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Journal articles on the topic 'Sequential nonparametric kernel regression'

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Author: Grafiati

Published: 4 June 2021

Last updated: 5 February 2022

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1

Zenkov, I. V., A. V. Lapko, V. A. Lapko, S. T. Im, V. P. Tuboltsev, and V. L. Аvdeenok. "A nonparametric algorithm for automatic classification of large multivariate statistical data sets and its application." Computer Optics 45, no. 2 (April 2021): 253–60. http://dx.doi.org/10.18287/2412-6179-co-801.

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A nonparametric algorithm for automatic classification of large statistical data sets is proposed. The algorithm is based on a procedure for optimal discretization of the range of values of a random variable. A class is a compact group of observations of a random variable corresponding to a unimodal fragment of the probability density. The considered algorithm of automatic classification is based on the «compression» of the initial information based on the decomposition of a multidimensional space of attributes. As a result, a large statistical sample is transformed into a data array composed of the centers of multidimensional sampling intervals and the corresponding frequencies of random variables. To substantiate the optimal discretization procedure, we use the results of a study of the asymptotic properties of a kernel-type regression estimate of the probability density. An optimal number of sampling intervals for the range of values of one- and two-dimensional random variables is determined from the condition of the minimum root-mean square deviation of the regression probability density estimate. The results obtained are generalized to the discretization of the range of values of a multidimensional random variable. The optimal discretization formula contains a component that is characterized by a nonlinear functional of the probability density. An analytical dependence of the detected component on the antikurtosis coefficient of a one-dimensional random variable is established. For independent components of a multidimensional random variable, a methodology is developed for calculating estimates of the optimal number of sampling intervals for random variables and their lengths. On this basis, a nonparametric algorithm for the automatic classification is developed. It is based on a sequential procedure for checking the proximity of the centers of multidimensional sampling intervals and relationships between frequencies of the membership of the random variables from the original sample of these intervals. To further increase the computational efficiency of the proposed automatic classification algorithm, a multithreaded method of its software implementation is used. The practical significance of the developed algorithms is confirmed by the results of their application in processing remote sensing data.

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2

Malik, Azeem, and Wing Lon Ng. "Intraday liquidity patterns in limit order books." Studies in Economics and Finance 31, no. 1 (February 25, 2014): 46–71. http://dx.doi.org/10.1108/sef-11-2011-0093.

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Purpose – Algorithmic trading attempts to reduce trading costs by selecting optimal trade execution and scheduling algorithms. Whilst many common approaches only consider the bid-ask spread when measuring market impact, the authors aim to analyse the detailed limit order book data, which has more informational content. Design/methodology/approach – Using data from the London Stock Exchange's electronic SETS platform, the authors transform limit order book compositions into volume-weighted average price curves and accordingly estimate market impact. The regression coefficients of these curves are estimated, and their intraday patterns are revealed using a nonparametric kernel regression model. Findings – The authors find that market impact is nonlinear, time-varying, and asymmetric. Inferences drawn from marginal probabilities regarding Granger-causality do not show a significant impact of slope coefficients on the opposite side of the limit order book, thus implying that each side of the market is simultaneously rather than sequentially influenced by prevailing market conditions. Research limitations/implications – Results show that intraday seasonality patterns of liquidity may be exploited through trade scheduling algorithms in an attempt to minimise the trading costs associated with large institutional trades. Originality/value – The use of the detailed limit order book to reveal intraday patterns in liquidity provision offers better insight into the interactions of market participants. Such valuable information cannot be fully recovered from the traditional transaction data-based approaches.

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3

Eichner, Gerrit, and Winfried Stute. "Kernel adjusted nonparametric regression." Journal of Statistical Planning and Inference 142, no. 9 (September 2012): 2537–44. http://dx.doi.org/10.1016/j.jspi.2012.03.011.

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4

Gyorfi, Laszlo, Gabor Lugosi, and Frederic Udina. "NONPARAMETRIC KERNEL-BASED SEQUENTIAL INVESTMENT STRATEGIES." Mathematical Finance 16, no. 2 (April 2006): 337–57. http://dx.doi.org/10.1111/j.1467-9965.2006.00274.x.

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5

Zhao, Ge, and Yanyuan Ma. "Robust nonparametric kernel regression estimator." Statistics & Probability Letters 116 (September 2016): 72–79. http://dx.doi.org/10.1016/j.spl.2016.04.010.

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6

Halconruy, H., and N. Marie. "Kernel Selection in Nonparametric Regression." Mathematical Methods of Statistics 29, no. 1 (January 2020): 32–56. http://dx.doi.org/10.3103/s1066530720010044.

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7

Kyung-Joon, Cha, and William R. Schucany. "Nonparametric kernel regression estimation near endpoints." Journal of Statistical Planning and Inference 66, no. 2 (March 1998): 289–304. http://dx.doi.org/10.1016/s0378-3758(97)00082-7.

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8

Subramanian, Sundarraman. "Median regression using nonparametric kernel estimation." Journal of Nonparametric Statistics 14, no. 5 (January 2002): 583–605. http://dx.doi.org/10.1080/10485250213907.

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9

Andrews, Donald W. K. "Nonparametric Kernel Estimation for Semiparametric Models." Econometric Theory 11, no. 3 (June 1995): 560–86. http://dx.doi.org/10.1017/s0266466600009427.

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This paper presents a number of consistency results for nonparametric kernel estimators of density and regression functions and their derivatives. These results are particularly useful in semiparametric estimation and testing problems that rely on preliminary nonparametric estimators, as in Andrews (1994, Econometrica 62, 43–72). The results allow for near-epoch dependent, nonidentically distributed random variables, data-dependent bandwidth sequences, preliminary estimation of parameters (e.g., nonparametric regression based on residuals), and nonparametric regression on index functions.

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10

Okumura, Hidenori, and Kanta Naito. "Weighted kernel estimators in nonparametric binomial regression." Journal of Nonparametric Statistics 16, no. 1-2 (February 2004): 39–62. http://dx.doi.org/10.1080/10485250310001624828.

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11

Slaoui, Yousri, and Salah Khardani. "Nonparametric relative recursive regression." Dependence Modeling 8, no. 1 (October 1, 2020): 221–38. http://dx.doi.org/10.1515/demo-2020-0013.

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AbstractIn this paper, we propose the problem of estimating a regression function recursively based on the minimization of the Mean Squared Relative Error (MSRE), where outlier data are present and the response variable of the model is positive. We construct an alternative estimation of the regression function using a stochastic approximation method. The Bias, variance, and Mean Integrated Squared Error (MISE) are computed explicitly. The asymptotic normality of the proposed estimator is also proved. Moreover, we conduct a simulation to compare the performance of our proposed estimators with that of the two classical kernel regression estimators and then through a real Malaria dataset.

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12

Poměnková, Jitka. "Nonparametric estimate remarks." Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis 54, no. 3 (2006): 93–100. http://dx.doi.org/10.11118/actaun200654030093.

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Kernel smoothers belong to the most popular nonparametric functional estimates. They provide a simple way of finding structure in data. The idea of the kernel smoothing can be applied to a simple fixed design regression model. This article is focused on kernel smoothing for fixed design regresion model with three types of estimators, the Gasser-Müller estimator, the Nadaraya-Watson estimator and the local linear estimator. At the end of this article figures for ilustration of desribed estimators on simulated and real data sets are shown.

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13

Wang, Qiying, and Ying Xiang Rachel Wang. "NONPARAMETRIC COINTEGRATING REGRESSION WITH NNH ERRORS." Econometric Theory 29, no. 1 (July 6, 2012): 1–27. http://dx.doi.org/10.1017/s0266466612000205.

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This paper studies a nonlinear cointegrating regression model with nonlinear nonstationary heteroskedastic error processes. We establish uniform consistency for the conventional kernel estimate of the unknown regression function and develop atwo-stage approach for the estimation of the heterogeneity generating function.

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14

Dias, Ronaldo. "Sequential adaptive nonparametric regression via h-splines." Communications in Statistics - Simulation and Computation 28, no. 2 (January 1999): 501–15. http://dx.doi.org/10.1080/03610919908813562.

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15

Hidayat, Rahmat, I. Nyoman Budiantara, Bambang Widjanarko Otok, and Vita Ratnasari. "Kernel-Spline Estimation of Additive Nonparametric Regression Model." IOP Conference Series: Materials Science and Engineering 546 (June 26, 2019): 052028. http://dx.doi.org/10.1088/1757-899x/546/5/052028.

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16

Härdle, WOLFGANG, and GABRIELLE Kelly. "Nonparametric Kernel Regression Estimation-Optimal Choice of Bandwidth." Statistics 18, no. 1 (January 1987): 21–35. http://dx.doi.org/10.1080/02331888708801986.

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17

Suparti, Suparti, Budi Warsito, Rukun Santoso, Hasbi Yasin, Rezzy Eko Caraka, and Sudargo Sudargo. "Biresponses Kernel Nonparametric Regression: Inflation and Economic Growth." International Journal of Criminology and Sociology 10 (December 31, 2020): 465–71. http://dx.doi.org/10.6000/1929-4409.2021.10.54.

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The relation between inflation and economic growth is interesting to observe. To maintain the inflation rate, two factors should be taken into account, namely keeping the economic pulse at its optimal rate and keeping people's purchasing power from decreasing. Many factors influence the inflation and economic growth of a nation; one of which is the national bank interest rate. Since the data of inflation are closely related to economic growth, this study aims at modelling the data of inflation rate and economic growth of Central Java Province in Indonesia using bi-response kernel regression. Employing the data from the first trimester of 2007 up to those from the second trimester of 2019 which were processed using kernel Gauss, the best model to minimise the value of GCV was obtained with optimum h for inflation model amounting to 0.12 and 81.75 for economic growth model. The model performance was excellent because the MAPE out sample was less than 10%. The biresponses kernel model is better than the linear biresponses model in terms of GCV, MSE, R2, and MAPE values.

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18

Linton, Oliver, and Qiying Wang. "NONPARAMETRIC TRANSFORMATION REGRESSION WITH NONSTATIONARY DATA." Econometric Theory 32, no. 1 (October 10, 2014): 1–29. http://dx.doi.org/10.1017/s026646661400070x.

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We examine a kernel regression estimator for time series that takes account of the error correlation structure as proposed by Xiao et al. (2003, Journal of the American Statistical Association 98, 980–992). We show that this method continues to improve estimation in the case where the regressor is a unit root or a near unit root process.

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19

Mzyk, Grzegorz. "Generalized Kernel Regression Estimate for the Identification of Hammerstein Systems." International Journal of Applied Mathematics and Computer Science 17, no. 2 (June 1, 2007): 189–97. http://dx.doi.org/10.2478/v10006-007-0018-z.

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Generalized Kernel Regression Estimate for the Identification of Hammerstein SystemsA modified version of the classical kernel nonparametric identification algorithm for nonlinearity recovering in a Hammerstein system under the existence of random noise is proposed. The assumptions imposed on the unknown characteristic are weak. The generalized kernel method proposed in the paper provides more accurate results in comparison with the classical kernel nonparametric estimate, regardless of the number of measurements. The convergence in probability of the proposed estimate to the unknown characteristic is proved and the question of the convergence rate is discussed. Illustrative simulation examples are included.

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20

Altman, N. S. "An Introduction to Kernel and Nearest-Neighbor Nonparametric Regression." American Statistician 46, no. 3 (August 1992): 175. http://dx.doi.org/10.2307/2685209.

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21

Müller, H. G. "Density adjusted kernel smoothers for random design nonparametric regression." Statistics & Probability Letters 36, no. 2 (December 1997): 161–72. http://dx.doi.org/10.1016/s0167-7152(97)00059-x.

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22

Altman, N. S. "An Introduction to Kernel and Nearest-Neighbor Nonparametric Regression." American Statistician 46, no. 3 (August 1992): 175–85. http://dx.doi.org/10.1080/00031305.1992.10475879.

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23

Cheng, C. B., and E. S. Lee. "Nonparametric fuzzy regression—k-NN and kernel smoothing techniques." Computers & Mathematics with Applications 38, no. 3-4 (August 1999): 239–51. http://dx.doi.org/10.1016/s0898-1221(99)00198-4.

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24

Moon, Young-Il. "A Nonparametric Kernel Regression Estimator for Flood Frequency Analysis." International Journal of Urban Sciences 3, no. 1 (April 1999): 14–22. http://dx.doi.org/10.1080/12265934.1999.9693433.

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25

Woodfield, Terry J. "The Selection of Window Widths in Kernel Nonparametric Regression." Journal of Statistical Computation and Simulation 23, no. 1-2 (December 1985): 113–22. http://dx.doi.org/10.1080/00949658508810861.

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26

Wang, N. "Marginal nonparametric kernel regression accounting for within-subject correlation." Biometrika 90, no. 1 (March 1, 2003): 43–52. http://dx.doi.org/10.1093/biomet/90.1.43.

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27

Gao, Jiti, Anthony N. Pettitt, and Rodney C. L. Wolff. "Local linear kernel estimation for discontinuous nonparametric regression functions." Communications in Statistics - Theory and Methods 27, no. 12 (January 1998): 2871–94. http://dx.doi.org/10.1080/03610929808832261.

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28

Girard, S., and J. Saracco. "An Introduction to Dimension Reduction in Nonparametric Kernel Regression." EAS Publications Series 66 (2014): 167–96. http://dx.doi.org/10.1051/eas/1466012.

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29

Zougab, Nabil, Smail Adjabi, and Célestin C. Kokonendji. "Bayesian Approach in Nonparametric Count Regression with Binomial Kernel." Communications in Statistics - Simulation and Computation 43, no. 5 (October 24, 2013): 1052–63. http://dx.doi.org/10.1080/03610918.2012.725145.

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30

Harms, Torsten, and Pierre Duchesne. "On kernel nonparametric regression designed for complex survey data." Metrika 72, no. 1 (March 12, 2009): 111–38. http://dx.doi.org/10.1007/s00184-009-0244-5.

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31

Lindström, T., U. Holst, P. Weibring, and H. Edner. "Analysis of lidar measurements using nonparametric kernel regression methods." Applied Physics B: Lasers and Optics 74, no. 2 (February 1, 2002): 155–65. http://dx.doi.org/10.1007/s003400100781.

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32

Dharmasena, L. Sandamali, Basil M. De Silva, and Panlop Zeephongsekul. "Two stage sequential procedure for nonparametric regression estimation." ANZIAM Journal 49 (August 15, 2008): 699. http://dx.doi.org/10.21914/anziamj.v49i0.337.

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33

Efromovich, Sam. "Sequential Design and Estimation in Heteroscedastic Nonparametric Regression." Sequential Analysis 26, no. 1 (January 22, 2007): 3–25. http://dx.doi.org/10.1080/07474940601109670.

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34

IVAN, KOMANG CANDRA, I. WAYAN SUMARJAYA, and MADE SUSILAWATI. "ANALISIS MODEL REGRESI NONPARAMETRIK SIRKULAR-LINEAR BERGANDA." E-Jurnal Matematika 5, no. 2 (May 31, 2016): 52. http://dx.doi.org/10.24843/mtk.2016.v05.i02.p121.

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Circular data are data which the value in form of vector is circular data. Statistic analysis that is used in analyzing circular data is circular statistics analysis. In regression analysis, if any of predictor or response variables or both are circular then the regression analysis used is called circular regression analysis. Observation data in circular statistic which use direction and time units usually don’t satisfy all of the parametric assumptions, thus making nonparametric regression as a good solution. Nonparametric regression function estimation is using epanechnikov kernel estimator for the linier variables and von Mises kernel estimator for the circular variable. This study showed that the result of circular analysis by using circular descriptive statistic is better than common statistic. Multiple circular-linier nonparametric regressions with Epanechnikov and von Mises kernel estimator didn’t create estimation model explicitly as parametric regression does, but create estimation from its observation knots instead.

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35

Lestari, Budi. "Estimasi Fungsi Regresi Dalam Model Regresi Nonparametrik Birespon Menggunakan Estimator Smoothing Spline dan Estimator Kernel." Jurnal Matematika Statistika dan Komputasi 15, no. 2 (December 20, 2018): 20. http://dx.doi.org/10.20956/jmsk.v15i2.5710.

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Abstract Regression model of bi-respond nonparametric is a regression model which is illustrating of the connection pattern between respond variable and one or more predictor variables, where between first respond and second respond have correlation each other. In this paper, we discuss the estimating functions of regression in regression model of bi-respond nonparametric by using different two estimation techniques, namely, smoothing spline and kernel. This study showed that for using smoothing spline and kernel, the estimator function of regression which has been obtained in observation is a regression linier. In addition, both estimators that are obtained from those two techniques are systematically only different on smoothing matrices. Keywords: kernel estimator, smoothing spline estimator, regression function, bi-respond nonparametric regression model. AbstrakModel regresi nonparametrik birespon adalah suatu model regresi yang menggambarkan pola hubungan antara dua variabel respon dan satu atau beberapa variabel prediktor dimana antara respon pertama dan respon kedua berkorelasi. Dalam makalah ini dibahas estimasi fungsi regresi dalam model regresi nonparametrik birespon menggunakan dua teknik estimasi yang berbeda, yaitu smoothing spline dan kernel. Hasil studi ini menunjukkan bahwa, baik menggunakan smoothing spline maupun menggunakan kernel, estimator fungsi regresi yang didapatkan merupakan fungsi linier dalam observasi. Selain itu, kedua estimator fungsi regresi yang didapatkan dari kedua teknik estimasi tersebut secara matematis hanya dibedakan oleh matriks penghalusnya.Kata Kunci : Estimator Kernel, Estimator Smoothing Spline, Fungsi Regresi, Model Regresi Nonparametrik Birespon.

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36

Hulle, Marc M. Van. "Kernel-Based Equiprobabilistic Topographic Map Formation." Neural Computation 10, no. 7 (October 1, 1998): 1847–71. http://dx.doi.org/10.1162/089976698300017179.

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We introduce a new unsupervised competitive learning rule, the kernel-based maximum entropy learning rule (kMER), which performs equiprobabilistic topographic map formation in regular, fixed-topology lattices, for use with nonparametric density estimation as well as nonparametric regression analysis. The receptive fields of the formal neurons are overlapping radially symmetric kernels, compatible with radial basis functions (RBFs); but unlike other learning schemes, the radii of these kernels do not have to be chosen in an ad hoc manner: the radii are adapted to the local input density, together with the weight vectors that define the kernel centers, so as to produce maps of which the neurons have an equal probability to be active (equiprobabilistic maps). Both an “online” and a “batch” version of the learning rule are introduced, which are applied to nonparametric density estimation and regression, respectively. The application envisaged is blind source separation (BSS) from nonlinear, noisy mixtures.

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37

Schmid, Wolfgang, and Ansgar Steland. "Sequential control of non-stationary processes by nonparametric kernel control charts." Allgemeines Statistisches Archiv 84, no. 3 (September 21, 2000): 315–36. http://dx.doi.org/10.1007/s101820000035.

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38

Mahmoud, Hamdy F. F. "Parametric Versus Semi and Nonparametric Regression Models." International Journal of Statistics and Probability 10, no. 2 (February 23, 2021): 90. http://dx.doi.org/10.5539/ijsp.v10n2p90.

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There are three common types of regression models: parametric, semiparametric and nonparametric regression. The model should be used to fit the real data depends on how much information is available about the form of the relationship between the response variable and explanatory variables, and the random error distribution that is assumed. Researchers need to be familiar with each modeling approach requirements. In this paper, differences between these models, common estimation methods, robust estimation, and applications are introduced. For parametric models, there are many known methods of estimation, such as least squares and maximum likelihood methods which are extensively studied but they require strong assumptions. On the other hand, nonparametric regression models are free of assumptions regarding the form of the response-explanatory variables relationships but estimation methods, such as kernel and spline smoothing are computationally expensive and smoothing parameters need to be obtained. For kernel smoothing there two common estimators: local constant and local linear smoothing methods. In terms of bias, especially at the boundaries of the data range, local linear is better than local constant estimator.  Robust estimation methods for linear models are well studied, however the robust estimation methods in nonparametric regression methods are limited. A robust estimation method for the semiparametric and nonparametric regression models is introduced.

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39

Dioggban, Jakperik. "A note non Nonparametric Regression Modeling using a Density Function." African Journal of Applied Statistics 7, no. 2 (July 1, 2020): 993–1000. http://dx.doi.org/10.16929/ajas/2020.993.252.

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The nonparametric regression offers alternative to classical regression analysis when the data are not well behaved or when the classical regression model shows significant lack of fit. In recent years, It has been applied using Kernel estimators and the smoothing splines, but these methods wields some bias of estimation. In this study, a semi-parametric multiplicative bias reduction density function was used to develop a non parametric regression model. Simulation studies conducted showed that the proposed estimator performs better than both the Kernel and the smoothing splines estimators especially with large samples

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40

Cadigan, N. G., and J. Chen. "Kernel Regression Estimators for Nonparametric Model Calibration in Survey Sampling." Journal of Statistical Theory and Practice 4, no. 1 (March 2010): 1–25. http://dx.doi.org/10.1080/15598608.2010.10411970.

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41

Scheike, Thomas H. "Nonparametric kernel regression when the regressor follows a counting process." Journal of Nonparametric Statistics 6, no. 4 (January 1996): 337–53. http://dx.doi.org/10.1080/10485259608832680.

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42

Wang, Lu, Andrea Rotnitzky, and Xihong Lin. "Nonparametric Regression With Missing Outcomes Using Weighted Kernel Estimating Equations." Journal of the American Statistical Association 105, no. 491 (September 1, 2010): 1135–46. http://dx.doi.org/10.1198/jasa.2010.tm08463.

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43

Tsybakov, A. B. "On the Choice of the Bandwidth in Kernel Nonparametric Regression." Theory of Probability & Its Applications 32, no. 1 (January 1988): 142–48. http://dx.doi.org/10.1137/1132018.

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44

Müller, Hans-Georg. "Weighted Local Regression and Kernel Methods for Nonparametric Curve Fitting." Journal of the American Statistical Association 82, no. 397 (March 1987): 231–38. http://dx.doi.org/10.1080/01621459.1987.10478425.

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45

Muller, Hans-Georg. "Weighted Local Regression and Kernel Methods for Nonparametric Curve Fitting." Journal of the American Statistical Association 82, no. 397 (March 1987): 231. http://dx.doi.org/10.2307/2289159.

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46

Linton, Oliver B., and David T. Jacho-Chávez. "On internally corrected and symmetrized kernel estimators for nonparametric regression." TEST 19, no. 1 (March 14, 2009): 166–86. http://dx.doi.org/10.1007/s11749-009-0145-y.

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47

Lin, Lu, and Xia Cui. "Stahel-Donoho kernel estimation for fixed design nonparametric regression models." Science in China Series A: Mathematics 49, no. 12 (December 2006): 1879–96. http://dx.doi.org/10.1007/s11425-006-2038-9.

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48

Rahmawati, Dyah P., I. N. Budiantara, Dedy D. Prastyo, and Made A. D. Octavanny. "Mixed Spline Smoothing and Kernel Estimator in Biresponse Nonparametric Regression." International Journal of Mathematics and Mathematical Sciences 2021 (March 11, 2021): 1–14. http://dx.doi.org/10.1155/2021/6611084.

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Mixed estimators in nonparametric regression have been developed in models with one response. The biresponse cases with different patterns among predictor variables that tend to be mixed estimators are often encountered. Therefore, in this article, we propose a biresponse nonparametric regression model with mixed spline smoothing and kernel estimators. This mixed estimator is suitable for modeling biresponse data with several patterns (response vs. predictors) that tend to change at certain subintervals such as the spline smoothing pattern, and other patterns that tend to be random are commonly modeled using kernel regression. The mixed estimator is obtained through two-stage estimation, i.e., penalized weighted least square (PWLS) and weighted least square (WLS). Furthermore, the proposed biresponse modeling with mixed estimators is validated using simulation data. This estimator is also applied to the percentage of the poor population and human development index data. The results show that the proposed model can be appropriately implemented and gives satisfactory results.

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49

Martinsek, Adam T. "Sequential methods for bounding the error in nonparametric regression." Sequential Analysis 13, no. 1 (January 1994): 63–75. http://dx.doi.org/10.1080/07474949408836294.

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50

Bolbolian Ghalibaf, Mohammad. "Kernel Function in Local Linear Peters-Belson Regression." Revista Colombiana de Estadística 41, no. 2 (July 1, 2018): 235–49. http://dx.doi.org/10.15446/rce.v41n2.65654.

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Determining the extent of a disparity, if any, between groups of people, for example, race or gender, is of interest in many fields, including public health for medical treatment and prevention of disease or in discrimination cases concerning equal pay to estimate the pay disparities between minority and majority employees. An observed difference in the mean outcome between a majority/advantaged group (AG) and minority/disadvantaged group (DG) can be due to differences in the distribution of relevant covariates. The Peters Belson (PB) method fits a regression model with covariates to the AG to predict, for each DG member, their outcome measure as if they had been from the AG. The difference between the mean predicted and the mean observed outcomes of DG members is the (unexplained) disparity of interest. PB regression is a form of statistical matching, akin in spirit to Bhattacharya's band-width matching. In this paper we review the use of PB regression in legal cases from Hikawa et al. (2010b) Parametric and nonparametric approaches to PB regression are described and we show that in nonparametric PB regression choose a kernel function can be better resulted, i.e. by selecting the appropriate kernel function we can reduce bias and variance of estimators, also increase power of test.

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