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Journal articles on the topic 'Density estimation'

Author: Grafiati

Published: 4 June 2021

Last updated: 9 June 2025

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1

Sugiyama, Masashi, Takafumi Kanamori, Taiji Suzuki, Marthinus Christoffel du Plessis, Song Liu, and Ichiro Takeuchi. "Density-Difference Estimation." Neural Computation 25, no. 10 (October 2013): 2734–75. http://dx.doi.org/10.1162/neco_a_00492.

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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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2

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.

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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.

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3

Yamane, Ikko, Hiroaki Sasaki, and Masashi Sugiyama. "Regularized Multitask Learning for Multidimensional Log-Density Gradient Estimation." Neural Computation 28, no. 7 (July 2016): 1388–410. http://dx.doi.org/10.1162/neco_a_00844.

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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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4

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.

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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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Yamada, Makoto, and Masashi Sugiyama. "Direct Density-Ratio Estimation with Dimensionality Reduction via Hetero-Distributional Subspace Analysis." Proceedings of the AAAI Conference on Artificial Intelligence 25, no. 1 (August 4, 2011): 549–54. http://dx.doi.org/10.1609/aaai.v25i1.7905.

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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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6

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.

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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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7

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.

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8

Sheather, Simon J. "Density Estimation." Statistical Science 19, no. 4 (November 2004): 588–97. http://dx.doi.org/10.1214/088342304000000297.

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9

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.

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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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10

Tayeb, Djebbouri, and Djerfi Kouider. "On Kernel density estimation on Product Riemannian manifolds." STUDIES IN ENGINEERING AND EXACT SCIENCES 5, no. 3 (December 31, 2024): e12912. https://doi.org/10.54021/seesv5n3-114.

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Estimating the kernel density function of a random vector taking values on Riemannian manifolds are considered. More Precisely, we consider the problem of the estimation of the probability density of n i.i.d. random objects on the product of two compact Riemannian manifolds without boundary. The proposed methodology adapts the Pelletier’s approach which is the kernel density estimation on non Euclidean setting. Under sufficient regularity assumptions on the underlying density, L2 convergence rates are obtained. Riemannian products are the most natural and fruitful generalization of Cartesian products. These manifold models are of interest to many fields of applied mathematics, natural sciences and technology. Let us recall here as examples, the use of these products in the domain of computer vision and in biomedical sciences. In this note, we construct an estimator of the density on a product manifold from one of the estimators existing in the literature. Among the estimation methods, we mainly cite a method based on Fourier developments proposed in (Hendriks, 1990). Another method based on geodesic distance is proposed later by B. Pelletier in (Pelletier, 2005). This method used in the present work, is a generalization of the kernel method in the affine case. Another method recently developed by Kim et al. in (Kim, 2013) use the concept of exponential map to define a kernel estimator of the density on a Riemannian manifold. The aim in the present work, is to show that Pelletier's method is the best suited to the case of the Riemannian product.

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11

Fujii, Takayuki. "Nonparametric Estimation for a Class of Piecewise-Deterministic Markov Processes." Journal of Applied Probability 50, no. 4 (December 2013): 931–42. http://dx.doi.org/10.1239/jap/1389370091.

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In this paper we study nonparametric estimation problems for a class of piecewise-deterministic Markov processes (PDMPs). Borovkov and Last (2008) proved a version of Rice's formula for PDMPs, which explains the relation between the stationary density and the level crossing intensity. From a statistical point of view, their result suggests a methodology for estimating the stationary density from observations of a sample path of PDMPs. First, we introduce the local time related to the level crossings and construct the local-time estimator for the stationary density, which is unbiased and uniformly consistent. Secondly, we investigate other estimation problems for the jump intensity and the conditional jump size distribution.

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Fujii, Takayuki. "Nonparametric Estimation for a Class of Piecewise-Deterministic Markov Processes." Journal of Applied Probability 50, no. 04 (December 2013): 931–42. http://dx.doi.org/10.1017/s0021900200013711.

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In this paper we study nonparametric estimation problems for a class of piecewise-deterministic Markov processes (PDMPs). Borovkov and Last (2008) proved a version of Rice's formula for PDMPs, which explains the relation between the stationary density and the level crossing intensity. From a statistical point of view, their result suggests a methodology for estimating the stationary density from observations of a sample path of PDMPs. First, we introduce the local time related to the level crossings and construct the local-time estimator for the stationary density, which is unbiased and uniformly consistent. Secondly, we investigate other estimation problems for the jump intensity and the conditional jump size distribution.

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13

Dai, Mingrui, Weifeng Shi, and Guohua Li. "Image based fog density estimation." PLOS One 20, no. 6 (June 2, 2025): e0323536. https://doi.org/10.1371/journal.pone.0323536.

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Although the application of image-based fog density estimation brings excellent convenience and low-cost methods, the accuracy of such methods still needs to be improved, and further research is encouraged on accuracy evaluation methods. To improve the accuracy and computational efficiency of fog density estimation in images, we first construct three image features based on the image dark channel information, the image saturation information, and the proportion of gray noise points, respectively. Then, we use a feature fusion method to estimate fog density in the images. In addition, two indicators have been constructed to evaluate the accuracy of various fog density estimation methods. These two indicators are the sequential error indicator and the proportional error indicator, which are calculated using fog image sequences with known density values. These two new indicators enable the evaluation of any fog density estimation method in terms of the ability to maintain order and ratio values. The experimental results show that the proposed method can effectively estimate the fog densities of images and display the best performance among the eight latest image-based methods for estimating fog density; the three features used in the proposed method significantly impact the effectiveness of image-based fog density estimation. The proposed method has been illustrated for fog density analysis of indoor and outdoor surveillance videos. The source code is available at https://github.com/Dai-MR/ImageFogDensityEsitmation.

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14

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.

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<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>

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15

Arefi, Mohsen, Reinhard Viertl, and S. Mahmoud Taheri. "Fuzzy density estimation." Metrika 75, no. 1 (April 29, 2010): 5–22. http://dx.doi.org/10.1007/s00184-010-0311-y.

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16

Locke, Judson B., and Adrian M. Peter. "Multiwavelet density estimation." Applied Mathematics and Computation 219, no. 11 (February 2013): 6002–15. http://dx.doi.org/10.1016/j.amc.2012.11.099.

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17

Kronenfeld, Barry J. "A Plotless Density Estimator Based on the Asymptotic Limit of Ordered Distance Estimation Values." Forest Science 55, no. 4 (August 1, 2009): 283–92. http://dx.doi.org/10.1093/forestscience/55.4.283.

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Abstract Estimation of tree density from point-tree distances is an attractive option for quick inventory of new sites, but estimators that are unbiased in clustered and dispersed situations have not been found. Noting that bias of an estimator derived from distances to the kth nearest neighbor from a random point tends to decrease with increasing k, a method is proposed for estimating the limit of an asymptotic function through a set of ordered distance estimators. A standard asymptotic model is derived from the limiting case of a clustered distribution. The proposed estimator is evaluated against 13 types of simulated generating processes, including random, clustered, dispersed, and mixed. Performance is compared with ordered distance estimation of the same rank and with fixed-area sampling with the same number of trees tallied. The proposed estimator consistently performs better than ordered distance estimation and nearly as well as fixed-area sampling in all but the most clustered situations. The estimator also provides information regarding the degree of clustering or dispersion.

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18

Zinde-Walsh, Victoria. "KERNEL ESTIMATION WHEN DENSITY MAY NOT EXIST." Econometric Theory 24, no. 3 (February 26, 2008): 696–725. http://dx.doi.org/10.1017/s0266466608080298.

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Nonparametric kernel estimation of density and conditional mean is widely used, but many of the pointwise and global asymptotic results for the estimators are not available unless the density is continuous and appropriately smooth; in kernel estimation for discrete-continuous cases smoothness is required for the continuous variables. Nonsmooth density and mass points in distributions arise in various situations that are examined in empirical studies; some examples and explanations are discussed in the paper. Generally, any distribution function consists of absolutely continuous, discrete, and singular components, but only a few special cases of nonparametric estimation involving singularity have been examined in the literature, and asymptotic theory under the general setup has not been developed. In this paper the asymptotic process for the kernel estimator is examined by means of the generalized functions and generalized random processes approach; it provides a unified theory because density and its derivatives can be defined as generalized functions for any distribution, including cases with singular components. The limit process for the kernel estimator of density is fully characterized in terms of a generalized Gaussian process. Asymptotic results for the Nadaraya–Watson conditional mean estimator are also provided.

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19

Kwasniok, Frank. "Semiparametric maximum likelihood probability density estimation." PLOS ONE 16, no. 11 (November 9, 2021): e0259111. http://dx.doi.org/10.1371/journal.pone.0259111.

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A comprehensive methodology for semiparametric probability density estimation is introduced and explored. The probability density is modelled by sequences of mostly regular or steep exponential families generated by flexible sets of basis functions, possibly including boundary terms. Parameters are estimated by global maximum likelihood without any roughness penalty. A statistically orthogonal formulation of the inference problem and a numerically stable and fast convex optimization algorithm for its solution are presented. Automatic model selection over the type and number of basis functions is performed with the Bayesian information criterion. The methodology can naturally be applied to densities supported on bounded, infinite or semi-infinite domains without boundary bias. Relationships to the truncated moment problem and the moment-constrained maximum entropy principle are discussed and a new theorem on the existence of solutions is contributed. The new technique compares very favourably to kernel density estimation, the diffusion estimator, finite mixture models and local likelihood density estimation across a diverse range of simulation and observation data sets. The semiparametric estimator combines a very small mean integrated squared error with a high degree of smoothness which allows for a robust and reliable detection of the modality of the probability density in terms of the number of modes and bumps.

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20

Hodgson, Douglas J. "ADAPTIVE ESTIMATION OF ERROR CORRECTION MODELS." Econometric Theory 14, no. 1 (February 1998): 44–69. http://dx.doi.org/10.1017/s0266466698141026.

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This paper considers adaptive maximum likelihood estimation of reduced rank vector error correction models. It is shown that such models can be asymptotically efficiently estimated even in the absence of knowledge of the shape of the density function of the innovation sequence, provided that this density is symmetric. The construction of the estimator, involving the nonparametric kernel estimation of the unknown density using the residuals of a consistent preliminary estimator, is described, and its asymptotic distribution is derived. Asymptotic efficiency gains over the Gaussian pseudo–maximum likelihood estimator are evaluated for elliptically symmetric innovations.

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21

Rathke, Fabian, and Christoph Schnörr. "A Computational Approach to Log-Concave Density Estimation." Analele Universitatii "Ovidius" Constanta - Seria Matematica 23, no. 3 (November 1, 2015): 151–66. http://dx.doi.org/10.1515/auom-2015-0053.

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Abstract Non-parametric density estimation with shape restrictions has witnessed a great deal of attention recently. We consider the maximum-likelihood problem of estimating a log-concave density from a given finite set of empirical data and present a computational approach to the resulting optimization problem. Our approach targets the ability to trade-off computational costs against estimation accuracy in order to alleviate the curse of dimensionality of density estimation in higher dimensions.

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22

Carbon, Michel, Lanh Tat Tran, and Berlin Wu. "Kernel density estimation for random fields (density estimation for random fields)." Statistics & Probability Letters 36, no. 2 (December 1997): 115–25. http://dx.doi.org/10.1016/s0167-7152(97)00054-0.

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23

Gao, Chao, Guorong Zhao, Jianhua Lu, and Shuang Pan. "Decentralized state estimation for networked spatial-navigation systems with mixed time-delays and quantized complementary measurements: The moving horizon case." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 232, no. 11 (June 8, 2017): 2160–77. http://dx.doi.org/10.1177/0954410017712277.

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In this paper, the navigational state estimation problem is investigated for a class of networked spatial-navigation systems with quantization effects, mixed time-delays, and network-based observations (i.e. complementary measurements and regional estimations). A decentralized moving horizon estimation approach, featuring complementary reorganization and recursive procedure, is proposed to tackle this problem. First, through the proposed reorganized scheme, a random delayed system with complementary observations is reconstructed into an equivalent delay-free one without dimensional augment. Second, with this equivalent system, a robust moving horizon estimation scheme is presented as a uniform estimator for the navigational states. Third, for the demand of real-time estimate, the recursive form of decentralized moving horizon estimation approach is developed. Furthermore, a collective estimation is obtained through the weighted fusion of two parts, i.e. complementary measurements based estimation, and regional estimations directly from the neighbors. The convergence properties of the proposed estimator are also studied. The obtained stability condition implicitly establishes a relation between the upper bound of the estimation error and two parameters, i.e. quantization density and delay occur probability. Finally, an application example to networked unmanned aerial vehicles is presented and comparative simulations demonstrate the main features of the proposed method.

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24

Cao, Kaikai, and Xiaochen Zeng. "Adaptive Wavelet Estimations in the Convolution Structure Density Model." Mathematics 8, no. 9 (August 19, 2020): 1391. http://dx.doi.org/10.3390/math8091391.

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Using kernel methods, Lepski and Willer study a convolution structure density model and establish adaptive and optimal Lp risk estimations over an anisotropic Nikol’skii space (Lepski, O.; Willer, T. Oracle inequalities and adaptive estimation in the convolution structure density model. Ann. Stat.2019, 47, 233–287). Motivated by their work, we consider the same problem over Besov balls by wavelets in this paper and first provide a linear wavelet estimate. Subsequently, a non-linear wavelet estimator is introduced for adaptivity, which attains nearly-optimal convergence rates in some cases.

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25

Ye, Renyu, Xinsheng Liu, and Yuncai Yu. "Pointwise Optimality of Wavelet Density Estimation for Negatively Associated Biased Sample." Mathematics 8, no. 2 (February 2, 2020): 176. http://dx.doi.org/10.3390/math8020176.

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This paper focuses on the density estimation problem that occurs when the sample is negatively associated and biased. We constructed a block thresholding wavelet estimator to recover the density function from the negatively associated biased sample. The pointwise optimality of this wavelet density estimation is shown as L p ( 1 ≤ p < ∞ ) risks over Besov space. To validate the effectiveness of the block thresholding wavelet method, we provide some examples and implement the numerical simulations. The results indicate that our block thresholding wavelet density estimator is superior in terms of the mean squared error (MSE) when comparing with the nonlinear wavelet density estimator.

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26

Bii, Nelson Kiprono, Christopher Ouma Onyango, and John Odhiambo. "Boundary Bias Correction Using Weighting Method in Presence of Nonresponse in Two-Stage Cluster Sampling." Journal of Probability and Statistics 2019 (June 2, 2019): 1–8. http://dx.doi.org/10.1155/2019/6812795.

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Kernel density estimators due to boundary effects are often not consistent when estimating a density near a finite endpoint of the support of the density to be estimated. To address this, researchers have proposed the application of an optimal bandwidth to balance the bias-variance trade-off in estimation of a finite population mean. This, however, does not eliminate the boundary bias. In this paper weighting method of compensating for nonresponse is proposed. Asymptotic properties of the proposed estimator of the population mean are derived. Under mild assumptions, the estimator is shown to be asymptotically consistent.

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27

Ogbeide, Efosa Michael, and Joseph Erunmwosa Osemwenkhae. "On the Use of a Modified Intersection of Confidence Intervals (MICIH) Kernel Density Estimation Approach." ATHENS JOURNAL OF SCIENCES 8, no. 4 (November 4, 2021): 309–32. http://dx.doi.org/10.30958/ajs.8-4-4.

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Density estimation is an important aspect of statistics. Statistical inference often requires the knowledge of observed data density. A common method of density estimation is the kernel density estimation (KDE). It is a nonparametric estimation approach which requires a kernel function and a window size (smoothing parameter H). It aids density estimation and pattern recognition. So, this work focuses on the use of a modified intersection of confidence intervals (MICIH) approach in estimating density. The Nigerian crime rate data reported to the Police as reported by the National Bureau of Statistics was used to demonstrate this new approach. This approach in the multivariate kernel density estimation is based on the data. The main way to improve density estimation is to obtain a reduced mean squared error (MSE), the errors for this approach was evaluated. Some improvements were seen. The aim is to achieve adaptive kernel density estimation. This was achieved under a sufficiently smoothing technique. This adaptive approach was based on the bandwidths selection. The quality of the estimates obtained of the MICIH approach when applied, showed some improvements over the existing methods. The MICIH approach has reduced mean squared error and relative faster rate of convergence compared to some other approaches. The approach of MICIH has reduced points of discontinuities in the graphical densities the datasets. This will help to correct points of discontinuities and display adaptive density. Keywords: approach, bandwidth, estimate, error, kernel density

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28

Gasparini, Mauro. "Bayesian density estimation via dirichlet density processes." Journal of Nonparametric Statistics 6, no. 4 (January 1996): 355–66. http://dx.doi.org/10.1080/10485259608832681.

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Kansanen, Kasper, Petteri Packalen, Timo Lähivaara, Aku Seppänen, Jari Vauhkonen, Matti Maltamo, and Lauri Mehtätalo. "Refining and evaluating a Horvitz–Thompson-like stand density estimator in individual tree detection based on airborne laser scanning." Canadian Journal of Forest Research 52, no. 4 (April 2022): 527–38. http://dx.doi.org/10.1139/cjfr-2021-0123.

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Horvitz–Thompson-like stand density estimation is a method for estimating the stand density from tree crown objects extracted from airborne laser scanning data through individual tree detection. The estimator is based on stochastic geometry and mathematical morphology of the (planar) set formed by the detected tree crowns. This set is used to approximate the detection probabilities of trees. These probabilities are then used to calculate the estimate. The method includes a tuning parameter, which needs to be known to apply the method. We present a refinement of the method to allow more general detection conditions than those of previous papers. We also present and discuss the methods for estimating the tuning parameter of the estimator using a functional k-nearest neighbors method. We test the model fitting and prediction in two spatially separate data sets and examine the plot-level accuracy of estimation. The estimator produced a 13% lower RMSE (root-mean-square error) than the benchmark method in an external validation data set. We also analyze the effects of similarity and dissimilarity of training and validation data on the results.

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30

Meynberg, O., and G. Kuschk. "Airborne Crowd Density Estimation." ISPRS Annals of Photogrammetry, Remote Sensing and Spatial Information Sciences II-3/W3 (October 8, 2013): 49–54. http://dx.doi.org/10.5194/isprsannals-ii-3-w3-49-2013.

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31

Mâacsse, Benoît R., and Young K. Truong. "Conditional logspline density estimation." Canadian Journal of Statistics 27, no. 4 (December 1999): 819–32. http://dx.doi.org/10.2307/3316133.

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32

Havet, A., M. Lerasle, and É. Moulines. "Density Estimation for RWRE." Mathematical Methods of Statistics 28, no. 1 (January 2019): 18–38. http://dx.doi.org/10.3103/s1066530719010022.

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33

Van Es, Bert, Peter Spreij, and Harry Van Zanten. "Nonparametric volatility density estimation." Bernoulli 9, no. 3 (June 2003): 451–65. http://dx.doi.org/10.3150/bj/1065444813.

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Kerkyacharian, Gérard, Dominique Picard, and Karine Tribouley. "Lp adaptive density estimation." Bernoulli 2, no. 3 (September 1996): 229–47. http://dx.doi.org/10.3150/bj/1178291720.

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35

Hall, Peter, and Brett Presnell. "Density Estimation under Constraints." Journal of Computational and Graphical Statistics 8, no. 2 (June 1999): 259. http://dx.doi.org/10.2307/1390636.

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36

Hansen, Bruce E. "Autoregressive Conditional Density Estimation." International Economic Review 35, no. 3 (August 1994): 705. http://dx.doi.org/10.2307/2527081.

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37

Hazelton, Martin L. "Variable kernel density estimation." Australian New Zealand Journal of Statistics 45, no. 3 (September 2003): 271–84. http://dx.doi.org/10.1111/1467-842x.00283.

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38

Barron, A. R., and T. M. Cover. "Minimum complexity density estimation." IEEE Transactions on Information Theory 37, no. 4 (July 1991): 1034–54. http://dx.doi.org/10.1109/18.86996.

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39

Wand, M. P., J. S. Marron, and D. Ruppert. "Transformations in Density Estimation." Journal of the American Statistical Association 86, no. 414 (June 1991): 343–53. http://dx.doi.org/10.1080/01621459.1991.10475041.

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Luedicke, Joerg, and Alberto Bernacchia. "Self-Consistent Density Estimation." Stata Journal: Promoting communications on statistics and Stata 14, no. 2 (June 2014): 237–58. http://dx.doi.org/10.1177/1536867x1401400201.

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Fortmann-Roe, Scott, Richard Starfield, and Wayne M. Getz. "Contingent Kernel Density Estimation." PLoS ONE 7, no. 2 (February 24, 2012): e30549. http://dx.doi.org/10.1371/journal.pone.0030549.

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Miao, Xu, Ali Rahimi, and Rajesh P. N. Rao. "Complementary Kernel Density Estimation." Pattern Recognition Letters 33, no. 10 (July 2012): 1381–87. http://dx.doi.org/10.1016/j.patrec.2012.02.019.

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Loader, Clive R. "Local likelihood density estimation." Annals of Statistics 24, no. 4 (August 1996): 1602–18. http://dx.doi.org/10.1214/aos/1032298287.

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Hall, Peter, and Brett Presnell. "Density Estimation under Constraints." Journal of Computational and Graphical Statistics 8, no. 2 (June 1999): 259–77. http://dx.doi.org/10.1080/10618600.1999.10474813.

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De Gooijer, Jan G., and Dawit Zerom. "On Conditional Density Estimation." Statistica Neerlandica 57, no. 2 (May 2003): 159–76. http://dx.doi.org/10.1111/1467-9574.00226.

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Srihera, Ramidha, and Winfried Stute. "Kernel adjusted density estimation." Statistics & Probability Letters 81, no. 5 (May 2011): 571–79. http://dx.doi.org/10.1016/j.spl.2011.01.013.

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Yang, Ying. "Penalized semiparametric density estimation." Statistics and Computing 19, no. 4 (September 23, 2008): 355–66. http://dx.doi.org/10.1007/s11222-008-9097-4.

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Gr�bel, Rudolf. "Estimation of density functionals." Annals of the Institute of Statistical Mathematics 46, no. 1 (1994): 67–75. http://dx.doi.org/10.1007/bf00773593.

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Faraway, Julian. "Implementing semiparametric density estimation." Statistics & Probability Letters 10, no. 2 (July 1990): 141–43. http://dx.doi.org/10.1016/0167-7152(90)90009-v.

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Priebe, Carey E., and David J. Marchette. "Adaptive mixture density estimation." Pattern Recognition 26, no. 5 (May 1993): 771–85. http://dx.doi.org/10.1016/0031-3203(93)90130-o.

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