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We address the problem of estimating the difference between two probability densities. A naive approach is a two-step procedure of first estimating two densities separately and then computing their difference. However, this procedure does not necessarily work well because the first step is performed without regard to the second step, and thus a small estimation error incurred in the first stage can cause a big error in the second stage. In this letter, we propose a single-shot procedure for directly estimating the density difference without separately estimating two densities. We derive a nonparametric finite-sample error bound for the proposed single-shot density-difference estimator and show that it achieves the optimal convergence rate. We then show how the proposed density-difference estimator can be used in L2-distance approximation. Finally, we experimentally demonstrate the usefulness of the proposed method in robust distribution comparison such as class-prior estimation and change-point detection.
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Sasaki, Hiroaki, Yung-Kyun Noh, Gang Niu, and Masashi Sugiyama. "Direct Density Derivative Estimation." Neural Computation 28, no. 6 (June 2016): 1101–40. http://dx.doi.org/10.1162/neco_a_00835.
Estimating the derivatives of probability density functions is an essential step in statistical data analysis. A naive approach to estimate the derivatives is to first perform density estimation and then compute its derivatives. However, this approach can be unreliable because a good density estimator does not necessarily mean a good density derivative estimator. To cope with this problem, in this letter, we propose a novel method that directly estimates density derivatives without going through density estimation. The proposed method provides computationally efficient estimation for the derivatives of any order on multidimensional data with a hyperparameter tuning method and achieves the optimal parametric convergence rate. We further discuss an extension of the proposed method by applying regularized multitask learning and a general framework for density derivative estimation based on Bregman divergences. Applications of the proposed method to nonparametric Kullback-Leibler divergence approximation and bandwidth matrix selection in kernel density estimation are also explored.
Log-density gradient estimation is a fundamental statistical problem and possesses various practical applications such as clustering and measuring nongaussianity. A naive two-step approach of first estimating the density and then taking its log gradient is unreliable because an accurate density estimate does not necessarily lead to an accurate log-density gradient estimate. To cope with this problem, a method to directly estimate the log-density gradient without density estimation has been explored and demonstrated to work much better than the two-step method. The objective of this letter is to improve the performance of this direct method in multidimensional cases. Our idea is to regard the problem of log-density gradient estimation in each dimension as a task and apply regularized multitask learning to the direct log-density gradient estimator. We experimentally demonstrate the usefulness of the proposed multitask method in log-density gradient estimation and mode-seeking clustering.
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Hovda, Sigve. "Properties of Transmetric Density Estimation." International Journal of Statistics and Probability 5, no. 3 (April 13, 2016): 63. http://dx.doi.org/10.5539/ijsp.v5n3p63.
Transmetric density estimation is a generalization of kernel density estimation that is proposed in Hovda(2014) and Hovda (2016), This framework involves the possibility of making assumptions on the kernel of the distribution to improve convergence orders and to reduce the number of dimensions in the graphical display. In this paper we show that several state-of-the-art nonparametric, semiparametric and even parametric methods are special cases of this formulation, meaning that there is a unified approach. Moreover, it is shown that parameters can be trained using unbiased cross-validation. When parameter estimation is included, the mean integrated squared error of the transmetric density estimator is lower than for the common kernel density estimator, when the number of dimensions is larger than two.
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Liu, Qing, David Pitt, Xibin Zhang, and Xueyuan Wu. "A Bayesian Approach to Parameter Estimation for Kernel Density Estimation via Transformations." Annals of Actuarial Science 5, no. 2 (April 18, 2011): 181–93. http://dx.doi.org/10.1017/s1748499511000030.
AbstractIn this paper, we present a Markov chain Monte Carlo (MCMC) simulation algorithm for estimating parameters in the kernel density estimation of bivariate insurance claim data via transformations. Our data set consists of two types of auto insurance claim costs and exhibits a high-level of skewness in the marginal empirical distributions. Therefore, the kernel density estimator based on original data does not perform well. However, the density of the original data can be estimated through estimating the density of the transformed data using kernels. It is well known that the performance of a kernel density estimator is mainly determined by the bandwidth, and only in a minor way by the kernel. In the current literature, there have been some developments in the area of estimating densities based on transformed data, where bandwidth selection usually depends on pre-determined transformation parameters. Moreover, in the bivariate situation, the transformation parameters were estimated for each dimension individually. We use a Bayesian sampling algorithm and present a Metropolis-Hastings sampling procedure to sample the bandwidth and transformation parameters from their posterior density. Our contribution is to estimate the bandwidths and transformation parameters simultaneously within a Metropolis-Hastings sampling procedure. Moreover, we demonstrate that the correlation between the two dimensions is better captured through the bivariate density estimator based on transformed data.
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Beaumont, Chris, and B. W. Silverman. "Density Estimation." Journal of the Operational Research Society 37, no. 11 (November 1986): 1102. http://dx.doi.org/10.2307/2582699.
Methods for estimating the ratio of two probability density functions have been actively explored recently since they can be used for various data processing tasks such as non-stationarity adaptation, outlier detection, feature selection, and conditional probability estimation. In this paper, we propose a new density-ratio estimator which incorporates dimensionality reduction into the density-ratio estimation procedure. Through experiments, the proposed method is shown to compare favorably with existing density-ratio estimators in terms of both accuracy and computational costs.
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Li, Rui, and Youming Liu. "Wavelet Optimal Estimations for Density Functions under Severely Ill-Posed Noises." Abstract and Applied Analysis 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/260573.
Motivated by Lounici and Nickl's work (2011), this paper considers the problem of estimation of a densityfbased on an independent and identically distributed sampleY1,…,Ynfromg=f*φ. We show a wavelet optimal estimation for a density (function) over Besov ballBr,qs(L)andLprisk (1≤p<∞) in the presence of severely ill-posed noises. A wavelet linear estimation is firstly presented. Then, we prove a lower bound, which shows our wavelet estimator optimal. In other words, nonlinear wavelet estimations are not needed in that case. It turns out that our results extend some theorems of Pensky and Vidakovic (1999), as well as Fan and Koo (2002).
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Hovda, Sigve. "Transmetric Density Estimation." International Journal of Statistics and Probability 5, no. 2 (February 10, 2016): 35. http://dx.doi.org/10.5539/ijsp.v5n2p35.
<div>A transmetric is a generalization of a metric that is tailored to properties needed in kernel density estimation. Using transmetrics in kernel density estimation is an intuitive way to make assumptions on the kernel of the distribution to improve convergence orders and to reduce the number of dimensions in the graphical display. This framework is required for discussing the estimators that are suggested by Hovda (2014).</div><div> </div><div>Asymptotic arguments for the bias and the mean integrated squared error is difficult in the general case, but some results are given when the transmetric is of the type defined in Hovda (2014). An important contribution of this paper is that the convergence order can be as high as $4/5$, regardless of the number of dimensions.</div>
With the advent of the information technology, the amount of data we are facing today is growing in both the scale and the dimensionality dramatically. It thus raises new challenges for some traditional machine learning tasks. This thesis is mainly concerned with manifold aligned density estimation problems. In particular, the work presented in this thesis includes efficiently learning the density distribution on very large-scale datasets and estimating the manifold aligned density through explicit manifold modeling. First, we propose an efficient and sparse density estimator: Fast Parzen Windows (FPW) to represent the density of large-scale dataset by a mixture of locally fitted Gaussians components. The Gaussian components in the model are estimated in a "sloppy" way, which can avoid very time-consuming "global" optimizations, keep the simplicity of the density estimator and also assure the estimation accuracy. Preliminary theoretical work shows that the performance of the local fitted Gaussian components is related to the curvature of the true density and the characteristic of Gaussian model itself. A successful application of our FPW on principled calibrating the galaxy simulations is also demonstrated in the thesis. Then, we investigate the problem of manifold (i.e., low dimensional structure) aligned density estimation through explicit manifold modeling, which aims to obtain the embedded manifold and the density distribution simultaneously. A new manifold learning algorithm is proposed to capture the non-linear low dimensional structure and provides an improved initialization to Generative Topographic Mapping (GTM) model. The GTM models are then employed in our proposed hierarchical mixture model to estimate the density of data aligned along multiple manifolds. Extensive experiments verified the effectiveness of the presented work.
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Rademeyer, Estian. "Bayesian kernel density estimation." Diss., University of Pretoria, 2017. http://hdl.handle.net/2263/64692.
This dissertation investigates the performance of two-class classi cation credit scoring data
sets with low default ratios. The standard two-class parametric Gaussian and naive Bayes
(NB), as well as the non-parametric Parzen classi ers are extended, using Bayes' rule, to
include either a class imbalance or a Bernoulli prior. This is done with the aim of addressing
the low default probability problem. Furthermore, the performance of Parzen classi cation
with Silverman and Minimum Leave-one-out Entropy (MLE) Gaussian kernel bandwidth
estimation is also investigated. It is shown that the non-parametric Parzen classi ers yield
superior classi cation power.
However, there is a longing for these non-parametric classi ers to posses a predictive power,
such as exhibited by the odds ratio found in logistic regression (LR). The dissertation therefore
dedicates a section to, amongst other things, study the paper entitled \Model-Free Objective
Bayesian Prediction" (Bernardo 1999). Since this approach to Bayesian kernel density
estimation is only developed for the univariate and the uncorrelated multivariate case, the
section develops a theoretical multivariate approach to Bayesian kernel density estimation.
This approach is theoretically capable of handling both correlated as well as uncorrelated
features in data. This is done through the assumption of a multivariate Gaussian kernel
function and the use of an inverse Wishart prior. Dissertation (MSc)--University of Pretoria, 2017. The financial assistance of the National Research Foundation (NRF) towards this research is hereby acknowledged. Opinions expressed and conclusions arrived at, are those of the authors and are not necessarily to be attributed to the NRF. Statistics MSc Unrestricted
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Stride, Christopher B. "Semi-parametric density estimation." Thesis, University of Warwick, 1995. http://wrap.warwick.ac.uk/109619/.
The local likelihood method of Copas (1995a) allows for the incorporation into our parametric model of influence from data local to the point t at which we are estimating the true density function g(t). This is achieved through an analogy with censored data; we define the probability of a data point being considered observed, given that it has taken value xi, as where K is a scaled kernel function with smoothing parameter h. This leads to a likelihood function which gives more weight to observations close to t, hence the term ‘local likelihood’. After constructing this local likelihood function and maximising it at t, the resulting density estimate f(tiOt) can be described as semi-parametric in terms of its limits with respect to h. As h--oo, it approximates a standard parametric' fit f(I.O) whereas when h decreases towards 0, it approximates the non - parametric kernel density estimate. My thesis develops this idea, initially proving its asymptotic superiority over the standard parametric estimate under certain conditions. We then consider the improvements possible by making smoothing parameter h a function of /, enabling our semi parametric estimate to vary from approximating y(l) in regions of high density to f(t,0) in regions where we believe the true density to be low. Our improvement in accuracy is demonstrated in both simulated and real data examples, and the limits with respect to h and the new adaption parameter oo are examined. Methods for choosing h and oo are given and evaluated, along with a procedure for incorporating prior belief about the true form of the density into these choices. Further practical examples illustrate the effectiveness of I these ideas when applied to a wide range of data sets.
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Rossiter, Jane E. "Epidemiological applications of density estimation." Thesis, University of Oxford, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.291543.
Sung, Iyue. "Importance sampling kernel density estimation /." The Ohio State University, 2001. http://rave.ohiolink.edu/etdc/view?acc_num=osu1486398528559777.
Kile, Håkon. "Bandwidth Selection in Kernel Density Estimation." Thesis, Norwegian University of Science and Technology, Department of Mathematical Sciences, 2010. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-10015.
In kernel density estimation, the most crucial step is to select a proper bandwidth (smoothing parameter). There are two conceptually different approaches to this problem: a subjective and an objective approach. In this report, we only consider the objective approach, which is based upon minimizing an error, defined by an error criterion. The most common objective bandwidth selection method is to minimize some squared error expression, but this method is not without its critics. This approach is said to not perform satisfactory in the tail(s) of the density, and to put too much weight on observations close to the mode(s) of the density. An approach which minimizes an absolute error expression, is thought to be without these drawbacks. We will provide a new explicit formula for the mean integrated absolute error. The optimal mean integrated absolute error bandwidth will be compared to the optimal mean integrated squared error bandwidth. We will argue that these two bandwidths are essentially equal. In addition, we study data-driven bandwidth selection, and we will propose a new data-driven bandwidth selector. Our new bandwidth selector has promising behavior with respect to the visual error criterion, especially in the cases of limited sample sizes.
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Achilleos, Achilleas. "Deconvolution kernal density and regression estimation." Thesis, University of Bristol, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.544421.
In the realm of high-dimensional statistics, regression and classification have received much attention, while density estimation has lagged behind. Yet there are compelling scientific questions which can only be addressed via density estimation using high-dimensional data, such as the paths of North Atlantic tropical cyclones. If we cast each track as a single high-dimensional data point, density estimation allows us to answer such questions via integration or Monte Carlo methods. In this dissertation, I present three new methods for estimating densities and intensities for high-dimensional data, all of which rely on a technique called diffusion maps. This technique constructs a mapping for high-dimensional, complex data into a low-dimensional space, providing a new basis that can be used in conjunction with traditional density estimation methods. Furthermore, I propose a reordering of importance sampling in the high-dimensional setting. Traditional importance sampling estimates high-dimensional integrals with the aid of an instrumental distribution chosen specifically to minimize the variance of the estimator. In many applications, the integral of interest is with respect to an estimated density. I argue that in the high-dimensional realm, performance can be improved by reversing the procedure: instead of estimating a density and then selecting an appropriate instrumental distribution, begin with the instrumental distribution and estimate the density with respect to it directly. The variance reduction follows from the improved density estimate. Lastly, I present some initial results in using climatic predictors such as sea surface temperature as spatial covariates in point process estimation.
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Lu, Shan. "Essays on volatility forecasting and density estimation." Thesis, University of Aberdeen, 2019. http://digitool.abdn.ac.uk:80/webclient/DeliveryManager?pid=240161.
This thesis studies two subareas within the forecasting literature: volatility forecasting and risk-neutral density estimation and asks the question of how accurate volatility forecasts and risk-neutral density estimates can be made based on the given information. Two sources of information are employed to make those forecasts: historical information contained in time series of asset prices, and forward-looking information embedded in prices of traded options. Chapter 2 tests the comparative performance of two volatility scaling laws - the square-root-of-time (√T) and an empirical law, TH, characterized by the Hurst exponent (H) - where volatility is measured by sample standard deviation of returns, for forecasting the volatility term structure of crude oil price changes and ten foreign currency changes. We find that the empirical law is overall superior for crude oil, whereas the selection of a superior model is currency-specific and relative performance substantially differs across currencies. Our results are particularly important for regulatory risk management using Value-at-Risk and suggest the use of empirical law for volatility and quantile scaling. Chapter 3 studies the predictive ability of corridor implied volatility (CIV) measure. By adding CIV measures to the modified GARCH specifications, we show that narrow and mid-range CIVs outperform the wide CIVs, market volatility index and the BlackScholes implied volatility for horizons up to 21 days under various market conditions. Results of simulated trading reinforce our statistical findings. Chapter 4 compares six estimation methods for extracting risk-neutral densities (RND) from option prices. By using a pseudo-price based simulation, we find that the positive convolution approximation method provides the best performance, while mixture of two lognormals is the worst; In addition, we show that both price and volatility jumps are important components for option pricing. Our results have practical applications for policymakers as RNDs are important indicators to gauge market sentiment and expectations.
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Chan, Kwokleung. "Bayesian learning in classification and density estimation /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC IP addresses, 2002. http://wwwlib.umi.com/cr/ucsd/fullcit?p3061619.
Devroye, Luc, and Gábor Lugosi. Combinatorial Methods in Density Estimation. New York, NY: Springer New York, 2001. http://dx.doi.org/10.1007/978-1-4613-0125-7.
Book chapters on the topic "Estimation of Density":
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Györfi, Lázió, Wolfgang Härdle, Pascal Sarda, and Philippe Vieu. "Density Estimation." In Nonparametric Curve Estimation from Time Series, 53–79. New York, NY: Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3686-3_4.
Webb, Geoffrey I., Johannes Fürnkranz, Johannes Fürnkranz, Johannes Fürnkranz, Geoffrey Hinton, Claude Sammut, Joerg Sander, et al. "Density Estimation." In Encyclopedia of Machine Learning, 270. Boston, MA: Springer US, 2011. http://dx.doi.org/10.1007/978-0-387-30164-8_210.
Kolassa, John E. "Density Estimation." In An Introduction to Nonparametric Statistics, 143–48. First edition. | Boca Raton : CRC Press, 2020. |: Chapman and Hall/CRC, 2020. http://dx.doi.org/10.1201/9780429202759-8.
Sammut, Claude. "Density Estimation." In Encyclopedia of Machine Learning and Data Mining, 348–49. Boston, MA: Springer US, 2017. http://dx.doi.org/10.1007/978-1-4899-7687-1_210.
Lee, Myoung-jae. "Nonparametric Density Estimation." In Methods of Moments and Semiparametric Econometrics for Limited Dependent Variable Models, 123–42. New York, NY: Springer New York, 1996. http://dx.doi.org/10.1007/978-1-4757-2550-6_7.
Gu, Chong. "Probability Density Estimation." In Smoothing Spline ANOVA Models, 177–210. New York, NY: Springer New York, 2002. http://dx.doi.org/10.1007/978-1-4757-3683-0_6.
Härdle, Wolfgang. "Kernel Density Estimation." In Springer Series in Statistics, 43–84. New York, NY: Springer New York, 1991. http://dx.doi.org/10.1007/978-1-4612-4432-5_2.
Simonoff, Jeffrey S. "Multivariate Density Estimation." In Springer Series in Statistics, 96–133. New York, NY: Springer New York, 1996. http://dx.doi.org/10.1007/978-1-4612-4026-6_4.
Härdle, Wolfgang, Axel Werwatz, Marlene Müller, and Stefan Sperlich. "Nonparametric Density Estimation." In Springer Series in Statistics, 39–83. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-642-17146-8_3.
Conference papers on the topic "Estimation of Density":
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Ram, Parikshit, and Alexander G. Gray. "Density estimation trees." In the 17th ACM SIGKDD international conference. New York, New York, USA: ACM Press, 2011. http://dx.doi.org/10.1145/2020408.2020507.
JooSeuk Kim and Clayton Scott. "Robust kernel density estimation." In ICASSP 2008 - 2008 IEEE International Conference on Acoustics, Speech and Signal Processing. IEEE, 2008. http://dx.doi.org/10.1109/icassp.2008.4518376.
Miao, Yun-Qian, Ahmed K. Farahat, and Mohamed S. Kamel. "Discriminative Density-ratio Estimation." In Proceedings of the 2014 SIAM International Conference on Data Mining. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2014. http://dx.doi.org/10.1137/1.9781611973440.95.
Sun, Ke, and Stéphane Marchand-Maillet. "Information geometric density estimation." In BAYESIAN INFERENCE AND MAXIMUM ENTROPY METHODS IN SCIENCE AND ENGINEERING (MAXENT 2014). AIP Publishing LLC, 2015. http://dx.doi.org/10.1063/1.4905982.
Ting, Kai Ming, Takashi Washio, Jonathan R. Wells, and Hang Zhang. "Isolation Kernel Density Estimation." In 2021 IEEE International Conference on Data Mining (ICDM). IEEE, 2021. http://dx.doi.org/10.1109/icdm51629.2021.00073.
Yilan, Mikail, and Mehmet Kemal Ozdemir. "A simple approach to traffic density estimation by using Kernel Density Estimation." In 2015 23th Signal Processing and Communications Applications Conference (SIU). IEEE, 2015. http://dx.doi.org/10.1109/siu.2015.7130220.
Takahashi, Hiroshi, Tomoharu Iwata, Yuki Yamanaka, Masanori Yamada, and Satoshi Yagi. "Student-t Variational Autoencoder for Robust Density Estimation." In Twenty-Seventh International Joint Conference on Artificial Intelligence {IJCAI-18}. California: International Joint Conferences on Artificial Intelligence Organization, 2018. http://dx.doi.org/10.24963/ijcai.2018/374.
We propose a robust multivariate density estimator based on the variational autoencoder (VAE). The VAE is a powerful deep generative model, and used for multivariate density estimation. With the original VAE, the distribution of observed continuous variables is assumed to be a Gaussian, where its mean and variance are modeled by deep neural networks taking latent variables as their inputs. This distribution is called the decoder. However, the training of VAE often becomes unstable. One reason is that the decoder of VAE is sensitive to the error between the data point and its estimated mean when its estimated variance is almost zero. We solve this instability problem by making the decoder robust to the error using a Bayesian approach to the variance estimation: we set a prior for the variance of the Gaussian decoder, and marginalize it out analytically, which leads to proposing the Student-t VAE. Numerical experiments with various datasets show that training of the Student-t VAE is robust, and the Student-t VAE achieves high density estimation performance.
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Suga, Norisato, Kazuto Yano, Julian Webber, Yafei Hou, Toshihide Higashimori, and Yoshinori Suzuki. "Estimation of Probability Density Function Using Multi-bandwidth Kernel Density Estimation for Throughput." In 2020 International Conference on Artificial Intelligence in Information and Communication (ICAIIC). IEEE, 2020. http://dx.doi.org/10.1109/icaiic48513.2020.9065033.
Krauthausen, Peter, and Uwe D. Hanebeck. "Regularized non-parametric multivariate density and conditional density estimation." In 2010 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2010). IEEE, 2010. http://dx.doi.org/10.1109/mfi.2010.5604457.
Charikar, Moses, Michael Kapralov, Navid Nouri, and Paris Siminelakis. "Kernel Density Estimation through Density Constrained Near Neighbor Search." In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS). IEEE, 2020. http://dx.doi.org/10.1109/focs46700.2020.00025.
Marchette, David J., Carey E. Priebe, George W. Rogers, and Jeffrey L. Solka. Filtered Kernel Density Estimation. Fort Belvoir, VA: Defense Technical Information Center, October 1994. http://dx.doi.org/10.21236/ada288293.
Marchette, David J., Carey E. Priebe, George W. Rogers, and Jefferey L. Solka. Filtered Kernel Density Estimation. Fort Belvoir, VA: Defense Technical Information Center, October 1994. http://dx.doi.org/10.21236/ada290438.
Collins, David H. Density estimation with trigonometric kernels. Office of Scientific and Technical Information (OSTI), February 2016. http://dx.doi.org/10.2172/1237269.
Yu, Bin. Optimal Universal Coding and Density Estimation. Fort Belvoir, VA: Defense Technical Information Center, November 1994. http://dx.doi.org/10.21236/ada290694.
Rakhlin, Alexander, Dmitry Panchenko, and Sayan Mukherjee. Risk Bounds for Mixture Density Estimation. Fort Belvoir, VA: Defense Technical Information Center, January 2004. http://dx.doi.org/10.21236/ada459846.
Smith, Richard J., and Vitaliy Oryshchenko. Improved density and distribution function estimation. The IFS, July 2018. http://dx.doi.org/10.1920/wp.cem.2018.4718.
Powell, James L., Fengshi Niu, and Bryan S. Graham. Kernel density estimation for undirected dyadic data. The IFS, August 2019. http://dx.doi.org/10.1920/wp.cem.2019.3919.
Chen, X. R., P. R. Krishnaiah, and W. Q. Liang. Estimation of Multivariate Binary Density Using Orthonormal Functions. Fort Belvoir, VA: Defense Technical Information Center, December 1986. http://dx.doi.org/10.21236/ada186386.
Mellinger, David K. Detection, Classification, and Density Estimation of Marine Mammals. Fort Belvoir, VA: Defense Technical Information Center, October 2012. http://dx.doi.org/10.21236/ada579344.
Mizera, Ivan, and Roger Koenker. Shape constrained density estimation via penalized Rényi divergence. The IFS, September 2018. http://dx.doi.org/10.1920/wp.cem.2018.5418.
