Stat Plan Inference Research Articles

The Asymptotic Number of Score Sequences

A tournament on a graph is an orientation of its edges. The score sequence lists the in-degrees in non-decreasing order. Works by Winston and Kleitman (J Comb Theory Ser A 35(2):208–230, 1983) and Kim and Pittel (J Comb Theory Ser A 92(2):197–206, 2000) showed that the number S_n of score sequences on the complete graph K_n satisfies S_n=Theta (4^n/n^{5/2}). By combining a recent recurrence relation for S_n in terms of the Erdős–Ginzburg–Ziv numbers N_n with the limit theory for discrete infinitely divisible distributions, we observe that n^{5/2}S_n/4^nrightarrow e^lambda /2sqrt{pi }, where lambda =sum _{k=1}^infty N_k/k4^k. This limit agrees numerically with the asymptotics of S_n conjectured by Takács (J Stat Plan Inference 14(1):123–142, 1986). We also identify the asymptotic number of strong score sequences, and show that the number of irreducible subscores in a random score sequence converges in distribution to a shifted negative binomial with parameters r=2 and p=e^{-lambda }.

Unforced errors in the matching problem

The solution of the Montmort's matching problem can be seen as the probability mass function of the number of matches in an experiment for assessing the agreement between nominal variables and gold standard classifications. [Vidal I, de Castro M. A Bayesian analysis of the matching problem. J Stat Plan Inference. 2021;212:194–200] presented a generalization of the Montmort's matching problem by considering the chronological order in what assignments are made and counting the number of unforced errors additionally to the number of matches. These authors carried out a Bayesian analysis of the problem, but in this paper we found a solution of this new approach from a frequentist point of view. We found the bivariate probability mass function of the number of matches and the number of unforced errors. The marginal distribution of the number of unforced errors is computed and expressed in terms of the Stirling numbers of the second kind. Also, some elementary properties were proven, including that the distribution of the unforced errors is equal to that of forced errors. As a practical consequence, we propose a new experiment and a new way of collecting and analysing the data in order to assess the agreement between nominal variables and gold standard classifications. Two real data sets were analysed using the proposed methodology.

Moments and random number generation for the truncated elliptical family of distributions

This paper proposes an algorithm to generate random numbers from any member of the truncated multivariate elliptical family of distributions with a strictly decreasing density generating function. Based on the ideas of Neal (Ann stat 31(3):705–767, 2003) and Ho et al. (J Stat Plan Inference 142(1):25–40, 2012), we construct an efficient sampling method by means of a slice sampling algorithm with Gibbs sampler steps. We also provide a faster approach to approximate the first and the second moment for the truncated multivariate elliptical distributions where Monte Carlo integration is used for the truncated partition and explicit expressions for the non-truncated part (Galarza et al., in J Multivar Anal 189(104):944, 2022). Examples and an application to environmental spatial data illustrate its usefulness. Methods are available for free in the new R library relliptical.

Use of primary decomposition of polynomial ideals arising from indicator functions to enumerate orthogonal fractions

A polynomial indicator function of designs is first introduced by Fontana et al. (J Stat Plan Inference 87:149–172, 2000) for two-level cases. They give the structure of the indicator functions, especially the relation to the orthogonality of designs. These results are generalized by Aoki (J Stat Plan Inference 203:91–105, 2019) for general multi-level cases. As an application of these results, we can enumerate all orthogonal fractional factorial designs with given size and orthogonality using computational algebraic software. For example, Aoki (2019) gives classifications of orthogonal fractions of \(2^4\times 3\) designs with strength 3, which is derived by simple eliminations of variables. However, the computational feasibility of this naive approach depends on the size of the problems. In fact, it is reported that the computation of orthogonal fractions of \(2^4\times 3\) designs with strength 2 fails to carry out in Aoki (2019). In this paper, using the theory of primary decomposition, we enumerate and classify orthogonal fractions of \(2^4\times 3\) designs with strength 2. We show there are 35,200 orthogonal half fractions of \(2^4\times 3\) designs with strength 2, classified into 63 equivalent classes.

Weighted least squares estimators for the Parzen tail index

Estimation of the tail index of heavy-tailed distributions and its applications are essential in many research areas. We propose a class of weighted least squares (WLS) estimators for the Parzen tail index. Our approach is based on the method developed by Holan and McElroy (J Stat Plan Inference 140(12):3693–3708, 2010). We investigate consistency and asymptotic normality of the WLS estimators. Through a simulation study, we make a comparison with the Hill, Pickands, DEdH (Dekkers, Einmahl and de Haan) and ordinary least squares (OLS) estimators using the mean square error as criterion. The results show that in a restricted model some members of the WLS estimators are competitive with the Pickands, DEdH and OLS estimators.

A prediction approach in adaptive sampling

Adaptive cluster sampling (ACS) due to Thompson (J Am Stat Assoc 85(412):1050–1059, 1990) is a tool to survey rare and hidden elements in a population as an improvement over traditional survey procedures. In ACS, if an observed sampled unit satisfies the given criterion of rarity, its neighboring units are added to the sample and this is continued until one is detected with no rarity. Chaudhuri (Calcutta Stat Assoc Bull 50(3–4):238–253, 2000) extended the above to unequal probability sampling as Adaptive sampling. In practice, often network sizes turn out too big demanding high cost and time. So, Chaudhuri et al. (J Stat Plan Inference 121: 175–189, 2004) gave a subsampling technique to restrict the sample size in Adaptive Sampling. Towards this end, Chaudhuri et al. (J Stat Plan Inference 134: 254–267, 2005) developed a sample size restriction technique. But in Adaptive sampling, capturing neighboring rare units turn out difficult because of various hazards. We propose to try Royall’s (Biometrika 57: 377–389, 1970) prediction approach here to model features of uncaptured network units. We employ Brewer’s (J Am Stat Assoc 74 (368): 911–915, 1979) model-assisted approach to derive a predictor with asymptotic design unbiasedness based on unequal probability samples and examine its efficacy by simulations.

Statistical inference based on a new weighted likelihood approach

We discuss a new weighted likelihood method for robust parametric estimation. The method is motivated by the need for generating a simple estimation strategy which provides a robust solution that is simultaneously fully efficient when the model is correctly specified. This is achieved by appropriately weighting the score function at each observation in the maximum likelihood score equation. The weight function determines the compatibility of each observation with the model in relation to the remaining observations and applies a downweighting only if it is necessary, rather than automatically downweighting a proportion of the observations all the time. This allows the estimators to retain full asymptotic efficiency at the model. We establish all the theoretical properties of the proposed estimators and substantiate the theory developed through simulation and real data examples. Our approach provides an alternative to the weighted likelihood method of Markatou et al. (J Stat Plan Inference 57(2):215–232, 1997; J Am Stat Assoc 93(442):740–750, 1998).

Analytic solutions for locally optimal designs for gamma models having linear predictors without intercept

The gamma model is a generalized linear model for gamma-distributed outcomes. The model is widely applied in psychology, ecology or medicine. Recently, Gaffke et al. (J Stat Plan Inference 203:199–214, 2019) established a complete class and an essentially complete class of designs for gamma models to obtain locally optimal designs in particular when the linear predictor includes an intercept term. In this paper we extend this approach to gamma models having linear predictors without intercept. For a specific scenario sets of locally D- and A-optimal designs are established. It turns out that the optimality problem can be transformed to one under gamma models with intercept leading to a reduction in the dimension of the experimental region. On that basis optimality results can be transferred from one model to the other and vice versa. Additionally by means of the general equivalence theorem optimality can be characterized for multiple regression by a system of polynomial inequalities which can be solved analytically or by computer algebra. Thus necessary and sufficient conditions can be obtained on the parameter values for the local D-optimality of specific designs. The robustness of the derived designs with respect to misspecification of the initial parameter values is examined by means of their local D-efficiencies.

Approximate and exact optimal designs for $$2^k$$ factorial experiments for generalized linear models via second order cone programming

Model-based optimal designs of experiments (M-bODE) for nonlinear models are typically hard to compute. The literature on the computation of M-bODE for nonlinear models when the covariates are categorical variables, i.e. factorial experiments, is scarce. We propose second order cone programming (SOCP) and Mixed Integer Second Order Programming (MISOCP) formulations to find, respectively, approximate and exact A- and D-optimal designs for $$2^k$$ factorial experiments for Generalized Linear Models (GLMs). First, locally optimal (approximate and exact) designs for GLMs are addressed using the formulation of Sagnol (J Stat Plan Inference 141(5):1684–1708, 2011). Next, we consider the scenario where the parameters are uncertain, and new formulations are proposed to find Bayesian optimal designs using the A- and log det D-optimality criteria. A quasi Monte-Carlo sampling procedure based on the Hammersley sequence is used for computing the expectation in the parametric region of interest. We demonstrate the application of the algorithm with the logistic, probit and complementary log–log models and consider full and fractional factorial designs.

Transformed symmetric generalized autoregressive moving average models

ABSTRACTCordeiro and Andrade [Transformed generalized linear models. J Stat Plan Inference. 2009;139:2970–2987] incorporated the idea of transforming the response variable to the generalized autoregressive moving average (GARMA) model, introduced by Benjamin et al. [Generalized autoregressive moving average models. J Am Stat Assoc. 2003;98:214–223], thus developing the transformed generalized autoregressive moving average (TGARMA) model. The goal of this article is to develop the TGARMA model for symmetric continuous conditional distributions with a possible nonlinear structure for the mean that enables the fitting of a wide range of models to several time series data types. We derive an iterative process for estimating the parameters of the new model by maximum likelihood and obtain a simple formula to estimate the parameter that defines the transformation of the response variable. Furthermore, we determine the moments of the original dependent variable which generalize previous published results. We illustrate the theory by means of real data sets and evaluate the results developed through simulation studies.

Multidimensional strong large deviation results

We establish strong large deviation results for an arbitrary sequence of random vectors under some assumptions on the normalized cumulant generating function. In other words, we give asymptotic approximations for a multivariate tail probability of the same kind as the one obtained by Bahadur and Rao (Ann Math Stat 31:1015–1027, 1960) for the sample mean (in the one-dimensional case). The proof of our results follows the same lines as in Chaganty and Sethuraman (J Stat Plan Inference, 55:265–280, 1996). We also present three statistical applications to illustrate our results, the first one dealing with a vector of independent sample variances, the second one with a Gaussian multiple linear regression model and the third one with the multivariate Nadaraya–Watson estimator. Some numerical results are also presented for the first two applications.

Improved heteroskedasticity likelihood ratio tests in symmetric nonlinear regression models

In this paper we address the issue of testing inference of the dispersion parameter in heteroscedastic symmetric nonlinear regression models considering small samples. We derive Bartlett corrections to improve the likelihood ratio as well modified profile likelihood ratio tests. Our results extend some of those obtained in Cordeiro (J Stat Comput Simul 74:609–620, 2004) and Ferrari et al. (J Stat Plan Inference 124:423–437, 2004), who consider a symmetric nonlinear regression model and normal linear regression model, respectively. We also present the bootstrap and bootstrap Bartlett corrected likelihood ratio tests. Monte Carlo simulations are carried out to compare the finite sample performances of the three corrected tests and their uncorrected versions. The numerical evidence shows that the corrected modified profile likelihood ratio test, the bootstrap and bootstrap Bartlett corrected likelihood ratio test perform better than the other ones. We also present an empirical application.

A Wald-type test statistic based on robust modified median estimator in logistic regression models

ABSTRACTIn this paper a new robust estimator, modified median estimator, is introduced and studied for the logistic regression model. This estimator is based on the median estimator considered in Hobza et al. [Robust median estimator in logistic regression. J Stat Plan Inference. 2008;138:3822–3840]. Its asymptotic distribution is obtained. Using the modified median estimator, we also consider a Wald-type test statistic for testing linear hypotheses in the logistic regression model and we obtain its asymptotic distribution under the assumption of random regressors. An extensive simulation study is presented in order to analyse the efficiency as well as the robustness of the modified median estimator and Wald-type test based on it.

Minimum distance estimation of the binormal ROC curve

The receiver operating characteristic (ROC) curve describes the performance of a diagnostic test, which classifies individuals into one of two categories. Many parametric, semiparametric and nonparametric estimation methods have been proposed for estimating the ROC curve and its functionals. In this paper the minimum distance estimation of the binormal ROC curve is considered. A modification of the estimator considered in the paper of Davidov and Nov (J Stat Plan Inference 142(4):872–877, 2012) and some new estimators are proposed. We compare the accuracy of the new estimators with known minimum distance estimators of the binormal ROC curve and we conclude that our estimators generally perform better than their competitors.

Test on the linear combinations of mean vectors in high-dimensional data

In this study, we propose a procedure for simultaneous testing $$l (l\ge 1)$$ linear relations on $$k(k\ge 2)$$ high-dimensional mean vectors with heterogeneous covariance matrices, which extends the result derived by Nishiyama et al. (J Stat Plan Inference 143(11):1898–1911, 2013) and does not need the normality assumption. The newly proposed test statistic is motivated by Bai and Saranadasa (Statistica Sinica 6(2):311–329, 1996) and Chen and Qin (Ann Stat 38(2):808–835, 2010). As a special case, our result could be applied to multivariate analysis of variance, that is, testing the equality of k high-dimensional mean vectors.

Periodically correlated modeling by means of the periodograms asymptotic distributions

In this paper, we introduce a test statistics to test whether a discrete time periodically correlated model with a given spectral density explains an observed time series. Our testing procedure is based on an application of the asymptotic distribution of the periodogram established in Soltani and Azimmohseni (Stat Plan Inference 137:1236–1242, 2007). We make comparisons between our procedure and the methods that are proposed by Broszkiewicz-Suwaj et al. (Physica A 336:196–205, 2004). It is observed that our testing procedure is more powerful. We illustrate the performance of the proposed methods in real and simulated data sets.

Confident difference criterion: a new Bayesian differentially expressed gene selection algorithm with applications.

BackgroundRecently, the Bayesian method becomes more popular for analyzing high dimensional gene expression data as it allows us to borrow information across different genes and provides powerful estimators for evaluating gene expression levels. It is crucial to develop a simple but efficient gene selection algorithm for detecting differentially expressed (DE) genes based on the Bayesian estimators.ResultsIn this paper, by extending the two-criterion idea of Chen et al. (Chen M-H, Ibrahim JG, Chi Y-Y. A new class of mixture models for differential gene expression in DNA microarray data. J Stat Plan Inference. 2008;138:387–404), we propose two new gene selection algorithms for general Bayesian models and name these new methods as the confident difference criterion methods. One is based on the standardized differences between two mean expression values among genes; the other adds the differences between two variances to it. The proposed confident difference criterion methods first evaluate the posterior probability of a gene having different gene expressions between competitive samples and then declare a gene to be DE if the posterior probability is large. The theoretical connection between the proposed first method based on the means and the Bayes factor approach proposed by Yu et al. (Yu F, Chen M-H, Kuo L. Detecting differentially expressed genes using alibrated Bayes factors. Statistica Sinica. 2008;18:783–802) is established under the normal-normal-model with equal variances between two samples. The empirical performance of the proposed methods is examined and compared to those of several existing methods via several simulations. The results from these simulation studies show that the proposed confident difference criterion methods outperform the existing methods when comparing gene expressions across different conditions for both microarray studies and sequence-based high-throughput studies. A real dataset is used to further demonstrate the proposed methodology. In the real data application, the confident difference criterion methods successfully identified more clinically important DE genes than the other methods.ConclusionThe confident difference criterion method proposed in this paper provides a new efficient approach for both microarray studies and sequence-based high-throughput studies to identify differentially expressed genes.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-015-0664-3) contains supplementary material, which is available to authorized users.

Automorphisms of strongly regular graphs with applications to partial difference sets

In this article we generalize a theorem of Benson (J Algebra 15:443---454, 1970) for generalized quadrangles to strongly regular graphs, deriving numerical restrictions on the number of fixed vertices and the number of vertices mapped to adjacent vertices under an automorphism. We then use this result to develop a few new techniques to study regular partial difference sets (PDS) in Abelian groups. Ma (Des Codes Cryptogr 4:221---261, 1994) provided a list of parameter sets of regular PDS with $$k\le 100$$k≤100 in Abelian groups for which existence was known or had not been excluded. In particular there were 32 parameter sets for which existence was not known. Ma (J Stat Plan Inference 62:47---56, 1997) excluded 13 of these parameter sets. As an application of our results we here exclude the existence of a regular partial difference set for all but two of the undecided parameter sets from Ma's list.

D-optimal designs based on the second-order least squares estimator

When the error distribution in a regression model is asymmetric, the second-order least squares estimator (SLSE) is more efficient than the ordinary least squares estimator. This result motivated the research in Gao and Zhou (J Stat Plan Inference 149:140–151, 2014), where A-optimal and D-optimal design criteria based on the SLSE were proposed and various design properties were studied. In this paper, we continue to investigate the optimal designs based on the SLSE and derive new results for the D-optimal designs. Using convex optimization techniques and moment theories, we can construct D-optimal designs for univariate polynomial and trigonometric regression models on any closed interval. Several theoretical results are obtained. The methodology is quite general. It can be applied to reduced polynomial models, reduced trigonometric models, and other regression models. It can also be extended to A-optimal designs based on the SLSE.

Generalized projection discrepancy and its applications in experimental designs

The objective of this paper is to study the issue of the generalized projection discrepancy along the line of Qin et al. (J Stat Plan Inference 142:1170–1177, 2012) based on generalized discrete discrepancy measure proposed by Chatterjee and Qin (J Stat Plan Inference 141:951–960, 2011). We shall study the projection properties for general asymmetric factorials and provide some analytic connections between minimum generalized projection uniformity and other optimality criteria. A new lower bound on the generalized projection discrepancy for asymmetric factorials is presented here.