Pointwise Wavelet Estimation of Density Function with Change-Points Based on NA and Biased Sample
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Pointwise Wavelet Estimation of Density Function with Change-Points Based on NA and Biased Sample Yuncai Yu Abstract. This paper is concerned with the density estimation problem of negatively associated biased sample with the presence of multiple change-points. We use the peaks-over-threshold approach to estimate the number and locations of change-points and give an equispaced design estimation to evaluate the jump sizes for the underlying density function. Subsequently, we propose a nonlinear wavelet change-point estimation of the underlying density and show the convergence rate under poinwise risk over Besov space. It should be pointed out that the convergence rate of wavelet change-point estimation is near optimal (up to a logarithmic term) and remains the same as that of the usual wavelet density estimation without change-points. Mathematics Subject Classification. 62G08, 62G20, 42C40. Keywords. nonlinear wavelet estimation, negatively biased sample, multiple change points, poinwise risk.
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1. Introduction Density estimation problem is the recovery of the density function of random variable from a given set of observations. But in practice, the given set of observations are not always drawn directly from the main phenomena of interest; they are sampled with some probabilities (biasing functions) that rely on the observation values (see Efromovich [9]). This sampling procedure leads to the observation data are biased. We formulate the above problem as the following model. 0123456789().: V,-vol
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Results Math
Suppose that {Xn ; n ≥ 1} is a sequence of unobserved realizations of random variable X with the density function denoted by g and {Yn ; n ≥ 1} is a stationary random process with the density function f (y) =
w (y) g (y) , μ
(1.1)
where w (y) stands for a weighting or biasing function and μ represents the 1 expectation of w (y) whose definition is μ = 0 w (y) g (y)dy (0 < μ < ∞). There are lots of practical settings may lead to a biased sample like the natural frequency of X be missing, visibility bias in aerial survey, larger (or smaller) observations are more likely to be drawn during the sampling procedure (see Efromovich [9]), and so on. A classic example which can be found in Cox [7] is the inter-event times that constitute a biased sample if they are sampled at some randomly chosen time instants. Another famous example from Efromovich [8] is to study the distribution of the blood alcohol concentration in the intoxicated drivers. Since it is more likely to arrest the drivers who have drunken more, thus the sample data are size-biased. In this situation, estimating the density g(y) from the realizations Y1 , Y2 , . . . , Yn directly may be not applicable since it means that the biasing function in the model (1.1) is always to be a positive constant, while the biased density estimation model (1.1) can well deal with the biased data since the weight w (y) allows the sampling procedure to be size-biased. To recover the unobserved density in m
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