Introduction:The score statisticZ(θ)and the maximin efficient robust test statisticZMERTare commonly used in genetic association study, but according to our knowledge there is no formal comparison of them.Methods:In this report, we compare the asymptotic behavior ofZ(θ)andZMERT, by computing their Asymptotic Relative Efficiencies (AREs) relative to each other. Four commonly used ARE measures, the Pitman ARE, Chernoff ARE, Hodges-Lehmann ARE and the Bahadur ARE are considered. Some modifications of these methods are made to simplify the computations. We found that the Chernoff, Hodges-Lehmann and Bahadur AREs are suitable for our setting.Results and Conclusion:Based on our study, the efficiencies of the two test statistic varies for different criterion used, and for different parameter values under the same criterion, so each test has its advantages and dis-advantages according to the criterion used and the parameters involved, which are described in the context. Numerical examples are given to illustrate the use of the two statistics in genetic association study.
Objective:We provide a new stochastic representation for a Kumaraswamy random variable with arbitrary non-negative parameters. The representation is in terms of maxima and minima of independent distributed standard uniform components and extends a similar representation for integer-valued parameters.Result:The result is further extended for generalized classes of distributions obtained from a “base” distribution functionFviz.G(x) =H(F(x)), whereHis the CDF of Kumaraswamy distribution.
In a 0 - 1 sequence of Markov dependent trials we consider a statistic which counts strings of a limited length run of 0s between subsequent 1s. Its probability mass function is used to determine the chance that a stochastic process remains or not in statistical control. Illustrative numerics are presented.
Objective: The main object of our study is to extend some entropy rate theorems to a Hidden Inhomogeneous Markov Chain (HIMC) and establish an entropy rate theorem under some mild conditions. Introduction: A hidden inhomogeneous Markov chain contains two different stochastic processes; one is an inhomogeneous Markov chain whose states are hidden and the other is a stochastic process whose states are observable. Materials and Methods: The proof of theorem requires some ergodic properties of an inhomogeneous Markov chain, and the flexible application of the properties of norm and the bounded conditions of series are also indispensable. Results: This paper presents an entropy rate theorem for an HIMC under some mild conditions and two corollaries for a hidden Markov chain and an inhomogeneous Markov chain. Conclusion: Under some mild conditions, the entropy rates of an inhomogeneous Markov chains, a hidden Markov chain and an HIMC are similar and easy to calculate.
Background:In the statistical analysis of directional data, the von Mises-Fisher distribution plays an important role to model unit vectors. The estimation of the parameters of a mixture of von Mises-Fisher distributions can be done through the Estimation-Maximization algorithm.Objective:In this paper we propose a dynamic clusters type algorithm based on the estimation of the parameters of a mixture of von Mises-Fisher distributions for clustering directions, and we compare this algorithm with the Estimation-Maximization algorithm. We also define the between-groups and within-groups variability measures to compare the solutions obtained with the algorithms through these measures.Results:The comparison of the clusters obtained with both algorithms is provided for a simulation study based on samples generated from a mixture of two Fisher distributions and for an illustrative example with spherical data.
Background:This paper considers three two-dimensional beta binomial models previously introduced in the literature. These were proposed as candidate models for modelling forms of correlated and overdispersed bivariate count data. However, the first model has a complicated form of bivariate probability mass function involving a generalized hypergeometric function and the remaining two do not have closed forms of probability mass functions and are not amenable to analysis using maximum likelihood. This limited their applicability.Objective:In this paper, we will discuss how the Bayesian analyses of these models may go forward using Markov chain Monte Carlo and data augmentation.Results:An illustrative example having to do with student achievement in two related university courses is included. Posterior and posterior predictive inferences and predictive information criteria are discussed.
Given a sequence of random variables ${\bf X}=X_1,X_2,\ldots$ suppose the aim is to maximize one's return by picking a `favorable' $X_i$. Obviously, the expected payoff crucially depends on the information at hand. An optimally informed person knows all the values $X_i=x_i$ and thus receives $E (\sup X_i)$. We will compare this return to the expected payoffs of a number of observers having less information, in particular $\sup_i (EX_i)$, the value of the sequence to a person who only knows the first moments of the random variables. In general, there is a stochastic environment (i.e. a class of random variables $\cal C$), and several levels of information. Given some ${\bf X} \in {\cal C}$, an observer possessing information $j$ obtains $r_j({\bf X})$. We are going to study `information sets' of the form $$ R_{\cal C}^{j,k} = \{ (x,y) | x = r_j({\bf X}), y=r_k({\bf X}), {\bf X} \in {\cal C} \}, $$ characterizing the advantage of $k$ relative to $j$. Since such a set measures the additional payoff by virtue of increased information, its analysis yields a number of interesting results, in particular `prophet-type' inequalities.
In this paper, we obtain certain expressions and recurrence relations for two general classes of distributions based on some conditional expectations of k-th lower record values. We consider the necessary and sufficient conditions such that these conditional expectations hold for some distribution functions. Furthermore, an expression of conditional expectation of other general class of distributions through truncated moments of some random variable is considered. Some distributions as examples of these general classes are shown in Tables1and2accordingly.
Measures of association play a role in selecting 2x2 tables exhibiting strong dependence in high-dimensional binary data. Several measures are in use differing on specific tables and in their dependence on the margins. We study a 2-dimensional group of margin transformations on the 3-dimensional manifold T of all 2x2 probability tables. The margin transformations allow introducing natural coordinates that identify T with the real 3-space such that the x-axis corresponds to log(sqrt(odds-ratio)) and margins vary on planes x=const. We use these coordinates to visualise and compare measures of association with respect to their dependence on the margins given the odds-ratio, their limit behaviour when cells approach zero and their weighting properties. We propose a novel measure of association in which tables with single small entries are up-weighted but those with skewed margins are down-weighted according to the relative entropy among the tables of the same odds-ratio.
This paper constructs a new family of distributions, which is based on the Hurwitz zeta function, which includes novel distributions as well important known distributions such as the normal, gamma, Weibull, Maxwell-Boltzmann and the exponential power distributions. We provide the n-th moment, the Esscher transform and premium and the tail conditional moments for this family.
Polarimetric SAR data provide information about the scattering of the area observed. The availability of data stacks allows the identification of stable targets and subsequent scattering analysis with a high degree of confidence at full resolution. A novel approach to find and evaluate polarimetric persistent target is presented, that is an extension of well-established analysis methods for single scenes. The use of the Cloude-Pottier distributed target decomposition analysis applied on the temporal averages (as opposed to spatial averaging), combined with a Cameron point target analysis applied on each layer separately to select pixels only, provides an efficient scattering classification of polarimetric persistent point targets in the stack. This method can also be used to analyze targets identified through other means, albeit at a lower degree of confidence. The approach retains the full resolution of the data set, though temporal changes between acquisitions add additional complexity. Result interpretation is therefore performed under consideration of a set of boundary conditions. Results from the analysis of two polarimetric data stacks acquired by RADARSAT-2 are shown.
We define equivalence of asymptotic Gaussian expectation tests when error probabilities of first kind are approaching zero at the same restricted speed for both tests and if the same holds true for the error probabilities of second type which are measured at a moderate locally chosen alternative. To ensure such equivalence, the influence of skewness and kurtosis parameters is studied.
Corrections are given to common misconceptions regarding the use of centered and uncentered regressors, and of meanings to be ascribed to A -, D -and E -criteria in evaluating two such designs.
In this paper, we consider the problem of modeling and predicting highway accidents in the presence of randomly changing winter driving conditions.Unlike most accident prediction models in the literature, which are typically formulated in a static (e.g.regression models) or discrete time (e.g.time-series models) setting, we propose a continuoustime stochastic model to describe the relation between highway accidents and winter weather dynamics.We believe this to be a more natural way to describe discrete-event highway accidents that occur in continuous-time.In particular, the accident counting process is viewed as a non-homogeneous Poisson process (NHPP) with an intensity function that depends on a (Markovian) weather process.Such a model is known in the stochastic process literature as a Markovmodulated Poisson process (MMPP) and has been successfully applied to queuing and telecommunications problems.One main advantage of such an approach, is its ability to provide explicit closed-form prediction formulae for both weather and accidents over any future time horizon (i.e.short or long-term predictions).To illustrate the effectiveness of the proposed stochastic model, we study a large winter data set provided by Ministry of Transportation of Ontario (MTO) that includes motor vehicle accidents on Highway 401, the busiest highway in North America.
Understanding of tissue growth is in its nature multidisciplinary, since it varies from cancer diagnostics, image processing, fractal analysis to regular and non-regular heat flows. By medical sciences it was requested to better under- stood the tissue grow relation to mathematical modelling (stochastic geometry, fractal growth, diffusions). It is clear, that deterministic fractal is not an appropriate model for cancer growth. Stochastic fractal is more appropriate, however, a validation measure should be developed for better comparability with advanced stochastic geometry model, e.g. Quer- mass-interaction process. Moreover, relation temperature-geometry of the tissue is studied. We have partial results, where it is observed, that benign alterations and malignant tumors originating from glandular tissues (e.g.mammary, prostatic, pancreatic) are naturally modelled by non-standard diffusions. For standard diffusions, fair approximation is provided by analytical models based on convective heat transfer in infinite tissues volume (e.g. model given by Perl 1962, later ex- tended by (1)).
A new lifetime class with decreasing failure rate which is obtained by compounding truncated Poisson distribution and a lifetime distribution, where the compounding procedure follows same way that was previously carried out by Adamidis and Loukas(1998). A general form of probability, distribution, survival and hazard rate functions as well as its properties will be presented for such a class. This new class of distributions generalizes several distributions which have been introduced and studied in the literature.
The usual domains for Cauchy distributions have been straight lines and unit circles.These domains are closed under arbitrary changes in location and scale, whether done sequentially or simultaneously.Such closure properties have been extended to spherical Cauchy distributions.Higher dimensional Cauchy based domains are created herein for unit hyperspheres and sets of straight lines of arbitrary dimension, and their Cauchy-like properties are determined and described.Cauchy distributions on these extended domains are shown to be closed under arbitrary transformations of location and scale, done singly or sequentially, but not generally closed when location and scale changes are done simultaneously.Stereographic projections are used to map the curved, finite surface of any hypersphere to a linear, infinite space of the same dimension as that of the hyperspherical surface.These mappings are one-one and onto, with no loss of information.These results show promise for uniting linear and directional mixtures of observations into a common domain linear or directional.
We provide a simple proof of Wald's second identity for a class of problems that can be formulated as a fair coin-tossing game. The identity provides a useful technique for computing the expected stopping time and allows us to explore the second-order moment of the so-called heads-minus-tails distribution. We also introduce some interesting applications related to this simple identity.
A new method is proposed for testing the mixed Poisson model which employs the well known identity between the probability generating function of the mixed Poisson distribution and the Laplace transform of the mixing distribution.The method utilizes a weighted L2-type statistic which may be easily calculated and the resulting test is consistent against general alternatives.A limit statistic is shown to be related to earlier methods which employ moment-based contrasts.Some simulation results are also shown.
For the analysis of square contingency tables with ordered categories, some models that the log odds for two symmetric cell probabilities is a linear function of the row and column values have been considered.This paper proposes a generalization of these models.This paper also proposes the model that the weighted sum of the probability that an observation will fall in one of the cells in upper right triangle of the table is equal to the weighted sum of the probability that it falls in one of the cells in lower left triangle of the table.In addition, this paper gives the theorem that the symmetry model is equivalent to both the proposed models holding simultaneously.Moreover, this paper shows that the likelihood ratio statistic for testing goodness-of-fit of the symmetry model is asymptotically equivalent to the sum of those for testing the proposed models.Examples are given.
