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Artigos de revistas sobre o tema "Estimation of Density"

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Autor: Grafiati

Publicado: 7 de julho de 2024

Última modificação: 7 de julho de 2024

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1

Sugiyama, Masashi, Takafumi Kanamori, Taiji Suzuki, Marthinus Christoffel du Plessis, Song Liu e Ichiro Takeuchi. "Density-Difference Estimation". Neural Computation 25, n.º 10 (outubro de 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 e Masashi Sugiyama. "Direct Density Derivative Estimation". Neural Computation 28, n.º 6 (junho de 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 e Masashi Sugiyama. "Regularized Multitask Learning for Multidimensional Log-Density Gradient Estimation". Neural Computation 28, n.º 7 (julho de 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

Hovda, Sigve. "Properties of Transmetric Density Estimation". International Journal of Statistics and Probability 5, n.º 3 (13 de abril de 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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5

Liu, Qing, David Pitt, Xibin Zhang e Xueyuan Wu. "A Bayesian Approach to Parameter Estimation for Kernel Density Estimation via Transformations". Annals of Actuarial Science 5, n.º 2 (18 de abril de 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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6

Beaumont, Chris, e B. W. Silverman. "Density Estimation." Journal of the Operational Research Society 37, n.º 11 (novembro de 1986): 1102. http://dx.doi.org/10.2307/2582699.

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7

Sheather, Simon J. "Density Estimation". Statistical Science 19, n.º 4 (novembro de 2004): 588–97. http://dx.doi.org/10.1214/088342304000000297.

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8

Yamada, Makoto, e Masashi Sugiyama. "Direct Density-Ratio Estimation with Dimensionality Reduction via Hetero-Distributional Subspace Analysis". Proceedings of the AAAI Conference on Artificial Intelligence 25, n.º 1 (4 de agosto de 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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9

Li, Rui, e 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

Hovda, Sigve. "Transmetric Density Estimation". International Journal of Statistics and Probability 5, n.º 2 (10 de fevereiro de 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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11

Arefi, Mohsen, Reinhard Viertl e S. Mahmoud Taheri. "Fuzzy density estimation". Metrika 75, n.º 1 (29 de abril de 2010): 5–22. http://dx.doi.org/10.1007/s00184-010-0311-y.

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12

Locke, Judson B., e Adrian M. Peter. "Multiwavelet density estimation". Applied Mathematics and Computation 219, n.º 11 (fevereiro de 2013): 6002–15. http://dx.doi.org/10.1016/j.amc.2012.11.099.

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13

Fujii, Takayuki. "Nonparametric Estimation for a Class of Piecewise-Deterministic Markov Processes". Journal of Applied Probability 50, n.º 4 (dezembro de 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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14

Fujii, Takayuki. "Nonparametric Estimation for a Class of Piecewise-Deterministic Markov Processes". Journal of Applied Probability 50, n.º 04 (dezembro de 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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15

Kwasniok, Frank. "Semiparametric maximum likelihood probability density estimation". PLOS ONE 16, n.º 11 (9 de novembro de 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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16

Zinde-Walsh, Victoria. "KERNEL ESTIMATION WHEN DENSITY MAY NOT EXIST". Econometric Theory 24, n.º 3 (26 de fevereiro de 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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17

Kronenfeld, Barry J. "A Plotless Density Estimator Based on the Asymptotic Limit of Ordered Distance Estimation Values". Forest Science 55, n.º 4 (1 de agosto de 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

Carbon, Michel, Lanh Tat Tran e Berlin Wu. "Kernel density estimation for random fields (density estimation for random fields)". Statistics & Probability Letters 36, n.º 2 (dezembro de 1997): 115–25. http://dx.doi.org/10.1016/s0167-7152(97)00054-0.

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19

Rathke, Fabian, e Christoph Schnörr. "A Computational Approach to Log-Concave Density Estimation". Analele Universitatii "Ovidius" Constanta - Seria Matematica 23, n.º 3 (1 de novembro de 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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20

Hodgson, Douglas J. "ADAPTIVE ESTIMATION OF ERROR CORRECTION MODELS". Econometric Theory 14, n.º 1 (fevereiro de 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

Cao, Kaikai, e Xiaochen Zeng. "Adaptive Wavelet Estimations in the Convolution Structure Density Model". Mathematics 8, n.º 9 (19 de agosto de 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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22

Gao, Chao, Guorong Zhao, Jianhua Lu e 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, n.º 11 (8 de junho de 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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23

Gasparini, Mauro. "Bayesian density estimation via dirichlet density processes". Journal of Nonparametric Statistics 6, n.º 4 (janeiro de 1996): 355–66. http://dx.doi.org/10.1080/10485259608832681.

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24

Ye, Renyu, Xinsheng Liu e Yuncai Yu. "Pointwise Optimality of Wavelet Density Estimation for Negatively Associated Biased Sample". Mathematics 8, n.º 2 (2 de fevereiro de 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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25

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

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26

Mâacsse, Benoît R., e Young K. Truong. "Conditional logspline density estimation". Canadian Journal of Statistics 27, n.º 4 (dezembro de 1999): 819–32. http://dx.doi.org/10.2307/3316133.

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27

Havet, A., M. Lerasle e É. Moulines. "Density Estimation for RWRE". Mathematical Methods of Statistics 28, n.º 1 (janeiro de 2019): 18–38. http://dx.doi.org/10.3103/s1066530719010022.

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28

Van Es, Bert, Peter Spreij e Harry Van Zanten. "Nonparametric volatility density estimation". Bernoulli 9, n.º 3 (junho de 2003): 451–65. http://dx.doi.org/10.3150/bj/1065444813.

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29

Kerkyacharian, Gérard, Dominique Picard e Karine Tribouley. "Lp adaptive density estimation". Bernoulli 2, n.º 3 (setembro de 1996): 229–47. http://dx.doi.org/10.3150/bj/1178291720.

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30

Hall, Peter, e Brett Presnell. "Density Estimation under Constraints". Journal of Computational and Graphical Statistics 8, n.º 2 (junho de 1999): 259. http://dx.doi.org/10.2307/1390636.

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31

Hansen, Bruce E. "Autoregressive Conditional Density Estimation". International Economic Review 35, n.º 3 (agosto de 1994): 705. http://dx.doi.org/10.2307/2527081.

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32

Hazelton, Martin L. "Variable kernel density estimation". Australian New Zealand Journal of Statistics 45, n.º 3 (setembro de 2003): 271–84. http://dx.doi.org/10.1111/1467-842x.00283.

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33

Barron, A. R., e T. M. Cover. "Minimum complexity density estimation". IEEE Transactions on Information Theory 37, n.º 4 (julho de 1991): 1034–54. http://dx.doi.org/10.1109/18.86996.

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34

Wand, M. P., J. S. Marron e D. Ruppert. "Transformations in Density Estimation". Journal of the American Statistical Association 86, n.º 414 (junho de 1991): 343–53. http://dx.doi.org/10.1080/01621459.1991.10475041.

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35

Luedicke, Joerg, e Alberto Bernacchia. "Self-Consistent Density Estimation". Stata Journal: Promoting communications on statistics and Stata 14, n.º 2 (junho de 2014): 237–58. http://dx.doi.org/10.1177/1536867x1401400201.

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36

Fortmann-Roe, Scott, Richard Starfield e Wayne M. Getz. "Contingent Kernel Density Estimation". PLoS ONE 7, n.º 2 (24 de fevereiro de 2012): e30549. http://dx.doi.org/10.1371/journal.pone.0030549.

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37

Miao, Xu, Ali Rahimi e Rajesh P. N. Rao. "Complementary Kernel Density Estimation". Pattern Recognition Letters 33, n.º 10 (julho de 2012): 1381–87. http://dx.doi.org/10.1016/j.patrec.2012.02.019.

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38

Loader, Clive R. "Local likelihood density estimation". Annals of Statistics 24, n.º 4 (agosto de 1996): 1602–18. http://dx.doi.org/10.1214/aos/1032298287.

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39

Hall, Peter, e Brett Presnell. "Density Estimation under Constraints". Journal of Computational and Graphical Statistics 8, n.º 2 (junho de 1999): 259–77. http://dx.doi.org/10.1080/10618600.1999.10474813.

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40

De Gooijer, Jan G., e Dawit Zerom. "On Conditional Density Estimation". Statistica Neerlandica 57, n.º 2 (maio de 2003): 159–76. http://dx.doi.org/10.1111/1467-9574.00226.

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41

Srihera, Ramidha, e Winfried Stute. "Kernel adjusted density estimation". Statistics & Probability Letters 81, n.º 5 (maio de 2011): 571–79. http://dx.doi.org/10.1016/j.spl.2011.01.013.

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42

Yang, Ying. "Penalized semiparametric density estimation". Statistics and Computing 19, n.º 4 (23 de setembro de 2008): 355–66. http://dx.doi.org/10.1007/s11222-008-9097-4.

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43

Gr�bel, Rudolf. "Estimation of density functionals". Annals of the Institute of Statistical Mathematics 46, n.º 1 (1994): 67–75. http://dx.doi.org/10.1007/bf00773593.

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44

Faraway, Julian. "Implementing semiparametric density estimation". Statistics & Probability Letters 10, n.º 2 (julho de 1990): 141–43. http://dx.doi.org/10.1016/0167-7152(90)90009-v.

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45

Priebe, Carey E., e David J. Marchette. "Adaptive mixture density estimation". Pattern Recognition 26, n.º 5 (maio de 1993): 771–85. http://dx.doi.org/10.1016/0031-3203(93)90130-o.

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46

Brown, Lawrence D., Edward I. George e Xinyi Xu. "Admissible predictive density estimation". Annals of Statistics 36, n.º 3 (junho de 2008): 1156–70. http://dx.doi.org/10.1214/07-aos506.

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47

Rodriguez, Abel, e Enrique ter Horst. "Bayesian dynamic density estimation". Bayesian Analysis 3, n.º 2 (junho de 2008): 339–65. http://dx.doi.org/10.1214/08-ba313.

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48

Bertin, K., E. Le Pennec e V. Rivoirard. "Adaptive Dantzig density estimation". Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 47, n.º 1 (fevereiro de 2011): 43–74. http://dx.doi.org/10.1214/09-aihp351.

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Van Kerm, Philippe. "Adaptive Kernel Density Estimation". Stata Journal: Promoting communications on statistics and Stata 3, n.º 2 (junho de 2003): 148–56. http://dx.doi.org/10.1177/1536867x0300300204.

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This insert describes the module akdensity. akdensity extends the official kdensity that estimates density functions by the kernel method. The extensions are of two types: akdensity allows the use of an “adaptive kernel” approach with varying, rather than fixed, bandwidths; and akdensity estimates pointwise variability bands around the estimated density functions.

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EL-SAYYAD, G. M., M. SAMUDDIN e A. A. AL-HARBEY. "On parametric density estimation". Biometrika 76, n.º 2 (1989): 343–48. http://dx.doi.org/10.1093/biomet/76.2.343.

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