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13
Estimating the Support of a HighDimensional Distribution
, 1999
"... Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S is bounded by some a priori specified between 0 and 1. We propo ..."
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Cited by 766 (29 self)
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Suppose you are given some dataset drawn from an underlying probability distribution P and you want to estimate a "simple" subset S of input space such that the probability that a test point drawn from P lies outside of S is bounded by some a priori specified between 0 and 1. We propose a method to approach this problem by trying to estimate a function f which is positive on S and negative on the complement. The functional form of f is given by a kernel expansion in terms of a potentially small subset of the training data; it is regularized by controlling the length of the weight vector in an associated feature space. The expansion coefficients are found by solving a quadratic programming problem, which we do by carrying out sequential optimization over pairs of input patterns. We also provide a preliminary theoretical analysis of the statistical performance of our algorithm. The algorithm is a natural extension of the support vector algorithm to the case of unlabelled d...
On the risk of estimates for block decreasing densities
 J. Mult. Anal
, 2003
"... Abstract. A density f = f(x1,..., x d) on [0, ∞) d is block decreasing if for each j ∈ {1,..., d}, it is a decreasing function of xj, when all other components are held fixed. Let us consider the class of all block decreasing densities on [0, 1] d bounded by B. We shall study the minimax risk over t ..."
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Cited by 5 (0 self)
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Abstract. A density f = f(x1,..., x d) on [0, ∞) d is block decreasing if for each j ∈ {1,..., d}, it is a decreasing function of xj, when all other components are held fixed. Let us consider the class of all block decreasing densities on [0, 1] d bounded by B. We shall study the minimax risk over this class using n i.i.d. observations, the loss being measured by the L1 distance between the estimate and the true density. We prove that if S = log(1 + B), lower bounds for the risk are of the form C(S d /n) 1/(d+2) , where C is a function of d only. We also prove that a suitable histogram with unequal bin widths as well as a variable kernel estimate achieve the optimal multivariate rate. We present a procedure for choosing all parameters in the kernel estimate automatically without loosing the minimax optimality, even if B and the support of f are unknown.
Testing For Monotonicity Of A Regression Mean Without Selecting A Bandwidth
, 1998
"... . A new approach to testing for monotonicity of a regression mean, not requiring computation of a curve estimator or a bandwidth, is suggested. It is based on the notion of `running gradients' over short intervals, although from some viewpoints it may be regarded as an analogue for monotonicity ..."
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Cited by 4 (3 self)
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. A new approach to testing for monotonicity of a regression mean, not requiring computation of a curve estimator or a bandwidth, is suggested. It is based on the notion of `running gradients' over short intervals, although from some viewpoints it may be regarded as an analogue for monotonicity testing of the dip/excess mass approach for testing modality hypotheses about densities. Like the latter methods, the new technique does not suffer difficulties caused by almostflat parts of the target function. In fact, it is calibrated so as to work well for flat response curves, and as a result it has relatively good power properties in boundary cases where the curve exhibits shoulders. In this respect, as well as in its construction, the `running gradients' approach differs from alternative techniques based on the notion of a critical bandwidth. KEYWORDS. Bootstrap, calibration, curve estimation, Monte Carlo, response curve, running gradient. SHORT TITLE. Testing for monotonicity. 1 The man...
Comparison of kernel density estimators with assumptions on number of modes
, 2012
"... Abstract. A datadriven bandwidth choice for a kernel density estimator called critical bandwidth is investigated. This procedure allows the estimation to have as many modes as assumed for the density to estimate. Both Gaussian and uniform kernels are considered. For the Gaussian kernel, asymptotic ..."
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Cited by 3 (1 self)
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Abstract. A datadriven bandwidth choice for a kernel density estimator called critical bandwidth is investigated. This procedure allows the estimation to have as many modes as assumed for the density to estimate. Both Gaussian and uniform kernels are considered. For the Gaussian kernel, asymptotic results are given. For the uniform kernel, an argument against these properties is mentioned. These theoretical results are illustrated with a simulation study which compare the kernel estimators that rely on critical bandwidth with another one which uses a plugin method to select its bandwidth.
ASSESSING EXTREMA OF EMPIRICAL PRINCIPAL COMPONENT FUNCTIONS
, 2006
"... The difficulties of estimating and representing the distributions of functional data mean that principal component methods play a substantially greater role in functional data analysis than in more conventional finitedimensional settings. Local maxima and minima in principal component functions are ..."
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The difficulties of estimating and representing the distributions of functional data mean that principal component methods play a substantially greater role in functional data analysis than in more conventional finitedimensional settings. Local maxima and minima in principal component functions are of direct importance; they indicate places in the domain of a random function where influence on the function value tends to be relatively strong but of opposite sign. We explore statistical properties of the relationship between extrema of empirical principal component functions, and their counterparts for the true principal component functions. It is shown that empirical principal component funcions have relatively little trouble capturing conventional extrema, but can experience difficulty distinguishing a “shoulder ” in a curve from a small bump. For example, when the true principal component function has a shoulder, the probability that the empirical principal component function has instead a bump is approximately equal to 1. We suggest and describe the 2 performance of bootstrap methods for assessing the strength of extrema. It is shown that the subsample bootstrap is more effective than the standard bootstrap in this regard. A “bootstrap likelihood” is proposed for measuring extremum strength. Exploratory numerical methods are suggested.
Oneclass SVM regularization path and comparison with alpha seeding
"... Abstract. Oneclass support vector machines (1SVMs) estimate the level set of the underlying density observed data. Aside the kernel selection issue, one difficulty concerns the choice of the ’level ’ parameter. In this paper, following the work by Hastie et. al (2004), we derive the entire regular ..."
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Abstract. Oneclass support vector machines (1SVMs) estimate the level set of the underlying density observed data. Aside the kernel selection issue, one difficulty concerns the choice of the ’level ’ parameter. In this paper, following the work by Hastie et. al (2004), we derive the entire regularization path for ν1SVMs. Since this regularization path is efficient for building different level sets estimate, we have empirically compared such approach to state of the art approach based on alpha seeding and we show that regularization path is far more efficient. 1
Feng D.: On a statistical framework for estimation from random set observations
 J. Theoret. Probab
"... Using the theory of random closed sets, we extend the statistical framework introduced by Schreiber (11) for inference based on setvalued observations from the case of finite sample spaces to compact metric spaces with continuous distributions. KEY WORDS: Statistical inference; conditional distrib ..."
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Using the theory of random closed sets, we extend the statistical framework introduced by Schreiber (11) for inference based on setvalued observations from the case of finite sample spaces to compact metric spaces with continuous distributions. KEY WORDS: Statistical inference; conditional distribution; random sets. 1.
Entropy estimate for high dimensional monotonic functions
 J. Multivariate Anal
, 2007
"... We establish upper and lower bounds for the metric entropy and bracketing entropy of the class of ddimensional bounded monotonic functions under Lp norms. It is interesting to see that both the metric entropy and bracketing entropy have different behaviors for p < d/(d − 1) and p> d/(d − 1). ..."
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We establish upper and lower bounds for the metric entropy and bracketing entropy of the class of ddimensional bounded monotonic functions under Lp norms. It is interesting to see that both the metric entropy and bracketing entropy have different behaviors for p < d/(d − 1) and p> d/(d − 1). We apply the new bounds for bracketing entropy to establish a global rate of convergence of the MLE of a ddimensional monotone density.
1 Entropy Estimate For High Dimensional Monotonic Functions
, 2005
"... We establish upper and lower bounds for the metric entropy and bracketing entropy of the class of ddimensional bounded monotonic functions under Lp norms. It is interesting to see that both the metric entropy and bracketing entropy have different behaviors for p<d/(d−1) and p>d/(d−1). We appl ..."
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We establish upper and lower bounds for the metric entropy and bracketing entropy of the class of ddimensional bounded monotonic functions under Lp norms. It is interesting to see that both the metric entropy and bracketing entropy have different behaviors for p<d/(d−1) and p>d/(d−1). We apply the new bounds for bracketing entropy to establish a global rate of convergence of the MLE of a ddimensional monotone density. 1