Central Chi-square Distribution Research Articles

Detection of radar signals scattered by acoustic disturbances generated by UAVs

The problem of UAV radar monitoring by its acoustic radiation is considered. It is shown that in a number of practical cases such an approach is preferable to observation by radar methods directly from the UAV airframe. It is noted that the radio signals scattered by acoustic packets from the UAV are characterized by an unknown in advance complex envelope, which does not allow the use of optimal filtering methods for their detection and estimation. It is shown that to solve these problems, it is advisable to use the principle of accumulation over the observation interval of the energy of a narrow-band random process reduced to the noise dispersion, using the statistical differences between noise fluctuations and the additive "signal-plus-noise" mixture. It is shown that the energy estimate reduced to noise has either a central or an off-center "chi-square" distribution with a certain number of degrees of freedom and an off-center parameter greater than or equal to zero. As a result of comparing the current value of the no centrality parameter with the threshold value, a decision is made on the presence or absence of a useful signal in the observation interval with minimal a priori information about its parameters. It is noted that the well-known expressions for the differential probability densities of the central and non-central chi-square distributions allow one to obtain qualitative estimates of the synthesized detector. A practical structural diagram of a detector using processing of received oscillations in quadrature channels is proposed.

Empirical likelihood inference and goodness-of-fit test for logistic regression model under two-phase case-control sampling

ABSTRACT Due to cost-effectiveness and high efficiency, two-phase case-control sampling has been widely used in epidemiology studies. We develop a semi-parametric empirical likelihood approach to two-phase case-control data under the logistic regression model. We show that the maximum empirical likelihood estimator has an asymptotically normal distribution, and the empirical likelihood ratio follows an asymptotically central chi-square distribution. We find that the maximum empirical likelihood estimator is equal to Breslow and Holubkov (1997)'s maximum likelihood estimator. Even so, the limiting distribution of the likelihood ratio, likelihood-ratio-based interval, and test are all new. Furthermore, we construct new Kolmogorov–Smirnov type goodness-of-fit tests to test the validation of the underlying logistic regression model. Our simulation results and a real application show that the likelihood-ratio-based interval and test have certain merits over the Wald-type counterparts and that the proposed goodness-of-fit test is valid.

Empirical likelihood meta-analysis with publication bias correction under Copas-like selection model

Meta-analysis is commonly used to synthesize multiple results from individual studies. However, its validation is usually threatened by publication bias and between-study heterogeneity, which can be captured by the Copas selection model. Existing inference methods under this model are all based on conditional likelihood and may not be fully efficient. In this paper, we propose a full likelihood approach to meta-analysis by integrating the conditional likelihood and a marginal semi-parametric empirical likelihood under a Copas-like selection model. We show that the maximum likelihood estimators (MLE) of all the underlying parameters have a jointly normal limiting distribution, and the full likelihood ratio follows an asymptotic central chi-square distribution. Our simulation results indicate that compared with the conditional likelihood method, the proposed MLEs have smaller mean squared errors and the full likelihood ratio confidence intervals have more accurate coverage probabilities. A real data example is analyzed to show the advantages of the full likelihood method over the conditional likelihood method.

An analysis of energy detector based on improved approximations of the chi-square distributions

Closed-form expressions for the detection probability, the false alarm probability and the energy detector constant threshold are derived using approximations of the central chi-square and non-central chi-square distributions. The approximations used show closer proximity to the original functions when compared to the expressions used in the literature. The novel expressions allow gains up to 6% and 16% in terms of measured false alarm and miss-detection probability, respectively, if compared to the Central Limit Theorem approach. The throughput of cognitive network is also enhanced when these novel expressions are implemented, providing gains up to 9%. New equations are also presented that minimize the total error rate to obtain the detection threshold and the optimal number of samples. The analytical results match the results of the simulation for a wide range of SNR values.

Likelihood ratio test for the adequacy of a regression model when the noise is given by the Slepian field

In this paper we establish a non parametric likelihood ratio (LR) test procedure for testing the mean of a regression or additive model when the disturbance is represented by the Slepian field. By utilizing the Cameron-Martin-Girsanov formula and the fundamental theorem of projection, we get the likelihood estimator of the mean function under either H 0 or H 1 as the projection of the random observation. By applying the fundamental characteristic of the Slepian field for a typical size of the moving rectangle, it is shown under H 0 that the probability distribution of the test statistic belongs to the family of a central chi-square distributions, whereas under H 1 it is distributed as a noncentral chi-square. Furthermore, by simulating a type of Fourier model, the empirical power function of the test shows that the proposed test is consistent in the sense it maximizes the probability of the rejection when the model is not true.

A Bayesian robust chi-squared test for testing simple hypotheses

In this paper, we introduce a new Bayesian chi-squared test based on an adjusted quadratic loss function for testing a simple null hypothesis. We show that the asymptotic null distribution of our suggested test is a central chi-squared distribution under some assumptions required for the Bayesian large sample theory. We refer to our test as the Bayesian robust chi-squared test, since it is robust to parametric misspecification in the alternative model. That is, the limiting null distribution of our test is a central chi-squared distribution irrespective of parametric misspecification in the alternative model. In addition to being robust to parametric misspecification, our test also shares properties of the test suggested by Li et al. (2015) based on a quadratic loss function. We provide four examples to illustrate the implementation of our suggested Bayesian test statistic.

An Efficient Multiple Imputation Approach for Estimating Equations with Response Missing at Random and High-Dimensional Covariates

Empirical-likelihood-based inference for parameters defined by the general estimating equations of Qin and Lawless (1994) remains an active research topic. When the response is missing at random (MAR) and the dimension of covariate is not low, the authors propose a two-stage estimation procedure by using the dimension-reduced kernel estimators in conjunction with an unbiased estimating function based on augmented inverse probability weighting and multiple imputation (AIPW-MI) methods. The authors show that the resulting estimator achieves consistency and asymptotic normality. In addition, the corresponding empirical likelihood ratio statistics asymptotically follow central chi-square distributions when evaluated at the true parameter. The finite-sample performance of the proposed estimator is studied through simulation, and an application to HIV-CD4 data set is also presented.

Secondary analysis of case-control association studies: Insights on weighting-based inference motivate a new specification test.

Case-control sampling is frequently used in genetic association studies to examine the relationship between disease and genetic exposures. Such designs usually collect extensive information on phenotypes beyond the primary disease, whose associations with the genetic exposures are also of great interest. Because the cases are over-sampled, appropriate analysis of secondary phenotypes should take into account this biased sampling design. We previously introduced a weighting-based estimator for appropriate secondary analysis, but have not thoroughly explored its statistical properties. In this article, we revisit our previous estimator to offer new insights and methodological extensions. Specifically, we extend our previous estimator and construct its more general form based on generalized least squares (GLS). Such an extension allows us to connect the GLS estimator with the generalized method of moments and motivates a new specification test designed to assess the adequacy of the disease model or the weights. The specification test statistic measures the weighted discrepancy between the case and control subsample estimators, and asymptotically follows a central Chi-squared distribution under correct disease model specification. We illustrate the GLS estimator and specification test using a case-control sample of peripheral arterial disease, and use simulations to further shed light on the operating characteristics of the specification test.

Empirical likelihood and variable selection for partially linear single-index EV models with missing censoring indicators

In this paper, we focus on the empirical likelihood inference for partially linear single-index errors-in-variables (EV) models when the data are right censored and the censoring indicator is missing at random (MAR). Two bias-corrected empirical log-likelihood ratio (BCELR) functions for the parameters by using regressing calibration and imputation methods are introduced. The limiting distributions of the BCELRs are shown to have a mixture of central chi-squared distribution. Based on this, the confidence regions of the parameters can be constructed by using bootstrap approximation. Furthermore, as there would be some spurious covariates in the linear and nonlinear parts, a penalized empirical likelihood (PEL) approach is proposed with the help of smoothly clipped absolute deviation penalty. Two proposed PEL estimators are shown to possess the oracle property. A simulation study and a real data analysis are conducted to evaluate the finite sample performance of the proposed methods.

Mixture representations of noncentral distributions

With any symmetric distribution μ on the real line we may associate a parametric family of noncentral distributions as the distributions of where X is a random variable with distribution μ. The classical case arises if μ is the standard normal distribution, leading to the noncentral chi-squared distributions. It is well known that these may be written as Poisson mixtures of the central chi-squared distributions with odd degrees of freedom. We obtain such mixture representations for the logistic distribution and for the hyperbolic secant distribution. We also derive alternative representations for chi-squared distributions and relate these to representations of the Poisson family. While such questions originated in parametric statistics they also appear in the context of the generalized second Ray–Knight theorem, which connects Gaussian processes and local times of Markov processes.

Penalized generalized empirical likelihood with a diverging number of general estimating equations for censored data

This article considers simultaneous variable selection and parameter estimation as well as hypothesis testing in censored survival models where a parametric likelihood is not available. For the problem, we utilize certain growing dimensional general estimating equations and propose a penalized generalized empirical likelihood, where the general estimating equations are constructed based on the semiparametric efficiency bound of estimation with given moment conditions. The proposed penalized generalized empirical likelihood estimators enjoy the oracle properties, and the estimator of any fixed dimensional vector of nonzero parameters achieves the semiparametric efficiency bound asymptotically. Furthermore, we show that the penalized generalized empirical likelihood ratio test statistic has an asymptotic central chi-square distribution. The conditions of local and restricted global optimality of weighted penalized generalized empirical likelihood estimators are also discussed. We present a two-layer iterative algorithm for efficient implementation, and investigate its convergence property. The performance of the proposed methods is demonstrated by extensive simulation studies, and a real data example is provided for illustration.

Generalized empirical likelihood specification test robust to local misspecification

It is well known that many of the standard specification tests may not be robust when the alternative is misspecified. This study analyzes a robust specification test for generalized empirical likelihood (GEL) estimators in a weakly dependent time series setting. We show that the usual score test statistic asymptotically follows a non-central chi-square distribution under the local misspecification in the GEL framework. Thus, it spuriously rejects the null hypothesis too frequently. We propose a robust score specification test that asymptotically follows a central chi-square distribution under the local misspecification. A Monte Carlo simulation verifies the usefulness of the proposed tests.

Spectrum sensing exploiting the maximum value of power spectrum density in wireless sensor network

Spectrum sensing plays a foundational role in cognitive radio sensor networks. However, only the methods with low computational complexity can be utilized due to energy restriction of sensor node. To this end, a novel frequency-domain spectrum sensing method is presented to satisfy corresponding requirements of cognitive radio sensor networks. Only the maximum value of power spectrum density is utilized as test statistic to reduce the computational complexity. According to the dependence of 2L real parts and imaginary parts of the maximum value of power spectrum density, we model the maximum value of power spectrum density as the central Chi-square distribution for the $$H_0$$ case and non-central Chi-square distribution for the $$H_1$$ case. Exploiting resulting distributions, we derive the analytic expressions for the detection probability and the false-alarm probability. Additionally, the computational complexity of the proposed method is quantitatively analyzed. Finally, we certify the proposed test statistic and the probability distribution of the maximum value of power spectrum density and evaluate the impact of some parameters on the detection performance experimentally. The theoretical analysis and simulation results demonstrate that the proposed algorithm can offer high performance gains over the existing time-domain detection method.

Performance Analysis of Locally Most Powerful Invariant Test for Sphericity of Gaussian Vectors in Coherent MIMO Radar

Locally most powerful invariant test (LMPIT) for sphericity of Gaussian vectors has been derived by Ramirez et al. Nevertheless, the decision threshold of the LMPIT is not accurate and its detection performance has not yet been addressed. In this paper, the LMPIT is performed for target detection in multiple-input multiple-output (MIMO) radar, and its theoretical decision threshold as well as detection probability are accurately determined. Utilizing asymptotic expansion approach, we calculate the asymptotic null distribution as a function of central chi-square distributions, resulting in precise closed-form formula for thresholding. On the other hand, the nonnull distribution is approximated by weighted sum of noncentral chi-square distributions and Gamma distribution for close and far hypotheses, respectively. This enables us to derive a closed-form formula to precisely evaluate the detection power of the LMPIT. Numerical results demonstrate that our theoretical computations are very accurate in determining the decision threshold and predicting the behaviors of the LMPIT. Moreover, the superiority of the LMPIT for MIMO radar target detection over state-of-the-art methods is demonstrated for spatially colored but temporarily white noise.

Proposed Distance-Based Test for Testing Multivariate Multiple Regression Coefficients under Restricted Alternatives

In constructing estimation and hypothesis testing procedures, it is important that all available information such as sign of parameter is used in order to maximize power of the test. Often prior information are known about the sign of regression coefficients (parameter) under test, the best example being that variances cannot be negative. Ignoring information about the signs of regression parameters can lead to loss of power in small samples. With this problem in mind, this paper concerned with developing restricted estimation and hypothesis testing approach in the context of multivariate multiple regression model. Developing the technique of estimating constraint regression coefficients and testing restricted parameters with the aid of information theoretic distance are the main contribution of this paper. The distribution of the existing two-sided test follows central chi-square distribution whereas the test statistic of our proposed distance-based one-sided test follows weighted mixture of chi-square distribution. Monte Carlo simulation indicates that our newly proposed test performs better than existing tests.

Model specification test in a semiparametric regression model for longitudinal data

We propose a model specification test for whether or not a postulated parametric model (null hypothesis) fits longitudinal data as well as a semiparametric model (alternative hypothesis) does. In the semiparametric model, we suppose that a baseline function of time is modeled nonparametrically, while the longitudinal covariate effect is assumed to be a parametric linear model. The existing kernel regression based likelihood ratio tests suffer from computing the likelihood function in the alternative hypothesis, because a specific parametric alternative is not desired. To circumvent this difficulty, we calibrate the semiparametric model to a regression model containing only the parametric parameters, and investigate the quadratic inference function in the calibrated model. The proposed approach yields an asymptotically unbiased parametric regression estimator without undersmoothing the baseline function. This provides us a simple and powerful test statistic that asymptotically follows a central chi-squared distribution with fixed degrees of freedom under the null hypothesis. Simulation studies show that the proposed test is able to identify the true parametric regression model consistently. We have also applied this test to real data and confirmed that the baseline function can be captured by a conjectured parametric form sufficiently well.

GMM gradient tests for spatial dynamic panel data models

In this study, we formulate adjusted gradient tests when the alternative model used to construct tests deviates from the true data generating process for a spatial dynamic panel data (SDPD) model. Following Bera et al. (2010), we introduce these adjusted gradient tests along with their standard counterparts within a generalized method of moments framework. These tests can be used to detect the presence of (i) the contemporaneous spatial lag terms, (ii) the time lag term, and (iii) the spatial time lag terms in a high order SDPD model. These adjusted tests have two advantages: (i) their null asymptotic distribution is a central chi-squared distribution irrespective of the mis-specified alternative model, and (ii) their test statistics are computationally simple and require only the ordinary least-squares estimates from a non-spatial two-way panel data model. We investigate the finite sample size and power properties of these tests through a Monte Carlo study. Our results indicates that the adjusted gradient tests have good finite sample properties. Finally, using an application from the empirical growth literature we complement our findings.

Reliable and More Powerful Methods for Power Analysis in Structural Equation Modeling

The normal-distribution-based likelihood ratio statistic is widely used for power analysis in structural Equation modeling (SEM). In such an analysis, power and sample size are computed by assuming that follows a central chi-square distribution under and a noncentral chi-square distribution under . However, with either violation of normality or not a large enough sample size, both empirical and analytical results indicate that the chi-square distribution assumptions are not realistic and consequently methods of power analysis based on such assumptions are not valid. This article describes a Monte Carlo (MC) method for power analysis. A measure of effect size for characterizing the power property of different rescaled statistics is also provided. Robust methods are proposed to increase the power of and other statistics. Simulation results show that the MC method reliably controls Type I errors and robust estimation methods effectively increase the power, and their combination is thus recommended for conducting power analysis in SEM.

A note on the misuses of the variance test in meteorological studies

Stochastic modeling of rainfall data is an important area in meteorology. The gamma distribution is a widely used probability model for non-zero rainfall. Typically the choice of the distribution for such meteorological studies is based on two goodness-of-fit tests—the Pearson’s Chi-square test and the Kolmogorov–Smirnov test. Inspired by the index of dispersion introduced by Fisher (Statistical methods for research workers. Hafner Publishing Company Inc., New York, 1925), Mooley (Mon Weather Rev 101:160–176, 1973) proposed the variance test as a goodness-of-fit measure in this context and a number of researchers have implemented it since then. We show that the asymptotic distribution of the test statistic for the variance test is generally not comparable to any central Chi-square distribution and hence the test is erroneous. We also describe a method for checking the validity of the asymptotic distribution for a class of distributions. We implement the erroneous test on some simulated, as well as real datasets and demonstrate how it leads to some wrong conclusions.

Comparison of Rigorous Design Procedure with Approximate Design Procedure for Variable Sampling Plans Indexed by Quality Loss

Traditional acceptance sampling plans have focused on the proportion of nonconforming items as an attribute criterion for quality. In today’s modern quality management under high quality production environments, the reduction of the deviation from a target value in a quality characteristic has become the most important purpose. In consequence, various inspection plans for the purpose of reducing the deviation from the target value in the quality characteristic have been investigated. In this case, a concept of the quality loss evaluated by the deviation from the target value has been accepted as the variable evaluation criterion of quality. Further, some quality measures based on the quality loss have been devised; e.g. the process loss and the process capability index. Then, as one of inspection plans based on the quality loss, the rigorous design procedure for the variable sampling plan having desired operating characteristics (VS-OC plan) indexed by the quality loss has been proposed by Yen and Chang in 2009. By the way, since the estimator of the quality loss obeys the non-central chi-square distribution, the rigorous design procedure for the VS-OC plan indexed by the quality loss is complicated. In particular, the rigorous design procedure for the VS-OC plan requires a large number of the repetitive and complicated numerical calculation about the non-central chi-square distribution. On the other hand, an approximate design procedure for the VS-OC plan has been proposed before the proposal of the above rigorous design procedure. The approximate design procedure for the VS-OC plan has been constructed by combining Patnaik approximation relating the non-central chi-square distribution to the central chi-square distribution and Wilson-Hilferty approximation relating the central chi-square distribution to the standard normal distribution. Then, the approximate design procedure has been devised as a convenient procedure without complicated and repetitive numerical calculations. In this study, through some comparisons between the rigorous and approximate design procedures, the applicability of the approximate design procedure has been confirmed.