THE INTEGRATED MEAN SQUARED ERROR OF SERIES REGRESSION AND A ROSENTHAL HILBERT-SPACE INEQUALITY

2014 ◽  
Vol 31 (2) ◽  
pp. 337-361 ◽  
Author(s):  
Bruce E. Hansen
Keyword(s):  
Hilbert Space ◽  
Least Squares ◽  
Mean Squared Error ◽  
Least Squares Estimators ◽  
Squared Error ◽  
Statistics And Probability ◽  
Uniform Approximations ◽  
Regression Estimators ◽  
Berkeley Symposium ◽  
Integrated Mean Squared Error

This paper develops uniform approximations for the integrated mean squared error (IMSE) of nonparametric series regression estimators, including both least-squares and averaging least-squares estimators. To develop these approximations, we also generalize an important probability inequality of Rosenthal (1970, Israel Journal of Mathematics 8, 273–303; 1972, Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 149–167. University of California Press) to the case of Hilbert-space valued random variables.

Use of reflectance spectroscopy to estimate the organic carbon and CaCO3 contents of soils

2012 ◽  
Vol 61 (2) ◽  
pp. 277-290 ◽  
Author(s):  
Ádám Csorba ◽  
Vince Láng ◽  
László Fenyvesi ◽  
Erika Michéli
Keyword(s):  
Organic Carbon ◽  
Least Squares ◽  
Partial Least Squares ◽  
Partial Least Squares Regression ◽  
Mean Squared Error ◽  
Reflectance Spectroscopy ◽  
Least Squares Regression ◽  
Root Mean Squared Error ◽  
Squared Error

Napjainkban egyre nagyobb igény mutatkozik olyan technológiák és módszerek kidolgozására és alkalmazására, melyek lehetővé teszik a gyors, költséghatékony és környezetbarát talajadat-felvételezést és kiértékelést. Ezeknek az igényeknek felel meg a reflektancia spektroszkópia, mely az elektromágneses spektrum látható (VIS) és közeli infravörös (NIR) tartományában (350–2500 nm) végzett reflektancia-mérésekre épül. Figyelembe véve, hogy a talajokról felvett reflektancia spektrum információban nagyon gazdag, és a vizsgált tartományban számos talajalkotó rendelkezik karakterisztikus spektrális „ujjlenyomattal”, egyetlen görbéből lehetővé válik nagyszámú, kulcsfontosságú talajparaméter egyidejű meghatározása. Dolgozatunkban, a reflektancia spektroszkópia alapjaira helyezett, a talajok ösz-szetételének meghatározását célzó módszertani fejlesztés első lépéseit mutatjuk be. Munkánk során talajok szervesszén- és CaCO3-tartalmának megbecslését lehetővé tévő többváltozós matematikai-statisztikai módszerekre (részleges legkisebb négyzetek módszere, partial least squares regression – PLSR) épülő prediktív modellek létrehozását és tesztelését végeztük el. A létrehozott modellek tesztelése során megállapítottuk, hogy az eljárás mindkét talajparaméter esetében magas R2értéket [R2(szerves szén) = 0,815; R2(CaCO3) = 0,907] adott. A becslés pontosságát jelző közepes négyzetes eltérés (root mean squared error – RMSE) érték mindkét paraméter esetében közepesnek mondható [RMSE (szerves szén) = 0,467; RMSE (CaCO3) = 3,508], mely a reflektancia mérési előírások standardizálásával jelentősen javítható. Vizsgálataink alapján arra a következtetésre jutottunk, hogy a reflektancia spektroszkópia és a többváltozós kemometriai eljárások együttes alkalmazásával, gyors és költséghatékony adatfelvételezési és -értékelési módszerhez juthatunk.

Estimating an Autoregressive Current Effects Model of Sales Response when Observations are Aggregated over Time: Least Squares versus Maximum Likelihood

1988 ◽  
Vol 25 (3) ◽  
pp. 301-307
Author(s):  
Wilfried R. Vanhonacker
Keyword(s):  
Maximum Likelihood ◽  
Least Squares ◽  
Serial Correlation ◽  
Mean Squared Error ◽  
Generalized Least Squares ◽  
Parameter Estimates ◽  
Squared Error ◽  
Positive Serial Correlation ◽  
Serial Correlation Coefficient ◽  
Over Time

Estimating autoregressive current effects models is not straightforward when observations are aggregated over time. The author evaluates a familiar iterative generalized least squares (IGLS) approach and contrasts it to a maximum likelihood (ML) approach. Analytic and numerical results suggest that (1) IGLS and ML provide good estimates for the response parameters in instances of positive serial correlation, (2) ML provides superior (in mean squared error) estimates for the serial correlation coefficient, and (3) IGLS might have difficulty in deriving parameter estimates in instances of negative serial correlation.

Alternative Bias Approximations in Regressions with a Lagged-Dependent Variable

1993 ◽  
Vol 9 (1) ◽  
pp. 62-80 ◽  
Author(s):  
Jan F. Kiviet ◽  
Garry D.A. Phillips
Keyword(s):  
Least Squares ◽  
Mean Squared Error ◽  
Multiple Linear Regression Model ◽  
Small Sample ◽  
Large Sample ◽  
Squared Error ◽  
Coefficient Vector ◽  
Lagged Dependent Variable ◽  
Biased Estimators ◽  
The One

The small sample bias of the least-squares coefficient estimator is examined in the dynamic multiple linear regression model with normally distributed whitenoise disturbances and an arbitrary number of regressors which are all exogenous except for the one-period lagged-dependent variable. We employ large sample (T → ∞) and small disturbance (σ → 0) asymptotic theory and derive and compare expressions to O(T−1) and to O(σ2), respectively, for the bias in the least-squares coefficient vector. In some simulations and for an empirical example, we examine the mean (squared) error of these expressions and of corrected estimation procedures that yield estimates that are unbiased to O(T−l) and to O(σ2), respectively. The large sample approach proves to be superior, easily applicable, and capable of generating more efficient and less biased estimators.

Rapid and Early Detection of Salmonella Serotypes with Hyperspectral Microscopy and Multivariate Data Analysis†

2015 ◽  
Vol 78 (4) ◽  
pp. 668-674 ◽  
Author(s):  
MATTHEW EADY ◽  
BOSOON PARK ◽  
SUN CHOI
Keyword(s):  
Least Squares ◽  
Partial Least Squares ◽  
Partial Least Squares Regression ◽  
Mean Squared Error ◽  
Pearson Correlation ◽  
Least Squares Regression ◽  
Root Mean Squared Error ◽  
Squared Error ◽  
Microscope Imaging ◽  
Salmonella Serotypes

This study was designed to evaluate hyperspectral microscope images for early and rapid detection of Salmonella serotypes Enteritidis, Heidelberg, Infantis, Kentucky, and Typhimurium at incubation times of 6, 8, 10, 12, and 24 h. Images were collected by an acousto-optical tunable filter hyperspectral microscope imaging system with a metal halide light source measuring 89 contiguous wavelengths every 4 nm between 450 and 800 nm. Pearson correlation values were calculated for incubation times of 8, 10, and 12 h and compared with data for 24 h to evaluate the change in spectral signatures from bacterial cells over time. Regions of interest were analyzed at 30% of the pixels in an average cell size. Spectral data were preprocessed by applying a global data transformation algorithm and then subjected to principal component analysis (PCA). The Mahalanobis distance was calculated from PCA score plots for analyzing serotype cluster separation. Partial least-squares regression was applied for calibration and validation of the model, and soft independent modeling of class analogy was utilized to classify serotype clusters in the training set. Pearson correlation values indicate very similar spectral patterns for reduced incubation times ranging from 0.9869 to 0.9990. PCA score plots indicated cluster separation at all incubation times, with incubation time Mahalanobis distances of 2.146 to 27.071. Partial least-squares regression had a maximum root mean squared error of calibration of 0.0025 and a root mean squared error of validation of 0.0030. Soft independent modeling of class analogy correctly classified values at 8 h (98.32%), 10 h (96.67%), 12 h (88.33%), and 24 h (98.67%) with the optimal number of principal components (four or five). The results of this study suggest that Salmonella serotypes can be classified by applying a PCA to hyperspectral microscope imaging data from samples after only 8 h of incubation.

MEAN SQUARED ERROR COMPARISONS OF SOME BIASED REGRESSION ESTIMATORS

2001 ◽  
Vol 30 (2) ◽  
pp. 347-361 ◽  
Author(s):  
Sadullah Sakallioğlu ◽  
Selabattin Kaçiranlar ◽  
Fikri Akdeniz
Keyword(s):  
Mean Squared Error ◽  
Squared Error ◽  
Regression Estimators ◽  
Biased Regression
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