Problems In Mathematical Statistics Research Articles

Algebraic Statistics in Practice: Applications to Networks

Algebraic statistics uses tools from algebra (especially from multilinear algebra, commutative algebra, and computational algebra), geometry, and combinatorics to provide insight into knotty problems in mathematical statistics. In this review, we illustrate this on three problems related to networks: network models for relational data, causal structure discovery, and phylogenetics. For each problem, we give an overview of recent results in algebraic statistics, with emphasis on the statistical achievements made possible by these tools and their practical relevance for applications to other scientific disciplines.

Analysis of the results of state land supervision by solving problems of mathematical statistics

The aim of the study was to identify the relationship between indicators of state land supervision, land use and the receipt of taxes, fees and other payments to budgets of different levels. In this study, when analyzing the results of supervision and control measures in the field of land relations, non-trivial conclusions were obtained. Each detected violation of land legislation is accompanied in the Tyumen region by the registration of the corresponding protocol. A close relationship was found between the number of issued instructions and the number of violations detected (r = 0.87); the number of issued orders is closely correlated with the number of issued protocols (r = 0.87). To a lesser extent, the revealed violations affect the amount of fines imposed (r = 0.63) and the area on which the violations were eliminated (r = 0.52). This suggests that the absolute increase in the number of identified violations does not have the proper impact on their elimination. The average strength of the relationship between the indicators of the amount of fines imposed for violations of land legislation and the area of the reserve land (r = 0.66) is established.

Ellipsoidal Approximation of Distributions and Its Applications

Analytical methods of the mathematical statistics of random vectors and matrices based on the parametrization of the distributions are widely used. These methods permit to design practically simple software when it is possible to have the definite information about analytical properties of the distributions under research. The main difficulty in practical applications of the methods based on the parametrization of the distributions is the rapid increase of the number of equations for the moments, the semiinvariants or the coefficients of the truncated orthogonal expansions of the dimension or the state vector (extended in the general case) and the maximal order of the moments involved. The number of equations for the parameters becomes exceedingly large in such cases. For structural parametrization and/or approximation of the probability densities of the random vectors we shall apply the ellipsoidal densities, i.e. the densities whose planes of the levels of equal probability are similar concentric ellipsoids (the ellipses for two-dimensional vectors, the ellipsoids for three-dimensional vectors, the hyperellipsoids for the vectors of the dimension more than three). In particular, a normal distribution in any finite-dimensional space has an ellipsoidal structure. The distinctive characteristics of such distributions consists in the fact that their probability densities are the functions of positively determined quadratic form where is an expectation of the random vector is some positively determined matrix. Ellipsoidal approximation method (EAM) cardinally reduces the number of parameters till ( ) where being the number of probabilistic moments. While using ellipsoidal linearization method (ELM) we get Basic EAM and ELM foundations and applications to problems of mathematical statistics and ellipsoidal distributions with invariant measure in populational Volterra differential stochastic nonlinear systems are considered.

Robust Nonparametric Methods of Statistical Analysis of Wind Velocity Components in Acoustic Sounding of the Lower Layer of the Atmosphere

Statistical analysis of the results of minisodar measurements of vertical profiles of wind velocity components in a 5–200 m layer of the atmosphere shows that this problem belongs to the class of robust nonparametric problems of mathematical statistics. In this work, a new consecutive nonparametric method of adaptive pendular truncation is suggested for outlier detection and selection in sodar data. The method is implemented in a censoring algorithm. The efficiency of the suggested algorithm is tested in numerical experiments. The algorithm has been used to calculate statistical characteristics of wind velocity components, including vertical profiles of the first four moments, the correlation coefficient, and the autocorrelation and structure functions of wind velocity components. The results obtained are compared with classical sample estimates.

Проблемы процесса обучения математической статистике: способ формирования мотивации у студентов

Проблемы процесса обучения математической статистике: способ формирования мотивации у студентов

Z-process method for change point problems with applications to discretely observed diffusion processes

The aim of this paper is to develop a general, unified approach, based on some partial estimation functions which we call “Z-process”, to some change point problems in mathematical statistics. The method proposed can be applied not only to ergodic models but also to some models where the Fisher information matrix is random. Applications to some concrete models, including a parametric model for volatilities of diffusion processes are presented. Simulations for randomly time-transformed Brownian bridge process appearing as the limit of the proposed test statistics are performed with computer intensive use.

A distribution-free [formula omitted]-out-of-[formula omitted] bootstrap approach to testing symmetry about an unknown median

Testing for symmetry about an unknown median is a ubiquitous problem in mathematical statistics, particularly, for nonparametric rank-based methods, and in a broad range of applied studies, from economics and business to biology, ecology, and medicine. However, the challenge still remains on how to derive a symmetry test with a good power performance and at the same time delivering a reliable Type I Error estimate. To overcome this problem, a new data-driven m-out-of-n bootstrap method is introduced for testing symmetry about an unknown median. The asymptotic properties of the developed m-out-of-n bootstrap tests are investigated along with their empirical finite-sample performance. The new tests are illustrated by applications to legal studies and wildlife monitoring.

Admissibility of the Usual Confidence Set for the Mean of a Univariate or Bivariate Normal Population: The Unknown Variance Case

Summary In the Gaussian linear regression model (with unknown mean and variance), we show that the standard confidence set for one or two regression coefficients is admissible in the sense of Joshi. This solves a long-standing open problem in mathematical statistics, and this has important implications on the performance of modern inference procedures post model selection or post shrinkage, particularly in situations where the number of parameters is larger than the sample size. As a technical contribution of independent interest, we introduce a new class of conjugate priors for the Gaussian location–scale model.

Estimation of entropy-type integral functionals

ABSTRACTEntropy-type integral functionals of densities are widely used in mathematical statistics, information theory, and computer science. Examples include measures of closeness between distributions (e.g., density power divergence) and uncertainty characteristics for a random variable (e.g., Rényi entropy). In this paper, we study U-statistic estimators for a class of such functionals. The estimators are based on ε-close vector observations in the corresponding independent and identically distributed samples. We prove asymptotic properties of the estimators (consistency and asymptotic normality) under mild integrability and smoothness conditions for the densities. The results can be applied in diverse problems in mathematical statistics and computer science (e.g., distribution identification problems, approximate matching for random databases, two-sample problems).

A Novel Approach to Finding Sampling Distributions for Truncated Laws Via Unbiasedness Equivalence Principle

Truncated distributions arise naturally in many practical situations. In this paper, the problem of finding sampling distributions for truncated laws is considered. This problem concerns the very important area of information processing in Industrial Engineering. It remains today perhaps the most difficult and important of all the problems of mathematical statistics that require considerable efforts and great skill for investigation. In a given problem, most would prefer to find a sampling distribution for truncated law by the simplest method available. For many situations encountered in textbooks and in the literature, the approach discussed here is simple and straightforward. It is based on use of the unbiasedness equivalence principle (UEP) that represents a new idea which often allows one to provide a neat method for finding sampling distributions for truncated laws. It avoids explicit integration over the sample space and the attendant Jacobian but at the expense of verifying completeness of the recognized family of densities. Fortunately, general results on completeness obviate the need for this verification in many problems involving exponential families. The proposed approach allows one to obtain results for truncated laws via the results obtained for non-truncated laws. It is much simpler than the known approaches. In many situations this approach allows one to find the results for truncated laws with known truncation points and to estimate system reliability in a simple way. The approach can also be used to find the sampling distribution for truncated law when some or all of its truncation parameters are left unspecified. The illustrative examples are given.

Examples and Problems in Mathematical Statistics by Shelemyahu Zacks

Examples and Problems in Mathematical Statistics by Shelemyahu Zacks

Antithetic Time Series Analysis and the Companyx Data

SummaryAntithetic time series analysis is the solution to a most perplexing problem in mathematical statistics. When a mathematical model is fitted to serially correlated data, the parameters of the model are unavoidably biased. All forecasts that are obtained from the model are unavoidably biased and therefore diverge. The forecast reliability worsens with the forecast horizon. It is shown that the forecast bias can be dynamically reduced. This is made possible by the entirely counterintuitive discovery of antithetic time series theory that permits unbiased forecast error convergence to a constant, independent of forecast origin. The forecast error variance in each time period is the same.

Do-Validation for Kernel Density Estimation

Bandwidth selection in kernel density estimation is one of the fundamental model selection problems of mathematical statistics. The study of this problem took major steps forward with the articles of Hall and Marron (1987) and Hall and Johnstone (1992). Since then, the focus seems to have been on various versions of implementing the so-called plug-in method aimed at estimating the minimum mean integrated squared error (MISE). The most successful of these efforts still seems to be the plug-in method of Sheather and Jones (1991) or Park and Marron (1990) that we also use as a benchmark in this article. In this article we derive a new theorem deriving the asymptotic theory for linear combinations of bandwidths obtained from different selectors as, for example, direct and indirect cross-validation and plug-in, where we take advantage of recent advances in the study of indirect cross-validation; see Hart and Yi (1998), Hart and Lee (2005), and Savchuk, Hart, and Sheather (2008, 2010). We conclude that the slow convergence of data-driven bandwidths implies that once asymptotic theory is close to that of the plug-in, then it is the practical implementation that counts. This insight led us to a bandwidth selector search with the symmetrized version of one-sided cross-validation as a clear winner.

Statistical inference for the [formula omitted]-entropy and the quadratic Rényi entropy

Entropy and its various generalizations are widely used in mathematical statistics, communication theory, physical and computer sciences for characterizing the amount of information in a probability distribution. We consider estimators of the quadratic Rényi entropy and some related characteristics of discrete and continuous probability distributions based on the number of coincident (or ϵ -close) vector observations in the corresponding independent and identically distributed sample. We show some asymptotic properties of these estimators (e.g., consistency and asymptotic normality). These estimators can be used in various problems in mathematical statistics and computer science (e.g., distribution identification problems, average case analysis for random databases, approximate pattern matching in bioinformatics, cryptography).

Nonparametric Statistical Inference for Ergodic Processes

In this work, a method for statistical analysis of time series is proposed, which is used to obtain solutions to some classical problems of mathematical statistics under the only assumption that the process generating the data is stationary ergodic. Namely, three problems are considered: goodness-of-fit (or identity) testing, process classification, and the change point problem. For each of the problems a test is constructed that is asymptotically accurate for the case when the data is generated by stationary ergodic processes. The tests are based on empirical estimates of distributional distance.

TWO‐SAMPLE PROBLEMS IN STATISTICAL DATA MODELLING

A common problem in mathematical statistics is to check whether two samples differ from each other. From modelling point of view it is possible to make a statistical test for the equality of two means or alternatively two distribution functions. The second approach allows to represent the two‐sample test graphically. This can be done by adding simultaneous confidence bands to the probability‐probability (P — P) or quantile‐quantile (Q — Q) plots. In this paper we compare empirically the accuracy of the classical two‐sample t‐test, empirical likelihood method and several bootstrap methods. For a real data example both Q — Q and P — P plots with simultaneous confidence bands have been plotted using the smoothed empirical likelihood and smoothed bootstrap methods.

A Unified Method for Checking Compatibility and Uniqueness for Finite Discrete Conditional Distributions

Checking compatibility for two given conditional distributions and identifying the corresponding unique compatible marginal distributions are important problems in mathematical statistics, especially in Bayesian inferences. In this article, we develop a unified method to check the compatibility and uniqueness for two finite discrete conditional distributions. By formulating the compatibility problem into a system of linear equations subject to constraints, it can be reduced to a quadratic optimization problem with box constraints. We also extend the proposed method from two-dimensional cases to higher-dimensional cases. Finally, we show that our method can be easily applied to checking compatibility and uniqueness for a regression function and a conditional distribution. Several numerical examples are used to illustrate the proposed method. Some comparisons with existing methods are also presented.

Comparing normal means: new methods for an old problem

Comparing the means of two normal populations is an old problem in mathematical statistics, but there is still no consensus about its most appropriate solution. In this paper we treat the problem of comparing two normal means as a Bayesian decision problem with only two alternatives: either to accept the hypothesis that the two means are equal, or to conclude that the observed data are, under the assumed model, incompatible with that hypothesis. The combined use of an information-theory based loss function, the intrinsic discrepancy (Bernardo and Rueda 2002}, and an objective prior function, the reference prior \citep{Bernardo 1979; Berger and Bernardo 1992), produces a new solution to this old problem which has the invariance properties one should presumably require.

On the unlinking conjecture of independent polynomial functions

The celebrated U-conjecture states that under the N n ( 0 , I n ) distribution of the random vector X = ( X 1 , … , X n ) in R n , two polynomials P ( X ) and Q ( X ) are unlinkable if they are independent [see Kagan et al., Characterization Problems in Mathematical Statistics, Wiley, New York, 1973]. Some results have been established in this direction, although the original conjecture is yet to be proved in generality. Here, we demonstrate that the conjecture is true in an important special case of the above, where P and Q are convex nonnegative polynomials with P ( 0 ) = 0 .

ON DELAY ESTIMATION FOR STOCHASTIC DIFFERENTIAL EQUATIONS

We present a review of some results concerning delay estimation by continuous time observations of solutions of stochastic differential equations in two asymptotics. The first one corresponds to small noise limit and the second to large samples limit. In both cases we describe the properties of the maximum likelihood estimator and Bayesian estimators with especial attention to asymptotic efficiency of the estimators. We show that the first asymptotic corresponds to regular problems of mathematical statistics and the second is close to non regular problems. In small noise asymptotics we give the next after the Gaussian term of asymptotic expansion of the maximum likelihood estimator.