Asymptotic Distribution Research Articles

Test of the Latent Dimension of a Spatial Blind Source Separation Model

We assume a spatial blind source separation model in which the observed multivariate spatial data is a linear mixture of latent spatially uncorrelated Gaussian random fields containing a number of pure white noise components. We propose a test on the number of white noise components and obtain the asymptotic distribution of its statistic for a general domain. We also demonstrate how computations can be facilitated in the case of gridded observation locations. Based on this test, we obtain a consistent estimator of the true dimension. Simulation studies and an environmental application demonstrate that our test is at least comparable to and often outperforms bootstrap-based techniques, which are also introduced in this paper.

On the asymptotic distribution of the maxima from Gaussian functions subject to missing observations

On the asymptotic distribution of the maxima from Gaussian functions subject to missing observations

Limit theorems for local polynomial estimation of regression for functional dependent data

<p>Local polynomial fitting exhibits numerous compelling statistical properties, particularly within the intricate realm of multivariate analysis. However, as functional data analysis gains prominence as a dynamic and pertinent field in data science, the exigency arises for the formulation of a specialized theory tailored to local polynomial fitting. We explored the intricate task of estimating the regression function operator and its partial derivatives for stationary mixing random processes, denoted as $ (Y_i, X_i) $, using local higher-order polynomial fitting. Our key contributions include establishing the joint asymptotic normality of the estimates for both the regression function and its partial derivatives, specifically in the context of strongly mixing processes. Additionally, we provide explicit expressions for the bias and the variance-covariance matrix of the asymptotic distribution. Demonstrating uniform strong consistency over compact subsets, along with delineating the rates of convergence, we substantiated these results for both the regression function and its partial derivatives. Importantly, these findings rooted in reasonably broad conditions that underpinned the underlying models. To demonstrate practical applicability, we leveraged our results to compute pointwise confidence regions. Finally, we extended our ideas to the nonparametric conditional distribution, and obtained its limiting distribution.</p>

Reliability estimation stress–strength dependent model based on copula function using ranked set sampling

In the stress-strength reliability analysis, it is usually assumed that both stress and strength are independent variables. In most cases, stress and strength depend on each other. In this paper, Clayton copula function is selected when stress and strength variables follow Kumaraswamy distribution and Weibull distribution respectively, and the maximum likelihood estimate and least squares estimate of unknown parameters of stress-strength dependent model are derived by two-step method under ranked set sample. At the same time, the approximate confidence intervals are constructed by using the asymptotic distribution of maximum likelihood and the parametric bootstrap percentile method. In addition, the simulation results of simple random sample and ranked set sample are compared by Monte Carlo simulation method. Finally, a real case is selected to illustrate the feasibility of the proposed model.

The asymptotic distribution of coefficients of the Dedekind zeta-function over a sparse sequence

The asymptotic distribution of coefficients of the Dedekind zeta-function over a sparse sequence

Inference of stress-strength reliability based on adaptive progressive type-Ⅱ censing from Chen distribution with application to carbon fiber data

<abstract><p>In this paper, we used the maximum likelihood estimation (MLE) and the Bayes methods to perform estimation procedures for the reliability of stress-strength $ R = P(Y < X) $ based on independent adaptive progressive censored samples that were taken from the Chen distribution. An approximate confidence interval of $ R $ was constructed using a variety of classical techniques, such as the normal approximation of the MLE, the normal approximation of the log-transformed MLE, and the percentile bootstrap (Boot-p) procedure. Additionally, the asymptotic distribution theory and delta approach were used to generate the approximate confidence interval. Further, the Bayesian estimation of $ R $ was obtained based on the balanced loss function, which came in two versions here, the symmetric balanced squared error (BSE) loss function and the asymmetric balanced linear exponential (BLINEX) loss function. When estimating $ R $ using the Bayesian approach, all the unknown parameters of the Chen distribution were assumed to be independently distributed and to have informative gamma priors. Additionally, a mixture of Gibbs sampling algorithm and Metropolis-Hastings algorithm was used to compute the Bayes estimate of $ R $ and the associated highest posterior density credible interval. In the end, simulation research was used to assess the general overall performance of the proposed estimators and a real dataset was provided to exemplify the theoretical results.</p></abstract>

On the asymptotic power of the “energy” test for equality of two distributions

In the paper the asymptotic distribution and the formula for the asymptotic power are found for the "energy" test in the case of alternative distributions differ from zero distribution by the parameter of shift and/or the parameter of scale. This criterion is an alternative for the well-known Mann-Whitney test but allows to compare distributions that differ by the scale parameter. The efficiency of the results obtained is demonstrated by the stochastic simulation for the normal distribution and the Cauchy distribution.

Bootstrap Inference for Fixed‐Effect Models

The maximum‐likelihood estimator of nonlinear panel data models with fixed effects is asymptotically biased under rectangular‐array asymptotics. The literature has devoted substantial effort to devising methods that correct for this bias as a means to salvage standard inferential procedures. The chief purpose of this paper is to show that the (recursive, parametric) bootstrap replicates the asymptotic distribution of the (uncorrected) maximum‐likelihood estimator and of the likelihood‐ratio statistic. This justifies the use of confidence sets and decision rules for hypothesis testing constructed via conventional bootstrap methods. No modification for the presence of bias needs to be made.

On Asymptotic Distribution of Sum of SINRs

On Asymptotic Distribution of Sum of SINRs

Комбинированные оценки линейного функционала с априорной догадкой о параметрическом классе распределений

The problem of estimating a linear functional from an unknown distribution, which expresses a certain numerical characteristic of an observed random variable, is considered. It is assumed that the researcher, based on his experience and knowledge, can make an assumption about the admissibility of a particular parametric class of distributions when estimating the functional. This assumption is usually called an a priori guess. This article proposes adaptive estimates of the linear functional that take into account an a priori guess about the parametric class of distributions. The resulting asymptotic distributions of such estimates made it possible to study the influence of an a priori guess on the accuracy of estimation analytically.

Application of Multivariate Logistic Regression and Decision Trees to Assess Factors Influencing Prevalence of Abortion and Stillbirth in Yankasa sheep

AbstractThis study was carried out to assess factors influencing prevalence of abortion and stillbirth in Yankasa sheep using multivariate logistic regression and decision trees. 191 kidding records of Ewes from a total of 50 traditional Yankasa sheep herders within Nasarawa South agro-ecological zone from the year 2020-2021 were utilized in the study. Sampling was restricted to only farmers that were able to give information on Lamb, Ram and Ewe identification as well as occurrence of abortion, stillbirth (defined as a kid born dead or dying within 24 hours after birth), kidding date or period, parity and number of foetuses. Three seasons of abortion or stillbirth were generated according to the month of the year: rainy season (from May to October), dry season (from February to April) and harmattan season (from November to January). The logit of the probability of an abortion or stillbirth was modelled using logistic regression assuming an asymptotic binomial distribution. The Chi-square goodness of fit test was performed to check if the multivariate logistic model fitted the data well (P>0.05). Chi-square automatic interaction detection (CHAID) algorithm was also employed to model incidence of abortion or stillbirth. The chi-square analysis revealed that parity did not significantly (P>0.05) affect the incidence of abortion. However, season of the year significantly (P<0.05) affected prevalence of abortion, which was higher in the harmattan (38.5%) compared to the rainy (11.5%) and dry season (16.5%), respectively. Parity and season of the year did not affect significantly (P>0.05) the prevalence of stillbirth. The binary logistic regression analysis showed that season affected significantly (P<0.05) the prevalence of abortion, especially in the harmattan period with a high odds ratio (4.539). This was also confirmed by the result of the CHAID analysis. The present information could be exploited in management practices to reduce to the incidence of abortion in order to improve the production level of the sheep farmers.

Predictive model averaging with parameter instability and heteroskedasticity

AbstractThis paper proposes a frequentist model averaging approach in the presence of parameter instability and heteroskedasticity. We derive optimal weights combining the stable and break specifications of a predictive model, with the weights from minimizing the leave‐one‐out cross‐validation information criterion (CV). We characterize the asymptotic distribution of the CV and provide the analytical expressions of the feasible optimal CV weights. Our simulations and applications forecasting the US and Taiwanese GDP growth demonstrate the superior performance of the CV model averaging relative to other methods such as the Mallows averaging, the approximate Bayesian averaging, and equal weighting.

An adaptive test based on Kendall's tau for independence in high dimensions

We consider testing the mutual independency for high-dimensional data. It is known that L 2 -type statistics have lower power under sparse alternatives and L ∞ -type statistics have lower power under dense alternatives in high dimensions. In this paper, we develop an adaptive test based on Kendall's tau to compromise both situations of the alternative, which can automatically be adapted to the underlying data. An adaptive test is very useful in practice as the sparsity or density for a data set is usually unknown. In addition, we establish the asymptotic joint distribution of L 2 -type and L ∞ -type statistics based on Kendall's tau under mild assumptions and the asymptotic null distribution of the proposed statistic. Simulation studies show that our adaptive test performs well in either dense or sparse cases. To illustrate the usefulness and effectiveness of the proposed test, real data sets are also analysed.

Maximum likelihood estimation of a change point for Poisson distributed data

In this study we develop a change point methodology to identify and estimate changes in the parameter of a Poisson distribution. The proposed methodology considers the case when the Poisson parameter changes abruptly at an unknown point of time. For this case, the maximum likelihood estimate of the change point and its asymptotic distribution are pursued. Mainly, we carry out a large scale simulation study for evaluating the appropriateness of the asymptotic distribution of the mle from the view point of finite samples, and also for evaluating the closeness under known and unknown parameters. The simulations study also compares the mle with that of a Bayesian estimate. Then, the methodology is applied to three examples. First, we uncover changes in the number of homicides in California using monthly data from January 2002 until December 2020. Secondly, data about deaths of females caused by stomach cancer is considered to detect possible changes in the numbers recorded from 1930 to 2011. Thirdly, British coal mining disasters from 1851 to 1962 in which more than 10 men were killed are analyzed.

Effective noisy dynamics within the phenotypic space of a growth-rate maximizing population

Microbial systems exhibit marked variability in metabolic phenotypes. A recently-proposed class of models explains this feature within a minimal mathematical setup which assumes that populations evolve towards maximum growth rate in a ‘phenotypic space’ subject to an intrinsic ‘diffusive’ stochasticity that causes small random changes in single-cell phenotypes. In such a framework, variability results from the exploration–exploitation balance between hardly accessible fast-growing phenotypes and easily accessible slow-growing ones. Here we extend the above scheme to include a degree of extrinsic noise, showing that the population dynamics over the phenotypic space is captured by an effective process that conflates both sources of randomness. This in turn leads to a simple approximation for the asymptotic distribution of the population over the phenotypic space, highlighting the connection between the strength of the noise that affects the dynamics and the degree of optimization. The theory thus obtained displays an excellent agreement with numerical simulations of low-dimensional systems.

Post-averaging inference for optimal model averaging estimator in generalized linear models

This article considers the problem of post-averaging inference for optimal model averaging estimators in a generalized linear model (GLM). We establish the asymptotic distributions of optimal model averaging estimators for GLMs. The asymptotic distributions of the model averaging estimators are nonstandard, depending on the configuration of the penalty term in the weight choice criterion. We also propose a feasible simulation-based confidence interval estimator and investigate its asymptotic properties rigorously. Monte Carlo simulations verify the usefulness of our theoretical results, and the proposed methods are employed to analyze a stock car racing dataset.

Central limit theorems for local network statistics

Summary Subgraph counts, in particular the number of occurrences of small shapes such as triangles, characterize properties of random networks. As a result, they have seen wide use as network summary statistics. Subgraphs are typically counted globally, making existing approaches unable to describe vertex-specific characteristics. In contrast, rooted subgraphs focus on vertex neighbourhoods, and are fundamental descriptors of local network properties. We derive the asymptotic joint distribution of rooted subgraph counts in inhomogeneous random graphs, a model that generalizes most statistical network models. This result enables a shift in the statistical analysis of graphs, from estimating network summaries to estimating models linking local network structure and vertex-specific covariates. As an example, we consider a school friendship network and show that gender and race are significant predictors of local friendship patterns.

Testing for coefficient distortion due to outliers with an application to the economic impacts of climate change

Outlying observations can bias regression estimates, requiring the use of outlier-robust estimators. Comparing robust estimates to those obtained using ordinary least squares (OLS) is a common robustness check, however, such comparisons have been mostly informal due to the lack of available tests. Here we introduce a formal test for coefficient distortion due to outliers in regression models. Our proposed test is based on the difference between OLS and robust estimates obtained using a class of Huber-skip M-type estimators (such as Impulse Indicator Saturation or Robustified Least Squares). We show that our distortion test has an asymptotic chi-squared distribution by establishing the asymptotics of the corresponding Huber-skip M-estimators using an empirical process Central Limit Theorem recently developed in the literature. The test is valid for cross-sectional, as well as panel, and stationary or deterministically-trending time series models. To improve finite sample performance and to alleviate concerns on distributional assumptions, we explore several bootstrap testing schemes. We apply our outlier distortion test to estimates of the macro-economic impacts of climate change allowing for adaptation.

The estimations of drift parameters for the Gaussian Vasicek process with time-varying volatility

This article mainly considers the parameter estimation problem for the Gaussian Vasicek process defined as d X t = ( α − β X t ) d t + σ t d G t , t≥0 with unknown parameters α∈ℝ and β < 0, where G t is a continuous centered Gaussian process. The volatility parameter σ t is allowed to vary with time t. Based on the time-continuous observations and least squares method, the Least squares estimations of drift parameters for the non ergodic Gaussian Vasicek model with time-varying volatility are obtained. The strong consistency and asymptotic distributions of proposed estimations are verified under some sufficient conditions. Finally, we verify that the results are still valid under different forms of the volatility function when G t is fractional Brownian motion, sub-fractional Brownian motion, and bi-fractional Brownian motion, respectively. In some sense, our results extend the findings of El Machkouri, Es-Sebaiy, and Ouknine (2016) and Es-Sebaiy and Sebaiy (2021).

Statistical inference of a step-stress model with competing risks under time censoring for alpha power exponential distribution

Accelerated life testing in a reliability experiment allows test units to be under more stress than normal (expected). Step-stress tests are a subset of accelerated life tests that raise the stress levels at specific predetermined time periods to gather data on lifetime parameters more quickly than they would under normal operating settings. Additionally, there are frequently a number of risk factors connected to the root cause of a test unit failure (mechanical, electrical, etc.). In this article, we discuss the statistical inference of a step-stress model in the presence of time censoring assuming independent alpha power exponential distribution. The point estimates of the unknown scale and shape parameters of the various causes are determined using the maximum likelihood technique under the assumption of cumulative damage. We also go through creating confidence intervals for the parameters using asymptotic distributions and the parametric bootstrap approach. Extensive Monte Carlo simulations are used to evaluate the accuracy of the estimates and the performance of the confidence intervals, and finally, an example is used to explain the methods of inference presented here.