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Register a new userReduced bias nonparametric lifetime density and hazard estimation
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Added by
Arthur Berg
Publication date
2018
fields
Mathematical Statistics
and research's language is
English
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Kernel-based nonparametric hazard rate estimation is considered with a special class of infinite-order kernels that achieves favorable bias and mean square error properties. A fully automatic and adaptive implementation of a density and hazard rate estimator is proposed for randomly right censored data. Careful selection of the bandwidth in the proposed estimators yields estimates that are more efficient in terms of overall mean squared error performance, and in some cases achieves a nearly parametric convergence rate. Additionally, rapidly converging bandwidth estimates are presented for use in second-order kernels to supplement such kernel-based methods in hazard rate estimation. Simulations illustrate the improved accuracy of the proposed estimator against other nonparametric estimators of the density and hazard function. A real data application is also presented on survival data from 13,166 breast carcinoma patients.
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A Bayesian nonparametric estimator to entropy is proposed. The derivation of the new estimator relies on using the Dirichlet process and adapting the well-known frequentist estimators of Vasicek (1976) and Ebrahimi, Pflughoeft and Soofi (1994). Several theoretical properties, such as consistency, of the proposed estimator are obtained. The quality of the proposed estimator has been investigated through several examples, in which it exhibits excellent performance.
Consider a Poisson point process with unknown support boundary curve $g$, which forms a prototype of an irregular statistical model. We address the problem of estimating non-linear functionals of the form $int Phi(g(x)),dx$. Following a nonparametric maximum-likelihood approach, we construct an estimator which is UMVU over Holder balls and achieves the (local) minimax rate of convergence. These results hold under weak assumptions on $Phi$ which are satisfied for $Phi(u)=|u|^p$, $pge 1$. As an application, we consider the problem of estimating the $L^p$-norm and derive the minimax separation rates in the corresponding nonparametric hypothesis testing problem. Structural differences to results for regular nonparametric models are discussed.
In this paper we consider multivariate Hawkes processes with baseline hazard and kernel functions that depend on time. This defines a class of locally stationary processes. We discuss estimation of the time-dependent baseline hazard and kernel functions based on a localized criterion. Theory on stationary Hawkes processes is extended to develop asymptotic theory for the estimator in the locally stationary model.
This paper studies nonparametric estimation of parameters of multivariate Hawkes processes. We consider the Bayesian setting and derive posterior concentration rates. First rates are derived for L1-metrics for stochastic intensities of the Hawkes process. We then deduce rates for the L1-norm of interactions functions of the process. Our results are exemplified by using priors based on piecewise constant functions, with regular or random partitions and priors based on mixtures of Betas distributions. Numerical illustrations are then proposed with in mind applications for inferring functional connec-tivity graphs of neurons.
We consider a model where the failure hazard function, conditional on a covariate $Z$ is given by $R(t,theta^0|Z)=eta_{gamma^0}(t)f_{beta^0}(Z)$, with $theta^0=(beta^0,gamma^0)^topin mathbb{R}^{m+p}$. The baseline hazard function $eta_{gamma^0}$ and relative risk $f_{beta^0}$ belong both to parametric families. The covariate $Z$ is measured through the error model $U=Z+epsilon$ where $epsilon$ is independent from $Z$, with known density $f_epsilon$. We observe a $n$-sample $(X_i, D_i, U_i)$, $i=1,...,n$, where $X_i$ is the minimum between the failure time and the censoring time, and $D_i$ is the censoring indicator. We aim at estimating $theta^0$ in presence of the unknown density $g$. Our estimation procedure based on least squares criterion provide two estimators. The first one minimizes an estimation of the least squares criterion where $g$ is estimated by density deconvolution. Its rate depends on the smoothnesses of $f_epsilon$ and $f_beta(z)$ as a function of $z$,. We derive sufficient conditions that ensure the $sqrt{n}$-consistency. The second estimator is constructed under conditions ensuring that the least squares criterion can be directly estimated with the parametric rate. These estimators, deeply studied through examples are in particular $sqrt{n}$-consistent and asymptotically Gaussian in the Cox model and in the excess risk model, whatever is $f_epsilon$.
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