3.ĭata tapers have also been used for stationary time series (in particular in spectral estimation, but also with Yule-Walker estimates and covariance estimation where they give positive definite autocovariances with a lower bias). This is straightforward for local covariance estimates and local Yule-Walker estimates and can usually also be applied to other estimation problems. In this context, this may be either achieved by using a kernel estimate or a data taper, which is asymptotically equivalent. 2.Ī first step toward a better estimate (as it is proved below) is to put higher weights in the middle and lower weights at the edges of the observation domain in order to cope in a better way with the nonstationarity of X t,T on the segment. The classical stationary method on a segment is in this case the estimator without data taper which is the same as the kernel estimator with a rectangular kernel. For reasons of clarity, a few remarks are in order: 1. If K ( x ) = h ( x ) 2, all three estimators are equivalent in the sense that they lead to the same asymptotic bias, variance, and mean-squared error. With i − j = k, which appears in least square regression – cf. (10) c ∼ ∼ T ( u 0, i, j ) : = 1 b T T ∑ t K u 0 − t ∕ T b T X t − i, T X t − j, T Again, simplifications are needed as the dimensions of X and/or Y increase. We approach these problems using the local Gaussian approximation in Chapters 10 and 11. Such conditional concepts are important, for instance, in causal networks. This is also the case for derived quantities like the partial correlation function. The above facts make it demanding to estimate conditional densities, and much theory and applications have been limited to the Gaussian case. This leads to instability in the ratio used to define the kernel estimate. The density f X ( x ) is in addition close to zero in its tails, so the kernel estimate will be expected to be close to zero as well. Here f X, Y ( x, y ) and f X ( x ) may be estimated by kernel estimation, but for moderate and large dimensions of X and/or Y, the curse of dimensionality comes into play. The obvious way to proceed in the nonparametric case is to use the definition of the conditional density as f Y | X ( y | x ) = f X, Y ( x, y ) / f X ( x ), where X and Y may be vector variables in general. The exception is, predictably, the multivariate normal, for which the conditional distribution has straightforward expressions, as seen in (2.2) and (2.3), and even stays within the same family of multivariate normal distributions. Even for the general elliptic family, the question of forming a conditional distribution is not trivial, since in general different generators have to be involved. It is more difficult to estimate conditional densities in both parametric and nonparametric cases, not in the least because simple expressions of conditional densities are not available for most parametric families of distributions. In all these approaches the problem is to find a point of balance to strike a compromise between accuracy and feasibility. Other ways of avoiding the curse of dimensionality in density estimation are discussed by Nagler and Czado (2016) using a pair copula construction and by Friedman et al. One such set of assumptions is made in Chapter 9, where a simplified local Gaussian framework is used based on Otneim and Tjøstheim (2017). It is in fact impossible to avoid the curse of dimensionality unless we want to make some additional restricting assumptions. Hayfield and Racine (2008) lessen the curse of dimensionality by clever bandwidth selection algorithms that work for discrete and mixed data types as well as in the continuous case. The corresponding smoothed kernel estimator will clearly also suffer from this problem, and this is one expression of the well-known curse of dimensionality. This happens because the number of bins N p quickly becomes a very big number compared to N, which is the number of bins in the one-dimensional case. Second, as p increases, there will be many empty cells in the p-dimensional histogram for a fixed sample size n. Extending this idea to the multivariate case, we face two problems: First, a histogram is not easy to visualize if p > 2. The histogram was used as a motivation for the univariate ( p = 1 ) kernel density estimator (2.11). Bård Støve, in Statistical Modeling Using Local Gaussian Approximation, 2022 2.5.3 Multivariate and conditional density estimation
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