Journal of Statistical Planning and Inference最新文献

The problem of computing an exact experimental design that is optimal for the least-squares estimation of the parameters of a regression model is considered. We show that this problem can be solved via mixed-integer linear programming (MILP) for a wide class of optimality criteria, including the criteria of A-, I-, G- and MV-optimality. This approach improves upon the current state-of-the-art mathematical programming formulation, which uses mixed-integer second-order cone programming. The key idea underlying the MILP formulation is McCormick relaxation, which critically depends on finite interval bounds for the elements of the covariance matrix of the least-squares estimator corresponding to an optimal exact design. We provide both analytic and algorithmic methods for constructing these bounds. We also demonstrate the unique advantages of the MILP approach, such as the possibility of incorporating multiple design constraints into the optimization problem, including constraints on the variances and covariances of the least-squares estimator.

We study some properties and relations for stochastic orders and aging classes related to the Laplace transform. In particular, we show that the NBU${}_{\text{Lt}}$ class of distributions is closed under convolution. We also obtain results for the ratio of derivatives of the Laplace transform between two distributions.

Convolutional neural networks (CNNs) trained with cross-entropy loss have proven to be extremely successful in classifying images. In recent years, much work has been done to also improve the theoretical understanding of neural networks. Nevertheless, it seems limited when these networks are trained with cross-entropy loss, mainly because of the unboundedness of the target function. In this paper, we aim to fill this gap by analysing the rate of the excess risk of a CNN classifier trained by cross-entropy loss. Under suitable assumptions on the smoothness and structure of the a posteriori probability, it is shown that these classifiers achieve a rate of convergence which is independent of the dimension of the image. These rates are in line with the practical observations about CNNs.

In this paper, two simple and effective construction methods are proposed to construct four-level design with large size via quaternary codes from some small two-level initial designs. Under the popular criteria for selecting optimal design, such as generalized minimum aberration, minimum moment aberration and uniformity measured by average Lee discrepancy, the close relationships between the constructed four-level design and its initial design are investigated, which provide the guidance for choosing the suitable initial design. Moreover, some lower bounds of average Lee discrepancy for the constructed four-level designs are obtained, which can be used as a benchmark for evaluating the uniformity of the constructed four-level designs. Some numerical examples show that the large four-level designs can be constructed with high efficiency.

Integrative analysis plays a critical role in integrating heterogeneous data from multiple datasets to provide a comprehensive view of the overall data features. However, in multiple datasets, outliers and heavy-tailed data can render least squares estimation unreliable. In response, we propose a Robust Integrative Analysis via Quantile Regression (RIAQ) that accounts for homogeneity and sparsity in multiple datasets. The RIAQ approach is not only able to identify latent homogeneous coefficient structures but also recover the sparsity of high-dimensional covariates via double penalty terms. The integration of sample information across multiple datasets improves estimation efficiency, while a sparse model improves model interpretability. Furthermore, quantile regression allows the detection of subgroup structures under different quantile levels, providing a comprehensive picture of the relationship between response and high-dimensional covariates. We develop an efficient alternating direction method of multipliers (ADMM) algorithm to solve the optimization problem and study its convergence. We also derive the parameter selection consistency of the modified Bayesian information criterion. Numerical studies demonstrate that our proposed estimator has satisfactory finite-sample performance, especially in heavy-tailed cases.

In studies on lifetimes, occasionally, the population contains statistical units that are born before the data collection has started. Left-truncated are units that deceased before this start. For all other units, the age at the study start often is recorded and we aim at testing whether this second measurement is independent of the genuine measure of interest, the lifetime. Our basic model of dependence is the one-parameter Gumbel–Barnett copula. For simplicity, the marginal distribution of the lifetime is assumed to be Exponential and for the age-at-study-start, namely the distribution of birth dates, we assume a Uniform. Also for simplicity, and to fit our application, we assume that units that die later than our study period, are also truncated. As a result from point process theory, we can approximate the truncated sample by a Poisson process and thereby derive its likelihood. Identification, consistency and asymptotic distribution of the maximum-likelihood estimator are derived. Testing for positive truncation dependence must include the hypothetical independence which coincides with the boundary of the copula’s parameter space. By non-standard theory, the maximum likelihood estimator of the exponential and the copula parameter is distributed as a mixture of a two- and a one-dimensional normal distribution. For the proof, the third parameter, the unobservable sample size, is profiled out. An interesting result is, that it differs to view the data as truncated sample, or, as simple sample from the truncated population, but not by much. The application are 55 thousand double-truncated lifetimes of German businesses that closed down over the period 2014 to 2016. The likelihood has its maximum for the copula parameter at the parameter space boundary so that the $p$-value of test is 0.5. The life expectancy does not increase relative to the year of foundation. Using a Farlie–Gumbel–Morgenstern copula, which models positive and negative dependence, finds that life expectancy of German enterprises even decreases significantly over time. A simulation under the condition of the application suggests that the tests retain the nominal level and have good power.

Row–column designs that provide unconfounded estimation of all main effects and the maximum number of two-factor interactions (2fi’s) are called 2fi-optimal. This issue has been paid great attention recently for its wide application in industrial or physical experiments. The constructions of 2fi-optimal two-level and three-level full factorial and fractional factorial row–column designs have been proposed. However, the results for higher prime levels have not been achieved yet. In this paper, we give theoretical constructions of 2fi-optimal $s^n$ full factorial row–column designs for any odd prime level $s$ and any parameter combination, and theoretical constructions of 2fi-optimal $s^{n-1}$ fractional factorial row–column designs for any prime level $s$ and any parameter combination.

The beta regression model is tailored for responses that assume values in the standard unit interval. It comprises two submodels, one for the mean response and another for the precision parameter. We develop tests of correct specification for such a model. The tests are based on the information matrix equality, which holds when the model is correctly specified. We establish the validity of the tests in the class of varying precision beta regressions, provide closed-form expressions for the quantities used in the test statistics, and present simulation evidence on the tests’ null and non-null behavior. We show that it is possible to achieve very good control of the type I error probability when data resampling is employed and that the tests are able to reliably detect incorrect model specification, especially when the sample size is not small. An empirical application is presented and discussed.

A self-normalized approach for testing the stationarity of a $d$-dimensional random field is considered in this paper. Because the discrete Fourier transforms (DFT) at fundamental frequencies of a second-order stationary random field are asymptotically uncorrelated (see Bandyopadhyay and Subba Rao, 2017), one can construct a stationarity test based on the sample covariance of the DFTs. Such a test is usually inferior because it involves an overestimated scale parameter that leads to low size and power. To circumvent this shortcoming, this paper proposes two self-normalized statistics based on extreme value and partial sum of the sample covariance of the DFTs. Under certain regularity conditions, it is shown that the proposed tests converge to functionals of Brownian motion. Simulations and a data analysis demonstrate the outstanding performance of the proposed tests.

Conditional tail expectation (CTE) is a coherent risk measure defined as the mean of the loss distribution above a high quantile. The existence of the CTE as well as the asymptotic properties of associated estimators however require integrability conditions that may be violated when dealing with heavy-tailed distributions. We introduce Box–Cox transforms of the CTE that have two benefits. First, they alleviate these theoretical issues. Second, they enable to recover a number of risk measures such as conditional tail expectation, expected shortfall, conditional value-at-risk or conditional tail variance. The construction of dedicated estimators is based on the investigation of the asymptotic relationship between Box–Cox transforms of the CTE and quantiles at extreme probability levels, as well as on an extrapolation formula established in the heavy-tailed context. We quantify and estimate the bias induced by the use of these approximations and then introduce reduced-bias estimators whose asymptotic properties are rigorously shown. Their finite-sample properties are assessed on a simulation study and illustrated on real data, highlighting the practical interest of both the bias reduction and the Box–Cox transform.