Statistical Methodology最新文献

The present paper deals with a nonparametric $M$-estimation for right censored regression model with stationary ergodic data. Defined as an implicit function, a kernel-type estimator of a family of robust regression is considered when the covariate takes its values in $R^d$ ($d\ge 1$) and the data are sampled from a stationary ergodic process. The strong consistency (with rate) and the asymptotic distribution of the estimator are established under mild assumptions. Moreover, a usable confidence interval is provided which does not depend on any unknown quantity. Our results hold without any mixing condition and do not require the existence of marginal densities. A comparison study based on simulated data is also provided.

In many situations, multiple copies of a software are tested in parallel with different test cases as input, and the detected errors from a particular round of testing are debugged together. In this article, we discuss a discrete time model of software reliability for such a scenario of periodic debugging. We propose likelihood based inference of the model parameters, including the initial number of errors, under the assumption that all errors are equally likely to be detected. The proposed method is used to estimate the reliability of the software. We establish asymptotic normality of the estimated model parameters. The performance of the proposed method is evaluated through a simulation study and its use is illustrated through the analysis of a dataset obtained from testing of a real-time flight control software. We also consider a more general model, in which different errors have different probabilities of detection.

Acute hypotensive episodes (AHEs) are serious clinical events in intensive care units (ICUs), and require immediate treatment to prevent patient injury. Reducing the risks associated with an AHE requires effective and efficient mining of data generated from multiple physiological time series. We propose HeartCast, a model that extracts essential features from such data to effectively predict AHE. HeartCast combines a non-linear support vector machine with best-feature extraction via analysis of the baseline threshold, quartile parameters, and window size of the physiological signals. Our approach has the following benefits: (a) it extracts the most relevant features; (b) it provides the best results for identification of an AHE event; (c) it is fast and scales with linear complexity over the length of the window; and (d) it can manage missing values and noise/outliers by using a best-feature extraction method. We performed experiments on data continuously captured from physiological time series of ICU patients (roughly 3 GB of processed data). HeartCast was found to outperform other state-of-the-art methods found in the literature with a 13.7% improvement in classification accuracy.

The testing of equality of several Pearson correlations can be found in a number of scientific fields. We surmise in many such cases that the alternatives of interest in practice are, in deed, order restricted, and therefore the researcher is best served by use of testing procedures developed for those specific alternatives. In this note we introduce a collection of tests for use in testing equality of $k$ correlation coefficients against order alternatives, with an emphasis on simple order. Specifically, we propose likelihood ratio tests and contrast tests based on the well known Fisher Z transformation as well as tests which make use of generalized variable methodologies. The proposed procedures are empirically compared with regard to type I and II error rates via Monte Carlo simulations studies, and the use of the approaches is illustrated using an example. These tests are found to be vastly superior to tests for the general alternative, and the contrast tests based on the Fisher Z transformation are recommended for practice based on the observed test properties and simplicity.

We consider two types of graphs based on a family of proximity catch digraphs (PCDs) and study their edge density. In particular, the PCDs we use are a parameterized digraph family called proportional-edge (PE) PCDs and the two associated graph types are the “underlying graphs” and the newly introduced “reflexivity graphs” based on the PE-PCDs. These graphs are extensions of random geometric graphs where distance is replaced with a dissimilarity measure and the threshold is not fixed but depends on the location of the points. PCDs and the associated graphs are constructed based on data points from two classes, say $X$ and $Y$, where one class (say class $X$) forms the vertices of the PCD and the Delaunay tessellation of the other class (i.e., class $Y$) yields the (Delaunay) cells which serve as the support of class $X$ points. We demonstrate that edge density of these graphs is a $U$-statistic, hence obtain the asymptotic normality of it for data from any distribution that satisfies mild regulatory conditions. The rate of convergence to asymptotic normality is sharper for the edge density of the reflexivity and underlying graphs compared to the arc density of the PE-PCDs. For uniform data in Euclidean plane where Delaunay cells are triangles, we demonstrate that the distribution of the edge density is geometry invariant (i.e., independent of the shape of the triangular support). We compute the explicit forms of the asymptotic normal distribution for uniform data in one Delaunay triangle in the Euclidean plane utilizing this geometry invariance property. We also provide various versions of edge density in the multiple triangle case. The approach presented here can also be extended for application to data in higher dimensions.

A general diagnostic approach to the evaluation of asymptotic approximation in likelihood based models is developed and applied to logistic regression. The expected asymptotic and observed log-likelihood functions are compared using a chi distribution in a directional Bayesian setting. This provides a general approach to assessing and visualizing non-convergence in higher dimensional models. Several well-known examples from the logistic regression literature are discussed.

In both statistical non-inferiority (NI) and superiority (S) tests, the critical region must be a Barnard convex set for two main reasons. One, being computational in nature, based on the fact that calculating test sizes is a computationally intensive problem due to the presence of a nuisance parameter. However, this calculation is considerably reduced when the critical region is a Barnard convex set. The other reason is that in order for the NI/S statistical tests to make sense, its critical regions must be Barnard convex sets. While it is indeed possible for NI/S tests’ critical regions to not be Barnard convex sets, for the reasons stated above, it is desirable that they are. Therefore, it is important to generate, from a given NI/S test, a test which guarantees that the critical regions are Barnard convex sets. We propose a method by which, from a given NI/S test, we construct another NI/S test, ensuring that the critical regions corresponding to the modified test are Barnard convex sets, we illustrate this through examples. This work is theoretical because the type of developments refers to the general framework of NI/S testing for two independent binomial proportions and it is applied because statistical tests that do not ensure that their critical regions are Barnard convex sets may appear in practice, particularly in the clinical trials area.

A problem for estimating the number of trials $n$ in the binomial distribution $B(n,p)$, is revisited by considering the large sample model $N(\mu ,c\mu )$ and the associated maximum likelihood estimator (MLE) and some sequential procedures. Asymptotic properties of the MLE of $n$ via the normal model $N(\mu ,c\mu )$ are briefly described. Beyond the asymptotic properties, our main focus is on the sequential estimation of $n$. Let $X_1,X_2,\dots ,X_m,\dots$ be iid $N(\mu ,c\mu )$$(c>0)$ random variables with an unknown mean $\mu =1,2,\dots$ and variance $c\mu$, where $c$ is known. The sequential estimation of $\mu$ is explored by a method initiated by Robbins (1970) and further pursued by Khan (1973). Various properties of the procedure including the error probability and the expected sample size are determined. An asymptotic optimality of the procedure is given. Sequential interval estimation and point estimation are also briefly discussed.

In statistical inference on the drift parameter $\theta$ in the process $X_t=\theta a\left(t\right)+{\int }_0^{t}b\left(s\right)d{W}_s$, where $a\left(t\right)$ and $b\left(t\right)$ are known, deterministic functions, there is known a large number of options how to do it. We may, for example, base this inference on the differences between the observed values of the process at discrete times and their normality. Although such methods are very simple, it turns out that it is more appropriate to use sequential methods. For the hypotheses testing about the drift parameter $\theta$, it is more proper to standardize the observed process and to use sequential methods based on the first exit time of the observed process of a pre-specified interval until some given time. These methods can be generalized to the case of random part being a symmetric Itô integral or continuous symmetric martingale.

Normal nonlinear regression models are applied in some areas of the sciences and engineering to explain or describe the phenomena under study. However, it is well known that several phenomena are not always represented by the normal model due to lack of symmetry or the presence of heavy- and light-tailed distributions related to the normal law in the data. This paper proposes an extension of nonlinear regression models using the skew-scale mixtures of normal (SSMN) distributions proposed by Ferreira et al. (2011). This class of models provides a useful generalization of the symmetrical nonlinear regression models since the random term distributions cover both asymmetric and heavy-tailed distributions, such as the skew-$t$-normal, skew-slash and skew-contaminated normal, among others. An expectation–maximization (EM) algorithm for maximum likelihood (ML) estimates is presented and the observed information matrix is derived analytically. Some simulation studies are presented to examine the performance of the proposed methods, with relation to robustness and asymptotic properties of the ML estimates. Finally, an illustration of the method is presented considering a dataset previously analyzed under normal and skew-normal (SN) nonlinear regression models. The main conclusion is that the ML estimates from the heavy tails SSMN nonlinear models are more robust against outlying observations compared to the corresponding SN estimates.