Adaptively weighted maximum likelihood estimation of discrete distributions

SummaryDiscrete data arise in various research fields, typically when the observations are count data.I propose a robust and efficient parametric procedure for estimation of discrete distributions. The estimation is done in two phases. First, a very robust, but possibly inefficient, estimate of the model parameters is computed and used to indentify outliers. Then the outliers are either removed from the sample or given low weights, and a weighted maximum likelihood estimate (WML) is computed.The weights are determined via an adaptive process such that if the data follow the model, then asymptotically no observation is downweighted.I prove that the final estimator inherits the breakdown point of the initial one, and that its influence function at the model is the same as the influence function of the maximum likelihood estimator, which strongly suggests that it is asymptotically fully efficient.The initial estimator is a minimum disparity estimator (MDE). MDEs can be shown to have full asymptotic efficiency, and some MDEs have very high breakdown points and very low bias under contamination. Several initial estimators are considered, and the performances of the WMLs based on each of them are studied.It results that in a great variety of situations the WML substantially improves the initial estimator, both in terms of finite sample mean square error and in terms of bias under contamination. Besides, the performances of the WML are rather stable under a change of the MDE even if the MDEs have very different behaviors.Two examples of application of the WML to real data are considered. In both of them, the necessity for a robust estimator is clear: the maximum likelihood estimator is badly corrupted by the presence of a few outliers.This procedure is particularly natural in the discrete distribution setting, but could be extended to the continuous case, for which a possible procedure is sketched.RésuméLes données discrètes sont présentes dans différents domaines de recherche, en particulier lorsque les observations sont des comptages.Je propose une méthode paramétrique robuste et efficace pour l'estimation de distributions discrètes. L'estimation est faite en deux phases. Tout d'abord, un estimateur très robuste des paramètres du modèle est calculé, et utilisé pour la détection des données aberrantes (outliers). Cet estimateur n'est pas nécessairement efficace. Ensuite, soit les outliers sont retirés de l'échantillon, soit des faibles poids leur sont attribués, et un estimateur du maximum de vraisemblance pondéré (WML) est calculé.Les poids sont déterminés via un processus adaptif, tel qu'asymptotiquement, si les données suivent le modèle, aucune observation n'est dépondérée.Je prouve que le point de rupture de l'estimateur final est au moins aussi élevé que celui de l'estimateur initial, et que sa fonction d'influence au modèle est la même que celle du maximum de vraisemblance, ce qui suggère que cet estimateur est pleinement efficace asymptotiquement.L'estimateur initial est un estimateur de disparité minimale (MDE). Les MDE sont asymptotiquement pleinement efficaces, et certains d'entre eux ont un point de rupture très élevé et un très faible biais sous contamination. J'étudie les performances du WML basé sur différents MDEs.Le résultat est que dans une grande variété de situations le WML améliore largement les performances de l'estimateur initial, autant en terme du carré moyen de l'erreur que du biais sous contamination. De plus, les performances du WML restent assez stables lorsqu'on change l'estimateur initial, même si les différents MDEs ont des comportements très différents.Je considère deux exemples d'application du WML à des données réelles, où la nécessité d'un estimateur robuste est manifeste : l'estimateur du maximum de vraisemblance est fortement corrompu par la présence de quelques outliers.La méthode proposée est particulièrement naturelle dans le cadre des distributions discrètes, mais pourrait être étendue au cas continu.

Optimum power allocation for parallel Gaussian channels with arbitrary input distributions

The mutual information of independent parallel Gaussian-noise channels is maximized, under an average power constraint, by independent Gaussian inputs whose power is allocated according to the waterfilling policy. In practice, discrete signalling constellations with limited peak-to-average ratios (m-PSK, m-QAM, etc) are used in lieu of the ideal Gaussian signals. This paper gives the power allocation policy that maximizes the mutual information over parallel channels with arbitrary input distributions. Such policy admits a graphical interpretation, referred to as mercury/waterfilling, which generalizes the waterfilling solution and allows retaining some of its intuition. The relationship between mutual information of Gaussian channels and nonlinear minimum mean-square error proves key to solving the power allocation problem.

Large-system analysis of a CDMA dynamic channel under a Markovian input process

We study the minimum mean square error (MMSE) and the multiuser efficiency η of large dynamic multiple access communication systems in which optimal multiuser detection is performed at the receiver as the number and the identities of active users is allowed to change at each transmission time. The system dynamics are ruled by a Markov model describing the evolution of the channel occupancy and a large-system analysis is performed when the number of observations grow large. Starting on the equivalent scalar channel and the fixed-point equation tying multiuser efficiency and MMSE, we extend it to the case of a dynamic channel, and derive lower and upper bounds for the MMSE (and, thus, for η as well) holding true in the limit of large signal–to–noise ratios and increasingly large observation time T.

Using composite estimators to improve both domain and total area estimation

In this article we propose using small area estimators to improve the estimatesof both the small and large area parameters. When the objective is to estimateparameters at both levels accurately, optimality is achieved by a mixed sampledesign of fixed and proportional allocations. In the mixed sample design, oncea sample size has been determined, one fraction of it is distributedproportionally among the different small areas while the rest is evenlydistributed among them. We use Monte Carlo simulations to assess theperformance of the direct estimator and two composite covariant-freesmall area estimators, for different sample sizes and different sampledistributions. Performance is measured in terms of Mean Squared Errors(MSE) of both small and large area parameters. It is found that the adoptionof small area composite estimators open the possibility of 1) reducingsample size when precision is given, or 2) improving precision for a givensample size.

Forecasting with missing data: Application to a real case

This paper presents a comparative analysis of linear and mixed modelsfor short term forecasting of a real data series with a high percentage of missing data. Data are the series of significant wave heights registered at regular periods of three hours by a buoy placed in the Bay of Biscay.The series is interpolated with a linear predictor which minimizes theforecast mean square error. The linear models are seasonal ARIMA models and themixed models have a linear component and a non linear seasonal component.The non linear component is estimated by a non parametric regression of dataversus time. Short term forecasts, no more than two days ahead, are of interestbecause they can be used by the port authorities to notice the fleet.Several models are fitted and compared by their forecasting behavior.

Estimadores compuestos en estadística regional: aplicación para la tasa de variación de la ocupación en la industria

This work is part of a project studying the performance of model basedestimators in a small area context. We have chosen a simple statisticalapplication in which we estimate the growth rate of accupation for severalregions of Spain. We compare three estimators: the direct one based onstraightforward results from the survey (which is unbiassed), and a thirdone which is based in a statistical model and that minimizes the mean squareerror.

Estimation of percentage body fat in 6- to 13-year-old children by skinfold thickness, body mass index and waist circumference.

We evaluated the accuracy of skinfold thicknesses, BMI and waist circumference for the prediction of percentage body fat (PBF) in a representative sample of 372 Swiss children aged 6-13 years. PBF was measured using dual-energy X-ray absorptiometry. On the basis of a preliminary bootstrap selection of predictors, seven regression models were evaluated. All models included sex, age and pubertal stage plus one of the following predictors: (1) log-transformed triceps skinfold (logTSF); (2) logTSF and waist circumference; (3) log-transformed sum of triceps and subscapular skinfolds (logSF2); (4) log-transformed sum of triceps, biceps, subscapular and supra-iliac skinfolds (logSF4); (5) BMI; (6) waist circumference; (7) BMI and waist circumference. The adjusted determination coefficient (R² adj) and the root mean squared error (RMSE; kg) were calculated for each model. LogSF4 (R² adj 0.85; RMSE 2.35) and logSF2 (R² adj 0.82; RMSE 2.54) were similarly accurate at predicting PBF and superior to logTSF (R² adj 0.75; RMSE 3.02), logTSF combined with waist circumference (R² adj 0.78; RMSE 2.85), BMI (R² adj 0.62; RMSE 3.73), waist circumference (R² adj 0.58; RMSE 3.89), and BMI combined with waist circumference (R² adj 0.63; RMSE 3.66) (P < 0.001 for all values of R² adj). The finding that logSF4 was only modestly superior to logSF2 and that logTSF was better than BMI and waist circumference at predicting PBF has important implications for paediatric epidemiological studies aimed at disentangling the effect of body fat on health outcomes.

Enhanced radar precipitation estimates using a combined clutter and beam blockage correction technique

Weather radar observations are currently the most reliable method for remote sensing of precipitation. However, a number of factors affect the quality of radar observations and may limit seriously automated quantitative applications of radar precipitation estimates such as those required in Numerical Weather Prediction (NWP) data assimilation or in hydrological models. In this paper, a technique to correct two different problems typically present in radar data is presented and evaluated. The aspects dealt with are non-precipitating echoes - caused either by permanent ground clutter or by anomalous propagation of the radar beam (anaprop echoes) - and also topographical beam blockage. The correction technique is based in the computation of realistic beam propagation trajectories based upon recent radiosonde observations instead of assuming standard radio propagation conditions. The correction consists of three different steps: 1) calculation of a Dynamic Elevation Map which provides the minimum clutter-free antenna elevation for each pixel within the radar coverage; 2) correction for residual anaprop, checking the vertical reflectivity gradients within the radar volume; and 3) topographical beam blockage estimation and correction using a geometric optics approach. The technique is evaluated with four case studies in the region of the Po Valley (N Italy) using a C-band Doppler radar and a network of raingauges providing hourly precipitation measurements. The case studies cover different seasons, different radio propagation conditions and also stratiform and convective precipitation type events. After applying the proposed correction, a comparison of the radar precipitation estimates with raingauges indicates a general reduction in both the root mean squared error and the fractional error variance indicating the efficiency and robustness of the procedure. Moreover, the technique presented is not computationally expensive so it seems well suited to be implemented in an operational environment.

Suboptimal filtering and nonlinear time scale transformation for the analysis of multiexponential decays

Multiexponential decays may contain time-constants differing in several orders of magnitudes. In such cases, uniform sampling results in very long records featuring a high degree of oversampling at the final part of the transient. Here, we analyze a nonlinear time scale transformation to reduce the total number of samples with minimum signal distortion, achieving an important reduction of the computational cost of subsequent analyses. We propose a time-varying filter whose length is optimized for minimum mean square error

On the consequences of misspecifing assumptions concerning residuals distribution in a repeated measures and nonlinear mixed modelling context

In this paper we describe the results of a simulation study performed to elucidate the robustness of the Lindstrom and Bates (1990) approximation method under non-normality of the residuals, under different situations. Concerning the fixed effects, the observed coverage probabilities and the true bias and mean square error values, show that some aspects of this inferential approach are not completely reliable. When the true distribution of the residuals is asymmetrical, the true coverage is markedly lower than the nominal one. The best results are obtained for the skew normal distribution, and not for the normal distribution. On the other hand, the results are partially reversed concerning the random effects. Soybean genotypes data are used to illustrate the methods and to motivate the simulation scenarios

Impact of incorrect assumptions on the covariance structure of random effects and/or residuals in nonlinear mixed models for repeated measures data

In this paper we analyse, using Monte Carlo simulation, the possible consequences of incorrect assumptions on the true structure of the random effects covariance matrix and the true correlation pattern of residuals, over the performance of an estimation method for nonlinear mixed models. The procedure under study is the well known linearization method due to Lindstrom and Bates (1990), implemented in the nlme library of S-Plus and R. Its performance is studied in terms of bias, mean square error (MSE), and true coverage of the associated asymptotic confidence intervals. Ignoring other criteria like the convenience of avoiding over parameterised models, it seems worst to erroneously assume some structure than do not assume any structure when this would be adequate.

Quantum fluctuations on domain walls, strings, and vacuum bubbles

We develop a covariant quantum theory of fluctuations on vacuum domain walls and strings. The fluctuations are described by a scalar field defined on the classical world sheet of the defects. We consider the following cases: straight strings and planar walls in flat space, true vacuum bubbles nucleating in false vacuum, and strings and walls nucleating during inflation. The quantum state for the perturbations is constructed so that it respects the original symmetries of the classical solution. In particular, for the case of vacuum bubbles and nucleating strings and walls, the geometry of the world sheet is that of a lower-dimensional de Sitter space, and the problem reduces to the quantization of a scalar field of tachyonic mass in de Sitter space. In all cases, the root-mean-squared fluctuation is evaluated in detail, and the physical implications are briefly discussed.

Revisiting Field Capacity (FC): variation of definition of FC and its estimation from pedotransfer functions

Taking into account the nature of the hydrological processes involved in in situ measurement of Field Capacity (FC), this study proposes a variation of the definition of FC aiming not only at minimizing the inadequacies of its determination, but also at maintaining its original, practical meaning. Analysis of FC data for 22 Brazilian soils and additional FC data from the literature, all measured according to the proposed definition, which is based on a 48-h drainage time after infiltration by shallow ponding, indicates a weak dependency on the amount of infiltrated water, antecedent moisture level, soil morphology, and the level of the groundwater table, but a strong dependency on basic soil properties. The dependence on basic soil properties allowed determination of FC of the 22 soil profiles by pedotransfer functions (PTFs) using the input variables usually adopted in prediction of soil water retention. Among the input variables, soil moisture content θ (6 kPa) had the greatest impact. Indeed, a linear PTF based only on it resulted in an FC with a root mean squared residue less than 0.04 m³ m-3 for most soils individually. Such a PTF proved to be a better FC predictor than the traditional method of using moisture content at an arbitrary suction. Our FC data were compatible with an equivalent and broader USA database found in the literature, mainly for medium-texture soil samples. One reason for differences between FCs of the two data sets of fine-textured soils is due to their different drainage times. Thus, a standardized procedure for in situ determination of FC is recommended.

Estimation of field capacity from ring infiltrometer-drainage data

Field capacity (FC) is a parameter widely used in applied soil science. However, its in situ method of determination may be difficult to apply, generally because of the need of large supplies of water at the test sites. Ottoni Filho et al. (2014) proposed a standardized procedure for field determination of FC and showed that such in situ FC can be estimated by a linear pedotransfer function (PTF) based on volumetric soil water content at the matric potential of -6 kPa [θ(6)] for the same soils used in the present study. The objective of this study was to use soil moisture data below a double ring infiltrometer measured 48 h after the end of the infiltration test in order to develop PTFs for standard in situ FC. We found that such ring FC data were an average of 0.03 m³ m- 3 greater than standard FC values. The linear PTF that was developed for the ring FC data based only on θ(6) was nearly as accurate as the equivalent PTF reported by Ottoni Filho et al. (2014), which was developed for the standard FC data. The root mean squared residues of FC determined from both PTFs were about 0.02 m³ m- 3. The proposed method has the advantage of estimating the soil in situ FC using the water applied in the infiltration test.