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Variability of shapes of bidimensional closed curves is a key matter in many fields of research. A statistical order of bidimensional
shape variability is introduced in this paper. For such a purpose a special class of random elements is considered. The order
is defined on such a class and the main properties of the order are analyzed. Such an order involves the curvature of a special
parameterization of bidimensional closed curves. The new order can be used as a basis for implementing statistical procedures,
such as hypothesis testing on variability of shapes. An example is developed by means of the image analysis of cell nuclei,
namely the shapes of cell nuclei in mastitis-affected cow milk and non-affected cow milk are compared.

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This manuscript introduces a criterion to compare interval valued random mappings by means of a stochastic order for such random elements. The comparison criterion is based on the distance to the origin on both sides of the values that those random elements assume. The proposed stochastic order is studied, providing characterizations and properties. One of those characterizations is based on a new stochastic order for bivariate random vectors when it is applied to the endpoints of the random intervals. Some examples with applications of the criterion to weather and economic problems are developed.

In this paper, we review some relations between partially ordered sets and stochastic orders. We focus our attention on analyzing if the property of being order-isomorphic is transferred from partially ordered sets to the stochastic orders generated by such partially ordered sets.

Given Χ a set endowed with a partial order ≤, we can consider the class of ≤-preserving real functions on Χ characterized by x ≤ y implies f(x) ≤ f(y). Such a class of functions generates a stochastic order ≤g on the set of probabilities associated with Χ by means of P ≤g Q when ∫ Χ f dP ≤ ∫ Χ f dQ for all f ≤-preserving functions. In this paper we analyze if the property of being order-isomorphic is transferred from partially ordered sets to the corresponding generated stochastic orders and conversely. We obtain that if two posets are order-isomorphic, the posets which are generated by means of the class of measurable preserving functions are also order-isomorphic. We prove that the converse is not true in general, and we obtain particular conditions under which the converse holds. The mathematical results in the paper are applied to the comparison of maritime areas with respect to chemical components of seaweeds. Moreover we show how the solution of the above comparison for specific components can lead to the solution of the comparison when we consider other components of seaweeds by applying the results on order-isomorphisms.

The study of shapes is a difficult topic, but it is becoming more and more important as computer vision techniques are already crucial in many research fields. Concretely, the variability of shapes is the basis for many criteria of symptom definition in medical diagnosis. This article introduces a stochastic order to address the variability of star-shaped sets. The main properties of the order are analyzed. An example of an application to hypothesis testing in medical diagnosis is also provided. Namely, we study if there are significant differences between healthy and diseased corneal endothelia with respect to cell shapes by means of ocular images.

The aim of the paper is to demonstrate possibilities of open software environment Maxima in educational process at technical universities whereby our attention is dedicated to the teaching of Bode plots. The developed procedure for drawing its asymptotic approximation can be used both for checking results on the base of the entered system transfer function and also for self testing purposes. In addition, the results were used for the building of web application that will be used in frame of the subject Control Theory.

The object of the study was to determine if apoptosis of neutrophils and their subsequent elimi- nation from the mammary gland by macrophages are modulated by an infection of Streptococcus uberis. The experiments were carried out in 5 clinically normal Holstein × Bohemian Red Pied crossbred heifers, aged 14 to 18 months. Before the experimental infection mammary glands were stimulated by PBS as a control. The samples of cell populations were obtained by lavages of the mammary glands in 4 intervals (24, 48, 72 and 168 h) after the PBS and after the experimental infection. Flow cytometry was used to determine the Annexin V positive and propidium jodide negative neutrophils (Annexin V +/PI-). The light microscopy was used to determine apoptotic neutrophils and myeloperoxidase (MPO) positive macrophages. After PBS and S. uberis administration the total number of both Annexin V +/PI- neutrophils and karyopycnotic neutrophils peaked at 24 hours. The highest per- centages of Annexin V +/PI- neutrophils were detected at 72 h after PBS and S. uberis, respectively. The highest percentages of karyopycnotic neutrophils were detected at 72 h after PBS and 168 h after S. uberis, respectively. The total number of MPO+ macrophages was the highest at 24 h after PBS and 72 h after S. uberis. The percent- age of MPO+ macrophages was the highest at 72 h after PBS and S. uberis. The results of this study demonstrate that during experimental infection of the mammary gland by S. uberis, the apoptosis of neutrophils is modulated. Apoptosis of neutrophils and the subsequent phagocytosis of apoptotic neutrophils by macrophages were delayed. This may cause the transition of the acute inflammatory reaction to a chronic state.

I Functional on Stochastic Processes.- 1. Stochastic Processes as Random Functions.- Notes.- Problems.- II Uniform Convergence of Empirical Measures.- 1. Uniformity and Consistency.- 2. Direct Approximation.- 3. The Combinatorial Method.- 4. Classes of Sets with Polynomial Discrimination.- 5. Classes of Functions.- 6. Rates of Convergence.- Notes.- Problems.- III Convergence in Distribution in Euclidean Spaces.- 1. The Definition.- 2. The Continuous Mapping Theorem.- 3. Expectations of Smooth Functions.- 4. The Central Limit Theorem.- 5. Characteristic Functions.- 6. Quantile Transformations and Almost Sure Representations.- Notes.- Problems.- IV Convergence in Distribution in Metric Spaces.- 1. Measurability.- 2. The Continuous Mapping Theorem.- 3. Representation by Almost Surely Convergent Sequences.- 4. Coupling.- 5. Weakly Convergent Subsequences.- Notes.- Problems.- V The Uniform Metric on Spaces of Cadlag Functions.- 1. Approximation of Stochastic Processes.- 2. Empirical Processes.- 3. Existence of Brownian Bridge and Brownian Motion.- 4. Processes with Independent Increments.- 5. Infinite Time Scales.- 6. Functional of Brownian Motion and Brownian Bridge.- Notes.- Problems.- VI The Skorohod Metric on D(0, ?).- 1. Properties of the Metric.- 2. Convergence in Distribution.- Notes.- Problems.- VII Central Limit Theorems.- 1. Stochastic Equicontinuity.- 2. Chaining.- 3. Gaussian Processes.- 4. Random Covering Numbers.- 5. Empirical Central Limit Theorems.- 6. Restricted Chaining.- Notes.- Problems.- VIII Martingales.- 1. A Central Limit Theorem for Martingale-Difference Arrays.- 2. Continuous Time Martingales.- 3. Estimation from Censored Data.- Notes.- Problems.- Appendix A Stochastic-Order Symbols.- Appendix B Exponential Inequalities.- Notes.- Problems.- Appendix C Measurability.- Notes.- Problems.- References.- Author Index.

This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.

This paper introduces a new distribution-free test for first-order stochastic dominance. It can be viewed as a modification of the well-known Wilcoxon-Mann-Whitney test. As it is based on the pp-plot of two distribution functions the new test statistic has a nice pictorial interpretation. Its finite sample distribution as well as its asymptotic distribution (which is a functional of the Brownian bridge) are derived and some relevant quantiles are tabulated. The new test is consistent on the whole set of alternatives-including those alternatives which the Wilcoxon-Mann-Whitney test is inconsistent against. Some Monte Carlo simulations indicate good power properties of the new test within selected families of alternative distributions.

We introduce a new interpolation method employing a closed algebraic curve. This note describes a trigonometric interpolation method to connect a given point sequence in Rd by an algebraic and therefore smooth closed curve. The interpolation formula is of Lagrangian type and seems to be new. In contrast to spline interpolation our method does not work piecewise but global. Nevertheless the curve connects the given points in a quite natural manner preserving the given order and without overshooting tendency. The curves are, like spline curves, stable in the following sense: changing the position of a single point of the given sequence only changes the curve in the neighbourhood of this point essentially. Besides its trigonometric parametrization the curve possesses a numerically efficient parametrization on the basis of Chebyshev polynomials of the second kind. To what extend the new interpolation method can replace the spline method, the experts may decide.

Let $x$ and $y$ be two random variables with continuous cumulative distribution functions $f$ and $g$. A statistic $U$ depending on the relative ranks of the $x$'s and $y$'s is proposed for testing the hypothesis $f = g$. Wilcoxon proposed an equivalent test in the Biometrics Bulletin, December, 1945, but gave only a few points of the distribution of his statistic. Under the hypothesis $f = g$ the probability of obtaining a given $U$ in a sample of $n x's$ and $m y's$ is the solution of a certain recurrence relation involving $n$ and $m$. Using this recurrence relation tables have been computed giving the probability of $U$ for samples up to $n = m = 8$. At this point the distribution is almost normal. From the recurrence relation explicit expressions for the mean, variance, and fourth moment are obtained. The 2rth moment is shown to have a certain form which enabled us to prove that the limit distribution is normal if $m, n$ go to infinity in any arbitrary manner. The test is shown to be consistent with respect to the class of alternatives $f(x) > g(x)$ for every $x$.

Let $F(x)$ be the continuous distribution function of a random variable $X,$ and $F_n(x)$ the empirical distribution function determined by a sample $X_1, X_2, \cdots, X_n$. It is well known that the probability $P_n(\epsilon)$ of $F(x)$ being everywhere majorized by $F_n(x) + \epsilon$ is independent of $F(x)$. The present paper contains the derivation of an explicit expression for $P_n(\epsilon)$, and a tabulation of the 10%, 5%, 1%, and 0.1% points of $P_n(\epsilon)$ for $n =$ 5, 8, 10, 20, 40, 50. For $n =$ 50 these values agree closely with those obtained from an asymptotic expression due to N. Smirnov.

A multivariate dispersion ordering is introduced in a weak and strong version. These arise naturally out of the consideration of two-sided versions of a well-known univariate dispersion ordering referred to as 'disp'. Various characterisations of the new orderings are given and some detailed results for the normal distribution. A final section points to applications.

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R: a language and environment for statistical computing. R Foundation for Statistical Computing

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