Optimality of the Maximum Likelihood estimator in Astrometry

The problem of astrometry is revisited from the perspective of analyzing the attainability of well-known performance limits (the Cramer-Rao bound) for the estimation of the relative position of light-emitting (usually point-like) sources on a CCD-like detector using commonly adopted estimators such as the weighted least squares and the maximum likelihood. Novel technical results are presented to determine the performance of an estimator that corresponds to the solution of an optimization problem in the context of astrometry. Using these results we are able to place stringent bounds on the bias and the variance of the estimators in close form as a function of the data. We confirm these results through comparisons to numerical simulations under a broad range of realistic observing conditions. The maximum likelihood and the weighted least square estimators are analyzed. We confirm the sub-optimality of the weighted least squares scheme from medium to high signal-to-noise found in an earlier study for the (unweighted) least squares method. We find that the maximum likelihood estimator achieves optimal performance limits across a wide range of relevant observational conditions. Furthermore, from our results, we provide concrete insights for adopting an adaptive weighted least square estimator that can be regarded as a computationally efficient alternative to the optimal maximum likelihood solution. We provide, for the first time, close-form analytical expressions that bound the bias and the variance of the weighted least square and maximum likelihood implicit estimators for astrometry using a Poisson-driven detector. These expressions can be used to formally assess the precision attainable by these estimators in comparison with the minimum variance bound.Comment: 24 pages, 7 figures, 2 tables, 3 appendices. Accepted by Astronomy & Astrophysic

ESTIMASI KURVA REGRESI PADA DATA LONGITUDINAL DENGAN WEIGHTED LEAST SQUARE

Model varying-coefficient pada data longitudinal akan dikaji dalam proposal ini. Hubungan antara variabel respon dan prediktor diasumsikan linier pada waktu tertentu, tapi koefisien-koefisiennya berubah terhadap waktu. Estimator spline berdasarkan Weighted least square (WLS) digunakan untuk mengestimasi kurva regresi dari Model Varying Coefficient. Generalized Cross-Validation (GCV) digunakan untuk memilih titik knot optimal. Aplikasi pada proposal ini diterapkan pada data ACTG yaitu hubungan antara HIV RNA dan sel CD4 pada orang yang terinfeksi HIV dengan menggunakan bantuan software Matlab 7

Enhancing Positioning Accuracy Through RSS Based Ranging And Weighted Least Square Approximation

International audienceIn this paper, localization based on Received Signal Strength (RSS) is investigated assuming a path loss log normal shadowing model. RSS-based estimation schemes of ranges are investigated; three different schemes are studied: Mean, median and mode. Estimation of position is performed using weighted least square approximation. We show that the positioning accuracy depends on the used estimator of ranges from RSS observables. We suggest that typical median estimator must be replaced by maximum likelihood estimator (mode) to enhance the positioning accuracy. Monte Carlo simulations show that the estimation scheme based on the mode estimator performs better than those based on the median or the mean estimator; and that the use of Weighted Least square approximation enhances the accuracy comparing to typical unweighted least square approximation

PENERAPAN METODE WEIGTHED LEAST SQUARE UNTUK MENGATASI HETEROSKEDASTISITAS PADA ANALISIS REGRESI LINEAR

Analisis regresi merupakan analisis statistik yang mempalajari bagaimana memodelkan regresi linear. Jika model regresi linear memenuhi uji asumsi klasik dengan metode OLS maka mempunyai sifat BLUE (Beast Linear Unbiased Estimator). Uji heteroskedastisitas,yaitu varian error pada setiap nilai variabel bebas bernilai tidak konstan. Akibat dari heteroskedastisitas yaitu nilai parameter yang diperoleh tetap tidak bias tetapi varian penaksir yang diperoleh menjadi tidak efisien, artinya uji hipotesis yang dilakukan tidak akan memberikan hasil yang baik (tidak valid) atau prediksi koefisien-koefisien populasinya akan keliru. Oleh karena itu untuk mengetahui apakah terdapat heteroskedastisitas dilakukan uji White. Karena terdapat heteroskedastisitas pada skripsi ini, maka harus dilakukan transformasi dengan metode kuadrat trkecil tertimbang (Weighted Least Square). Kata Kunci: Uji Asumsi Klasik, Weighted least Square, Uji White. Regression analysis is a statistical analysis that learn how to model linear regression. If a linear regression model meets the Classic Assumption Test by OLS method, it has the nature of BLUE (Best Linear Unbiased Estimator). Error variance at each independent variable value is not constant. It means that heteroskedasticity test is unfulfilled and the classical assumption is not met.The result of heteroskedastisitas is that the parameter value remains biased but variance estimator becomes inefficient. It means thata hypothesis test wouldn’t give good results (not valid) or predictions coefficients of the population would be mislead. Therefore, to know whether there are heteroskedasticity, White test is conducted. Because heteroskedasticity exists in this thesis, transformation with weighted least squares method (Weighted Least Square) must be carried out. Keyword: Classic Assumption Test, Weighted least Square, White Test

Analisis Pendapatan Menggunakan Metode Weighted Least Square (Wls) dengan Fungsi Pembobot Huber

The Weighted Least Square (WLS) method is a development of the OLS method which will provide a more accurate solution than the OLS method when outliers are identified in the data. It is possible that the model generated by the OLS method still contains outliers, while the WLS method minimizes outliers in the data. The income result determined using the OLS method is less than the income result using the WLS method, which means that if a fixed value is given (according to the standard) for each result of the factors that affect the income result in the OLS method, the average fisherman income is IDR 2,225 .220. While the average income of fishermen using the WLS method is Rp. 1,015,840. There were two outliers identified using the OLS method and after using the WLS method there were no outliers