A number of parametric and nonparametric methods for estimating cognitive diagnosis models (CDMs) have been developed and applied in a wide range of contexts. However, in the literature, a wide chasm exists between these two families of methods, and their relationship to each other is not well understood. In this paper, we propose a unified estimation framework to bridge the divide between parametric and nonparametric methods in cognitive diagnosis to better understand their relationship. We also develop iterative joint estimation algorithms and establish consistency properties within the proposed framework. Lastly, we present comprehensive simulation results to compare different methods and provide practical recommendations on the appropriate use of the proposed framework in various CDM contexts.

In this paper, we consider some dividend problems in the perturbed compound Poisson model under a constant barrier dividend strategy. We approximate the expected present value of dividend payments before ruin and the expected discounted penalty function based on the COS method, and construct some nonparametric estimators by using a random sample on claim number and individual claim sizes. Under a large sample size setting, we perform an error analysis of the estimators. We also provide some simulation results to verify the effectiveness of this estimation method when the sample size is finite.

There is an enormous literature on the so-called Grenander estimator, which is merely the nonparametric maximum likelihood estimator of a nonincreasing probability density on [0, 1] (see, for instance, Grenander (1981)), but unfortunately, there is no nonasymptotic (i.e. for arbitrary finite sample size n) explicit upper bound for the quadratic risk of the Grenander estimator readily applicable in practice by statisticians. In this paper, we establish, for the first time, a simple explicit upper bound 2n−1/2 for the latter quadratic risk. It turns out to be a straightforward consequence of an inequality valid with probability one and bounding from above the integrated squared error of the Grenander estimator by the Kolmogorov–Smirnov statistic.

Complex models are of increasing interest to social scientists. Researchers interested in prediction generally favor flexible, robust approaches, while those interested in causation are often interested in modeling nuanced treatment structures and confounding relationships. Unfortunately, estimators of complex models often scale poorly, especially if they seek to maintain interpretability. In this paper, we present an example of such a conundrum and show how optimization can alleviate the worst of these concerns. Specifically, we introduce bigKRLS, which offers a variety of statistical and computational improvements to the Hainmueller and Hazlett (2013) Kernel-Regularized Least Squares (KRLS) approach. As part of our improvements, we decrease the estimator’s single-core runtime by 50% and reduce the estimator’s peak memory usage by an order of magnitude. We also improve uncertainty estimates for the model’s average marginal effect estimates—which we test both in simulation and in practice—and introduce new visual and statistical tools designed to assist with inference under the model. We further demonstrate the value of our improvements through an analysis of the 2016 presidential election, an analysis that would have been impractical or even infeasible for many users with existing software.

Existing approaches to estimating ideal points offer no method for consistent estimation or inference without relying on strong parametric assumptions. In this paper, I introduce a nonparametric approach to ideal-point estimation and inference that goes beyond these limitations. I show that some inferences about the relative positions of two pairs of legislators can be made with minimal assumptions. This information can be combined across different possible choices of the pairs to provide estimates and perform hypothesis tests for all legislators without additional assumptions. I demonstrate the usefulness of these methods in two applications to Supreme Court data, one testing for ideological movement by a single justice and the other testing for multidimensional voting behavior in different decades.

We investigate the estimation of the perimeter of a set by a graph cut of a random geometric graph. For Ω ⊆ D = (0, 1)d with d ≥ 2, we are given n random independent and identically distributed points on D whose membership in Ω is known. We consider the sample as a random geometric graph with connection distance ε > 0. We estimate the perimeter of Ω (relative to D) by the, appropriately rescaled, graph cut between the vertices in Ω and the vertices in D ∖ Ω. We obtain bias and variance estimates on the error, which are optimal in scaling with respect to n and ε. We consider two scaling regimes: the dense (when the average degree of the vertices goes to ∞) and the sparse one (when the degree goes to 0). In the dense regime, there is a crossover in the nature of the approximation at dimension d = 5: we show that in low dimensions d = 2, 3, 4 one can obtain confidence intervals for the approximation error, while in higher dimensions one can obtain only error estimates for testing the hypothesis that the perimeter is less than a given number.

The subject of this paper is the problem of estimating the service time distribution of the M/G/∞ queue from incomplete data on the queue. The goal is to estimate G from observations of the queue-length process at the points of the regular grid on a fixed time interval. We propose an estimator and analyze its accuracy over a family of target service time distributions. An upper bound on the maximal risk is derived. The problem of estimating the arrival rate is considered as well.

In this paper we consider the stationary Poisson Boolean model with spherical grains and propose a family of nonparametric estimators for the radius distribution. These estimators are based on observed distances and radii, weighted in an appropriate way. They are ratio unbiased and asymptotically consistent for a growing observation window. We show that the asymptotic variance exists and is given by a fairly explicit integral expression. Asymptotic normality is established under a suitable integrability assumption on the weight function. We also provide a short discussion of related estimators as well as a simulation study.

In this paper, we investigate a nonparametric approach to provide a recursive estimatorof the transition density of a piecewise-deterministic Markov process, from only oneobservation of the path within a long time. In this framework, we do not observe a Markovchain with transition kernel of interest. Fortunately, one may write the transitiondensity of interest as the ratio of the invariant distributions of two embedded chains ofthe process. Our method consists in estimating these invariant measures. We state a resultof consistency and a central limit theorem under some general assumptions about the mainfeatures of the process. A simulation study illustrates the well asymptotic behavior ofour estimator.

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.

In this article, our aim is to estimate the successive derivatives of the stationarydensity f of a strictly stationary and β-mixing process (Xt)t≥0. This process is observed at discrete timest = 0,Δ,...,nΔ. Thesampling interval Δ can be fixed or small. We use a penalizedleast-square approach to compute adaptive estimators. If the derivativef(j) belongs to the Besov space \hbox{$\rond{B}_{2,\infty}^{\alpha}$}$B_{\mathrm{2}\mathit{,}\mathrm{\infty }}^{\mathit{\alpha }}$, then our estimator converges at rate (nΔ)−α/(2α+2j+1). Then we consider a diffusion with known diffusioncoefficient. We use the particular form of the stationary density to compute an adaptiveestimator of its first derivative f′. When the sampling intervalΔ tends to 0, and when the diffusion coefficient is known, theconvergence rate of our estimator is (nΔ)−α/(2α+1). When the diffusion coefficient is known, we alsoconstruct a quotient estimator of the drift for low-frequency data.

Our aim is to estimate the joint distribution of a finite sequence of independent categorical variables. We consider the collection of partitions into dyadic intervals and the associated histograms, and we select from the data the best histogram by minimizing a penalized least-squares criterion. The choice of the collection of partitions is inspired from approximation results due to DeVore and Yu. Our estimator satisfies a nonasymptotic oracle-type inequality and adaptivity properties in the minimax sense. Moreover, its computational complexity is only linear in the length of the sequence. We also use that estimator during the preliminary stage of a hybrid procedure for detecting multiple change-points in the joint distribution of the sequence. That second procedure still satisfies adaptivity properties and can be implemented efficiently. We provide a simulation study and apply the hybrid procedure to the segmentation of a DNA sequence.

Let {Z t }t≥0 be a Lévy process with Lévy measure ν, and let τ(t)=∫0 t r(u) d u, where {r(t)}t≥0 is a positive ergodic diffusion independent from Z. Based upon discrete observations of the time-changed Lévy process X t ≔Z τt during a time interval [0,T], we study the asymptotic properties of certain estimators of the parameters β(φ)≔∫φ(x)ν(d x), which in turn are well known to be the building blocks of several nonparametric methods such as sieve-based estimation and kernel estimation. Under uniform boundedness of the second moments of r and conditions on φ necessary for the standard short-term ergodic property limt→ 0 E φ(Zt )/t = β(φ) to hold, consistency and asymptotic normality of the proposed estimators are ensured when the time horizon T increases in such a way that the sampling frequency is high enough relative to T.

We consider a failure hazard function,conditional on a time-independent covariate Z,given by $\eta_{\gamma^0}(t)f_{\beta^0}(Z)$ . The baseline hazardfunction $\eta_{\gamma^0}$ and the relative risk $f_{\beta^0}$ both belong to parametricfamilies with $\theta^0=(\beta^0,\gamma^0)^\top\in \mathbb{R}^{m+p}$ . The covariate Z has an unknown density and is measured with an error through anadditive error model U = Z + ε where ε is a random variable, independent from Z, withknown density $f_\varepsilon$ .We observe a n-sample (Xi, Di, Ui), i = 1, ..., n, where X i isthe minimum between the failure time and the censoring time,and D i is the censoring indicator.Using least square criterion and deconvolution methods, we propose a consistent estimator of θ 0 using the observationsn-sample (Xi, Di, Ui), i = 1, ..., n.
We give an upper bound for its riskwhich depends on the smoothness properties of $f_\varepsilon$ and $f_\beta(z)$ as afunction of z, and we derive sufficient conditionsfor the $\sqrt{n}$ -consistency.We give detailed examples consideringvarious type of relative risks $f_{\beta}$ and various types of errordensity $f_\varepsilon$ . In particular, in the Cox model and inthe excess risk model, the estimator of θ 0 is $\sqrt{n}$ -consistent and asymptotically Gaussianregardless of the form of $f_\varepsilon$ .