, on arbitrary domains.

On estimating the scale parameter of the selected uniform population under the entropy loss function

Let $\pi_{1},\ldots,\pi_{k}$ be $k$ ($\geq2$) independent populations, where $\pi_{i}$ denotes the uniform distribution over the interval $(0,\theta_{i})$ and $\theta_{i}>0$ ($i=1,\ldots,k$) is an unknown scale parameter. Let $\theta_{[1]}\leq\cdots\leq\theta_{[k]}$ be the ordered values of $\theta_{1},\ldots,\theta_{k}$. The population $\pi_{(k)}$ ($\pi_{(1)}$) associated with the unknown parameter $\theta_{[k]}$ ($\theta_{[1]}$) is called the best (worst) population. For selecting the best population, we consider a general class of selection rules based on the natural estimators of $\theta_{i},i=1,\ldots,k$. Under the entropy loss function, we consider the problem of estimating the scale parameter $\theta_{S}$ of the population selected using a fixed selection rule from this class. We derive the uniformly minimum risk unbiased estimator of $\theta_{S}$ and two natural estimators of $\theta_{S}$ are also considered. We derive a general result for improving a scale invariant estimator of $\theta_{S}$ under the entropy loss function. A simulation study on the performances of various competing estimators of $\theta_{S}$ is also reported. Finally, we provide similar results for the problem of estimating the scale parameter of selected population when the selection goal is that of selecting the worst uniform population.

Asymptotic distributions of some robust scale estimators in explosive AR(1) model

This note establishes the asymptotic normality of the median of the absolute residuals and the median of the absolute differences of pairwise residuals in the first order explosive autoregressive time series. These estimators are useful for obtaining some scale invariant estimators of the autoregressive parameter.

Large Dimensional Analysis of Robust M-Estimators of Covariance With Outliers

A large dimensional characterization of robust M-estimators of covariance (or scatter) is provided under the assumption that the dataset comprises independent (essentially Gaussian) legitimate samples as well as arbitrary deterministic samples, referred to as outliers. Building upon recent random matrix advances in the area of robust statistics, we specifically show that the so-called Maronna M-estimator of scatter asymptotically behaves similar to well-known random matrices when the population and sample sizes grow together to infinity. The introduction of outliers leads the robust estimator to behave asymptotically as the weighted sum of the sample outer products, with a constant weight for all legitimate samples and different weights for the outliers. A fine analysis of this structure reveals importantly that the propensity of the M-estimator to attenuate (or enhance) the impact of outliers is mostly dictated by the alignment of the outliers with the inverse population covariance matrix of the legitimate samples. Thus, robust M-estimators can bring substantial benefits over more simplistic estimators such as the per-sample normalized version of the sample covariance matrix, which is not capable of differentiating the outlying samples. The analysis shows that, within the class of Maronna's estimators of scatter, the Huber estimator is most favorable for rejecting outliers. On the contrary, estimators more similar to Tyler's scale invariant estimator (often preferred in the literature) run the risk of inadvertently enhancing some outliers.

Estimation After Selection From Uniform Populations With Unequal Sample Sizes

SYNOPTIC ABSTRACTConsider k (⩾ 2) independent uniform populations π1, …, πk, where πi ≡ U(0, θi), and θi > 0 (i = 1, …, k) is an unknown scale parameter. For selecting the unknown population having the largest scale parameter, we consider a class of selection rules based on the natural estimators of θi, i = 1, …, k. We consider the problem of estimating the scale parameter θS of the selected population, using a fixed selection rule from this class, under the scaled-squared error loss function. We derive the uniformly minimum variance unbiased estimator (UMVUE) of θS. We also consider three natural estimators ϕN, 1, ϕN, 2, and ϕN, 3 of θS which are, respectively, based on the maximum likelihood estimators, UMVUEs, and minimum risk equivariant estimators for component estimation problems. The natural estimator ϕN, 3 is shown to be a generalized Bayes estimator with respect to a non informative prior. Further, we derive a general result for improving a scale-invariant estimator of θS. Using this result, the estimators better than the UMVUE and the natural estimator ϕN, 1 are obtained. It is also shown that a subclass of natural-type estimators, which contains the natural estimator ϕN, 2, is inadmissible for estimating θS under the scaled-squared error loss function. Finally, we provide a simulation study on the performances of various competing estimators of θS.

Admissible and minimax estimation of the parameter of the selected Pareto population under squared log error loss function

The problem of estimation after selection arises when we select a population from the given k populations by a selection rule, and estimate the parameter of the selected population. In this paper we consider the problem of estimation of the scale parameter of the selected Pareto population \(\theta _{M}\) (or \(\theta _{J}\)) under squared log error loss function. The uniformly minimum risk unbiased (UMRU) estimator of \(\theta _{M}\) and \(\theta _{J}\) are obtained. In the case of \(k=2,\) we give a sufficient condition for minimaxity of an estimator of \(\theta _{M}\) and \(\theta _{J},\) and show that the UMRU and natural estimators of \(\theta _{J}\) are minimax. Also the class of linear admissible estimators of \(\theta _{M}\) and \(\theta _{J}\) are obtained which contain the natural estimator. By using the Brewester–Ziedeck technique we find sufficient condition for inadmissibility of some scale and permutation invariant estimators of \(\theta _{J},\) and show that the UMRU estimator of \(\theta _{J}\) is inadmissible. Finally, we compare the risk of the obtained estimators numerically, and discuss the results for selected uniform population.

Estimation After Selection From Gamma Populations With Unequal Known Shape Parameters

Let be independent gamma populations, where the population has an unknown scale parameter and known shape parameter . We call the population associated with the best population. For the goal of selecting the best population, Misra and Arshad (2014) proposed a class of selection rules for the case of (possibly) unequal shape parameters. In this article, we consider the problem of estimating the mean of the population selected by a fixed selection rule , under a scale-invariant loss function. We derive the uniformly minimum variance unbiased estimator (UMVUE). Two other natural estimators and , which are respectively the analogs of the UMVUE and the best scale invariant estimators of for the component problem, are studied. We show that is generalized Bayes with respect to a noninformative prior distribution, and is also minimax when k = 2. The UMVUE and the natural estimator are shown to be inadmissible, and better estimators are obtained. A numerical study on the performance of various estimators indicates...

Adaptive PORT–MVRB estimation: an empirical comparison of two heuristic algorithms

In this article, we deal with an empirical comparison of two data-driven heuristic procedures of estimation of a positive extreme value index (EVI), working thus with heavy right tails. The semi-parametric EVI-estimators under consideration, the so-called peaks over random threshold (PORT)–minimum-variance reduced-bias (MVRB) EVI-estimators, are location and scale-invariant estimators, based on the PORT methodology applied to second-order MVRB EVI-estimators. Trivial adaptations of these algorithms make them work for a similar estimation of other parameters of extreme events, such as the Value-at-Risk at a level p, the expected shortfall and the probability of exceedance of a high level x, among others. Applications to simulated data sets and to real data sets in the field of finance are provided.

Expanding solitons with non-negative curvature operator coming out of cones

We consider Ricci flow of complete Riemannian manifolds which have bounded non-negative curvature operator, non-zero asymptotic volume ratio and no boundary. We prove scale invariant estimates for these solutions. Using these estimates, we show that there is a limit solution, obtained by scaling down this solution at a fixed point in space. This limit solution is an expanding soliton coming out of the asymptotic cone at infinity.

Optimal rules and robust Bayes estimation of a Gamma scale parameter

For estimating an unknown scale parameter of Gamma distribution, we introduce the use of an asymmetric scale invariant loss function reflecting precision of estimation. This loss belongs to the class of precautionary loss functions. The problem of estimation of scale parameter of a Gamma distribution arises in several theoretical and applied problems. Explicit form of risk-unbiased, minimum risk scale-invariant, Bayes, generalized Bayes and minimax estimators are derived. We characterized the admissibility and inadmissibility of a class of linear estimators of the form $$cX\,{+}\,d$$ , when $$X\sim \varGamma (\alpha ,\eta )$$ . In the context of Bayesian statistical inference any statistical problem should be treated under a given loss function by specifying a prior distribution over the parameter space. Hence, arbitrariness of a unique prior distribution is a critical and permanent question. To overcome with this issue, we consider robust Bayesian analysis and deal with Gamma minimax, conditional Gamma minimax, the stable and characterize posterior regret Gamma minimax estimation of the unknown scale parameter under the asymmetric scale invariant loss function in detail.

Stock Return Volatility Surrounding Management Earnings Forecasts

The primary aim of this study is to investigate the stock return volatility surrounding management earnings forecasts. Disclosure by managers of expected earnings are particularly important communications, and as such, it is important to understand the capital market implications surrounding them. In doing so, the research questions are essentially aimed at examining the stock return volatility, first, at the release of a management earnings forecast, and second, at the eventual announcement of the realised earnings for that period. The first test investigates whether there is an increase in volatility surrounding a management earnings forecast for those firms who release them compared to a matched-firm sample of firms without a management earnings forecast at that date, and then further examines that result based on different forecast antecedents and forecast characteristics. In brief, the evidence using the Garman and Klass (1980) ‘best analytic scale-invariant estimator’ of volatility in an Australian context, between 1993 and 2003, finds that stock return volatility is greater for bad news forecasts, forecasts of low specificity, and forecasts issued by firms perceived ex ante as being of lower credibility using both permutation analysis and modelling daily volatility.

Scale-invariant extinction time estimates for some singular diffusion equations

We study three singular parabolic evolutions: the second-order total variation flow, the fourth-order total variation flow, and a fourth-order surface diffusion law. Each has the property that the solution becomes identically zero in finite time. We prove scale-invariant estimates for the extinction time, using a simple argument which combines an energy estimate with a suitable Sobolev-type inequality.

The speed of time

Event Abstract Back to Event The speed of time Misha Ahrens1, 2* and Maneesh Sahani3 1 Cambridge University, United Kingdom 2 Janelia Farm Research Campus , United States 3 UCL, Gatsby Computational Neuroscience Unit, Wellcome Trust Neuroimaging Centre and Medical School, United Kingdom The content of a visual stimulus affects its perceived duration. In particular, stimuli that change rapidly generally appear to last longer than equal-duration stimuli that change more slowly. Exactly why this is, and how stimulus-driven biases interact with internal estimates of elapsed time, is not known. One possible account is that observers expect a certain rate of discrete "events" in the world (Fraisse, 1964). The rapidly changing stimuli contain more of these events per unit time, and so observers believe that more time has elapsed. We have proposed a different view as part of a general framework for psychological time estimation (Cosyne, 2008). In our account, observers have an expection regarding the temporal correlations in the environment, which they use to translate the amount by which the stimulus changes during presentation into probabilistic infomation about its duration. Here, we contrast these two accounts, showing that they make very different predictions regarding the variability of timing estimates, and testing these predictions experimentally. Internal duration estimates are known to follow a Scalar law, whereby the distribution of estimates scales in proportion to its mean (this, in turn, suggests that the variability in estimates follows a Weber law). We measured the variability of observers’ estimates of the duration of colored (i.e. space-time smoothed) Gaussian noise stimuli for a range of different true durations, and found (1) that estimates of the duration of these dynamic stimuli were less variable that estimates of the durations of static stimuli; and (2) that the Scalar law continued to hold in the presence of these random dynamic stimuli. Both models are consistent with observation (1), as the added information in the stimulus should help to reduce variability. We have previously shown that the change-based approach also yields scale-invariant estimates, and it is thus consistent with the second observation too. By contrast, we show here that the event-counting model leads to sub-Weberian growth in variability for these stimuli, and that even when combined with an internal estimate that retains the Scalar property it cannot account for observers’ behavior. Our results thus (1) provide further evidence that the reason the psychological speed of time is affected systematically by stimuli is that observers exploit these stimuli to derive additional information about the passage of time; and (2) argue against simple counting-based sources of this additional information, lending support to the idea that the information derives from more extensive knowledge of the statistics. Conference: Computational and Systems Neuroscience 2010, Salt Lake City, UT, United States, 25 Feb - 2 Mar, 2010. Presentation Type: Poster Presentation Topic: Poster session I Citation: Ahrens M and Sahani M (2010). The speed of time. Front. Neurosci. Conference Abstract: Computational and Systems Neuroscience 2010. doi: 10.3389/conf.fnins.2010.03.00066 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 19 Feb 2010; Published Online: 19 Feb 2010. * Correspondence: Misha Ahrens, Cambridge University, Paris, United Kingdom, mba22@cam.ac.uk Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Misha Ahrens Maneesh Sahani Google Misha Ahrens Maneesh Sahani Google Scholar Misha Ahrens Maneesh Sahani PubMed Misha Ahrens Maneesh Sahani Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Lidar remote sensing of forest biomass: A scale-invariant estimation approach using airborne lasers

Researchers in lidar (Light Detection And Ranging) strive to search for the most appropriate laser-based metrics as predictors in regression models for estimating forest structural variables. Many previously developed models are scale-dependent that need to be fitted and then applied both at the same scale or pixel size. The objective of this paper is to develop methods for scale-invariant estimation of forest biomass using lidar data. We proposed two scale-invariant models for biomass: a linear functional model and an equivalent nonlinear model that use lidar-derived canopy height distributions (CHD) and canopy height quantile functions (CHQ) as predictors, respectively. The two models are called functional regression models because the predictors CHD and CHQ are themselves functions or functional data. The model formulation was justified mathematically under moderate assumptions. We also created a fine-resolution biomass map by mapping individual tree component biomass in a temperate forest of eastern Texas with a lidar tree-delineation approach. The map was used as reference data to synthesize training and test datasets at multiple scales for validating the two scale-invariant models. Results suggest that the models can accurately predict biomass and yield consistent predictive performances across a variety of scales with an R 2 ranging from 0.80 to 0.95 (RMSE: from 14. 3 Mg/ha to 33.7 Mg/ha) among all the fitted models. Results also show that a training data size of around 50 plots or less was enough to guarantee a good fitting of the linear functional model. Our findings demonstrate the effectiveness of CHD and CHQ as lidar metrics for estimating biomass as well as the capability of lidar for mapping biomass at a range of scales. The functional regression models of this study are useful for lidar-based forest inventory tasks where the analysis units vary in size and shape. They also hold promise for estimating other forest characteristics such as below-ground biomass, timber volume, crown fuel weight, and Leaf Area Index.

A two-step estimator of the extreme value index

In this paper, we build a two-step estimator $\hat{\gamma}_{\rm STEP}$ , which satisfies $\sqrt{k}(\hat{\gamma}_{\rm STEP}-\hat{\gamma}_{ML})\stackrel{P}{\rightarrow} 0$ , where $\hat{\gamma}_{ML}$ is the well-known maximum likelihood estimator of the extreme value index. Since the two-step estimator $\hat{\gamma}_{\rm STEP}$ can be calculated easily as a function of the observations, it is much simpler to use in practice. By properly choosing the first step estimator, such as the Pickands estimator, we can even get a shift and scale invariant estimator with the above property.