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The article focuses on the application of mathematical methods in economics, in particular discussing economic problems that are easily solved using derivatives. The purpose of the article is to show students the way and opportunity to use mathematical methods to solve economic problems. To this end, the article discusses and analyzes several economic tasks in detail, which will be interesting and easy for students to master. I considered the derivative of a function as the rate of change and introduced the definition: The instantaneous rate of change of the function f with respect to x at a point is called the derivative if it exists. With the help of this definition I have discussed and explained Task 1: Suppose that the increase in production of a certain product over a period of time is described by a function And population growth is described by the following function: Where is the number of years from the initial period, then the production of these products per capita is given by the function: Find the growth rate of product production. By solving this task I came to the conclusion that after a year the production of products per capita increases. In the following tasks I used the method of finding the extremum values of a function using a derivative, namely I equated the first-order derivative to 0, found the critical points, and with the help of the second-order derivative I determined the extremes of the function. I discussed task 2: For the production of X volume of products, the firm plans a cost that is calculated by the formulan . For what volume of production will the average cost be the smallest? Find the numerical value of this small expense. Solving this problem, I came to the conclusion that the given function of the average cost takes on the least value when the output volume is a unit , and this value is equal to: which is the marginal cost when producing the volume output. I discussed Task 3: How many products should be sold in order for a firm to profit maximally if the derivative cost function is known: And return function: Here I came to the conclusion: if 600 units of the product are sold, then the firm's profit will be maximum and it will be numerically equal At the end of the article I discussed such a task 4 of applied optimization. What is the minimum amount of material needed to make a 2 liter cylindrical jar? Where I came to the conclusion: the smallest amount of material will be spent to get a cylindrical vessel if we take the height 2 times the radius of the base.

In dichotomous item response theory (IRT) framework, the asymptotic standard error (ASE) is the most common statistic to evaluate the precision of various ability estimators. Easy-to-use ASE formulas are readily available; however, the accuracy of some of these formulas was recently questioned and new ASE formulas were derived from a general asymptotic theory framework. Furthermore, exact standard errors were suggested to better evaluate the precision of ability estimators, especially with short tests for which the asymptotic framework is invalid. Unfortunately, the accuracy of exact standard errors was assessed so far only in a very limiting setting. The purpose of this article is to perform a global comparison of exact versus (classical and new formulations of) asymptotic standard errors, for a wide range of usual IRT ability estimators, IRT models, and with short tests. Results indicate that exact standard errors globally outperform the ASE versions in terms of reduced bias and root mean square error, while the new ASE formulas are also globally less biased than their classical counterparts. Further discussion about the usefulness and practical computation of exact standard errors are outlined.

International audience; In the case of ordered categorical data, the concepts of minimum and maximum inequality are not straightforward. In this chapter, the authors consider the Cowell and Flachaire (2017) indices of inequality. The authors show that the minimum and maximum inequality depend on preliminary choices made before using these indices, on status and the sensitivity parameter. Specifically, maximum inequality can be given by the distribution which is the most concentrated in the top or bottom category, or by the uniform distribution.

Bessel functions of the first kind are ubiquitous in the sciences and engineering in solutions to cylindrical problems including electrostatics, heat flow, and the Schrodinger equation. The roots of the Bessel functions are often quoted and calculated, but the maxima and minima for each Bessel function, used to match Neumann boundary conditions, have not had the same treatment. Here we compute 10000 extrema for the first 600 orders of the Bessel function J . To do this, we employ an adaptive root solver bounded by the roots of the Bessel function and solve to an accuracy of 10 − 19 . We compare with the existing literature (to 30 orders and 5 maxima and minima) and the results match exactly. It is hoped that these data provide values needed for orthogonal function expansions and numerical expressions including the calculation of geometric correction factors in the measurement of resistivity of materials, as is done in the original paper using these data.

Background: Some medical and technological tasks lead to the geometrical problem of how to cover the unit circle as much as possible by n congruent circles of given radius r, while r varies from the radius in the maximum packing to the radius in the minimum covering. Proven or conjectural solutions to this partial covering problem are known only for n = 2 to 5. In the present paper, numerical solutions are given to this problem for n = 6 and 7. Method: The method used transforms the geometrical problem to a mechanical one, where the solution to the geometrical problem is obtained by finding the self-stress positions of a generalised tensegrity structure. This method was developed by the authors and was published in an earlier publication. Results: The method applied results in locally optimal circle arrangements. The numerical data for the special circle arrangements are presented in a tabular form, and in drawings of the arrangements. Conclusion: It was found that the case of n = 6 is very complicated, whilst the case n = 7 is very simple. It is shown in this paper that locally optimal arrangements may exhibit different types of symmetry, and equilibrium paths may bifurcate.

International audience; A Gaussian field $X$ defined on a square $S$ of $\mathds R^2$ is considered. We assume that this field is only observed at some points of a regular grid with spacing $\frac{1}{n}$. We are interested in the normalized discretization error $n^2(M - M_n)$, with $M$ the global maximum of $X$ over $S$ and $M_n$ the maximum of $X$ over the observation grid. The density of the location of the maximum is given using Rice formulas and its regularity is studied. Joint densities with the value of the field and the value of the second derivative are also given.Then, a kind of Slepian model is used to study the field behavior around the unique point where the maximum is attained, called $t^*$. We show that the normalized discretization error can be bounded by a quantity that converges in distribution to a uniform variable. The set where this uniform variable lies principally depends on the second derivative of the field at $t^*$. The bound is a function of this quantity which is approached by finite differences in practice. The bound is applied both on simulated and real data. Real data are used in positioning by satellite systems quality assessment.

We give expansions for the distribution function, density function and moments of the sample minimum when sampling from a distribution on 0 , x 2 that is nearly analytic at zero. Similar results are given for the sample maximum when sampling from a distribution on x 1 , x 2 that is nearly analytic at x 2 . When these distributions are analytic, the expansions are in inverse powers of the sample size n . If not, they require a double expansion.

In the framework of nonlinear elasticity, we consider the model energy F ( u ) = ∫ Ω [ | D u ( x ) | p + h ( det ⁡ D u ( x ) ) ] d x , where u : Ω ⊂ R n → R n with det ⁡ D u > 0 and h : ( 0 , + ∞ ) → [ 0 , + ∞ ) is convex; moreover h ( t ) blows up when t → 0 + . We study the problem: fix the mapping u with finite energy F ( u ) and find a mapping v with the same boundary values, with det ⁡ D v > 0 and energy F ( v ) not higher than F ( u ) , such that every component v β of v enjoys a kind of maximum and minimum principle. Due to the constraint det ⁡ D v > 0 , a truncation argument does not work. On the contrary, the constraint det ⁡ D v > 0 makes the problem easy when p ≥ n since it is known that every component is weakly monotone, so we can take v = u . In the present work we address the case 2 ≤ p n and we give a positive answer if the blow up of h ( t ) is of logarithmic type.

In this article we study imaginary Gaussian multiplicative chaos-namely a family of random generalized functions which can formally be written as e(iX(x)), where X is a log-correlated real-valued Gaussian field on R-d, that is, it has a logarithmic singularity on the diagonal of its covariance. We study basic analytic properties of these random generalized functions, such as what spaces of distributions these objects live in, along with their basic stochastic properties, such as moment and tail estimates., After this, we discuss connections between imaginary multiplicative chaos and the critical planar Ising model, namely that the scaling limit of the spin field of the critical planar XOR-Ising model can be expressed in terms of the cosine of the Gaussian free field, that is, the real part of an imaginary multiplicative chaos distribution. Moreover, if one adds a magnetic perturbation to the XOR-Ising model, then the scaling limit of the spin field can be expressed in terms of the cosine of the sine-Gordon field, which can also be viewed as the real part of an imaginary multiplicative chaos distribution., The first sections of the article have been written in the style of a review, and we hope that the text will also serve as an introduction to imaginary chaos for an uninitiated reader.

Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The power triangle inequality represents an uncountably large class of inequalities, and includes the triangle inequality, relaxed triangle inequality, and inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the "power triangle function". The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean as special cases., This submission has been typeset with the assistance of XeLaTeX

Regression models for proportions are frequently encountered in applied work. The conditional expectation is bound between 0 and 1 and, therefore, must be non-linear which requires non-standard panel data extensions. The quasi-maximum likelihood estimator of Papke and Wooldridge (1996) suffers from the incidental parameters problem when including fixed effects. In this paper, we re-consider the binomial panel logit model with fixed effects (Machado, 2004). We show that the conditional maximum likelihood estimator is very easy to implement using standard software. We investigate the properties of the estimator under misspecification and derive a new test for overdispersion in the binomial fixed effects logit model. Models and test are applied in a study of contracted work-time percentage, measured as proportion of full-time work, for women in Switzerland.

The definition of a Detour&ndash, Harary index is &omega, H ( G ) = 1 2 &sum, u , v &isin, V ( G ) 1 l ( u , v | G ) , where G is a simple and connected graph, and l ( u , v | G ) is equal to the length of the longest path between vertices u and v. In this paper, we obtained the maximum Detour&ndash, Harary index about unicyclic graphs, bicyclic graphs, and cacti, respectively.

This paper explores the joint extreme-value behavior of discontinuous random variables. It is shown that as in the continuous case, the latter is characterized by the weak limit of the normalized componentwise maxima and the convergence of any compatible copula. Illustrations are provided and an extension to the case of triangular arrays is considered which sheds new light on recent work of Coles and Pauli (Stat Probab Lett 54:373–379, 2001) and Mitov and Nadarajah (Extremes 8:357–370, 2005). This leads to considerations on the meaning of the bivariate upper tail dependence coefficient of Joe (Comput Stat Data Anal 16:279–297, 1993) in the discontinuous case., Extremes, 13 (1), ISSN:1386-1999, ISSN:1572-915X

Let (Formula presented.) be a multivariate normal random vector. We derive explicit expressions for the cumulative distribution function, probability density function and the moments of (Formula presented.). Each expression involves single infinite sums of known special functions. Computational issues like accuracy, convergence, time and simulations are investigated.

This paper establishes an upper bound for the Kolmogorov distance between the maximum of a high-dimensional vector of smooth Wiener functionals and the maximum of a Gaussian random vector. As a special case, we show that the maximum of multiple Wiener-It\^o integrals with common orders is well-approximated by its Gaussian analog in terms of the Kolmogorov distance if their covariance matrices are close to each other and the maximum of the fourth cumulants of the multiple Wiener-It\^o integrals is close to zero. This may be viewed as a new kind of fourth moment phenomenon, which has attracted considerable attention in the recent studies of probability. This type of Gaussian approximation result has many potential applications to statistics. To illustrate this point, we present two statistical applications in high-frequency financial econometrics: One is the hypothesis testing problem for the absence of lead-lag effects and the other is the construction of uniform confidence bands for spot volatility., Comment: 48 pages. There were a few errors in Corollary 3.1 and Remark 3.7 of the previous version, and they have been corrected. To appear in Annals of Statistics

Dans cet article, nous developpons une methodologie permettant de prouver l’unicite faible pour des equations differentielles stochastiques dont les coefficients dependent de certaines fonctionnelles de la trajectoire du processus. Dans le prolongement de la technique developpee par Bass & Perkins (In From Probability to Geometry (I): Volume in Honor of the 60th Birthday of Jean-Michel Bismut (2009) 47–53) dans le cas des processus de diffusions standards, la methologie proposee permet de traiter le cas de processus dont la loi est singuliere par rapport a la mesure de Lebesgue. Afin d’illustrer notre methodologie, nous prouvons l’existence et l’unicite faible dans les deux exemples suivants: un processus de diffusion dont les coefficients dependent du temps local en zero courant et un processus de diffusion dont les coefficients dependent du maximum courant. Dans chaque exemple, nous prouvons egalement l’existence d’une densite de transition associee et etablissons des estimees Gaussiennes.

We give the distribution of $M_n$, the maximum of a sequence of $n$ observations from a moving average of order 1. Solutions are first given in terms of repeated integrals and then for the case where the underlying independent random variables have an absolutely continuous density. When the correlation is positive, $$ P(M_n %\max^n_{i=1} X_i \leq x) =\ \sum_{j=1}^\infty \beta_{jx} \nu_{jx}^{n} \approx B_{x} \nu_{1x}^{n} $$ where %$\{X_i\}$ is a moving average of order 1 with positive correlation, and $\{\nu_{jx}\}$ are the eigenvalues (singular values) of a Fredholm kernel and $\nu_{1x}$ is the eigenvalue of maximum magnitude. A similar result is given when the correlation is negative. The result is analogous to large deviations expansions for estimates, since the maximum need not be standardized to have a limit. % there are more terms, and $$P(M_n, Comment: 15 A4 pages. Version 4 corrects (3.8). Version 3 expands Section 2. Version 2 corrected recurrence relation (2.5)

We introduce logarithmic generalized Maxwell distribution motivated by Vodă (Math. Rep. 11:171-179, 2009), which is an extension of the generalized Maxwell distribution. Some interesting properties of this distribution are studied and the asymptotic distribution of the partial maximum of an i.i.d. sequence from the logarithmic generalized Maxwell distribution is gained. The expansion of the limit distribution from the normalized maxima is established under the optimal norming constants, which shows the rate of convergence of the distribution for normalized maximum to the extreme limit.

Let M n denote the partial maximum of an independent and identically distributed lognormal random sequence. In this paper, we derive the exact uniform convergence rate of the distribution of the normalized maximum ( M n − b n ) / a n to its extreme value distribution, where the constants a n and b n are chosen by an optimal way.

The coherence of a random matrix, which is defined to be the largest magnitude of the Pearson correlation coefficients between the columns of the random matrix, is an important quantity for a wide range of applications including high-dimensional statistics and signal processing. Inspired by these applications, this paper studies the limiting laws of the coherence of n×p random matrices for a full range of the dimension p with a special focus on the ultra high-dimensional setting. Assuming the columns of the random matrix are independent random vectors with a common spherical distribution, we give a complete characterization of the behavior of the limiting distributions of the coherence. More specifically, the limiting distributions of the coherence are derived separately for three regimes: 1nlogp→0, 1nlogp→β∈(0,∞), and 1nlogp→∞. The results show that the limiting behavior of the coherence differs significantly in different regimes and exhibits interesting phase transition phenomena as the dimension p grows as a function of n. Applications to statistics and compressed sensing in the ultra high-dimensional setting are also discussed.

Let G be a simple graph with the vertex set V(G) and @a be a real number with @a 0. The zeroth-order general Randic index of G is defined as R"@a^0(G)=@?"v"@?"V"("G")d^@a(v), where d(v) denotes the degree of the vertex v in G. A graph G is called a quasi-tree graph, if there exists a vertex u@?V(G) such that G[V(G)@?{u}] is a tree. In this paper, we characterize the extremal quasi-tree graphs containing cycles with the minimum and maximum values of the zeroth-order general Randic index for @a in different intervals.

The concept of upper and lower semicontinuity of fuzzy mappings introduced by Bao and Wu [Y.E. Bao, C.X. Wu, Convexity and semicontinuity of fuzzy mappings, Comput. Math. Appl., 51 (2006) 1809–1816] is redefined by using the concept of parameterized triples of fuzzy numbers. On the basis of the linear ordering of fuzzy numbers proposed by Goetschel and Voxman [R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems 18], we prove that an upper semicontinuous fuzzy mapping attains a maximum (with respect to this linear ordering) on a nonempty closed and bounded subset of the n -dimensional Euclidean space R n , and that a lower semicontinuous fuzzy mapping attains a minimum (with respect to this linear ordering) on a nonempty closed and bounded subset of R n .

We prove that if $\omega$ is uniformly distributed on $[0,1]$, then as $T\to\infty$, $t\mapsto \zeta(i\omega T+it+1/2)$ converges to a non-trivial random generalized function, which in turn is identified as a product of a very well behaved random smooth function and a random generalized function known as a complex Gaussian multiplicative chaos distribution. This demonstrates a novel rigorous connection between number theory and the theory of multiplicative chaos -- the latter is known to be connected to many other areas of mathematics. We also investigate the statistical behavior of the zeta function on the mesoscopic scale. We prove that if we let $\delta_T$ approach zero slowly enough as $T\to\infty$, then $t\mapsto \zeta(1/2+i\delta_T t+i\omega T)$ is asymptotically a product of a divergent scalar quantity suggested by Selberg's central limit theorem and a strictly Gaussian multiplicative chaos. We also prove a similar result for the characteristic polynomial of a Haar distributed random unitary matrix, where the scalar quantity is slightly different but the multiplicative chaos part is identical. This essentially says that up to scalar multiples, the zeta function and the characteristic polynomial of a Haar distributed random unitary matrix have an identical distribution on the mesoscopic scale., Comment: Supersedes arXiv:1604.08378. Version 2 contains a limit theorem describing the mesoscopic behavior of the characteristic polynomial of random unitary matrices

We study the exact distribution of linear combinations of order statistics of arbitrary (absolutely continuous) dependent random variables. In particular, we examine the case where the random variables have a joint elliptically contoured distribution and the case where the random variables are exchangeable. We investigate also the particular L-statistics that simply yield a set of order statistics, and study their joint distribution. We present the application of our results to genetic selection problems, design of cellular phone receivers, and visual acuity. We give illustrative examples based on the multivariate normal and multivariate Student t distributions.

The paper gives the joint distribution of maxima of contralateral observations taken from the same individual at several occasions, when the data are normal with given constraints on the parameters. Different constraints lead to different well-known generalizations of the normal distribution. As an immediate consequence, evaluation of several features of these maxima is greatly simplified. The results of the paper are also useful in modeling other phenomena, such as measures of physical fitness based on the best of two trials. Further applications deal with statistical inference, for example, testing whether an identified treatment is best.

In this paper we study the asymptotic joint behavior of the maximum and the partial sum of a multivariate Gaussian sequence. The multivariate maximum is defined to be the coordinatewise maximum. Results extend univariate results of McCormick and Qi. We show that, under regularity conditions, if the maximum has a limiting distribution it is asymptotically independent of the partial sum. We also prove that the maximum of a stationary sequence, when normalized in a special sense which includes subtracting the sample mean, is asymptotically independent of the partial sum (again, under regularity conditions). The limiting distributions are also obtained.

Using Rice's method to obtain the distribution of the maximum of Gaussian random fields we give the distribution of the length of the projection of a Gaussian variable onto any convex or non-convex spherical cone.

We study an extremal problem related to “splitted” Jacobi weights: for α,β>0, find the largest value of maxx∈[−1,1][(1+x)βpm(x)2+(1−x)αqn(x)2] among all polynomials pm and qn of degree at most m and n, respectively, satisfying∫−11[(1+x)βpm(x)2+(1−x)αqn(x)2]dx=1.We show that the solution of this problem is related to an estimation of the Christoffel functions and the Wronskians associated with certain Jacobi polynomials.

We present a methodology for extracting processes of curves in images, using a statistical summary of the directional information given in measures of location, curvature and direction associated with the pixels that compose each curve. The main purpose is to obtain measures that serve as input for the reconstruction, in vector format, of a process of curves which are of interest, so that the extracted curves can be easily stored and reconstructed based on few parameters conserving representative information of its curvature at each pixel. As starting point, the directional information obtained from a methodology of consistent curves detection is used, which includes the decomposition of the image in a directional domain contained in ℜˆ(2ˆk), with k ∈ ℵ. Basic summary measures criteria are proposed for this type of data and the application to four cases of satellite images for extraction of sections of rivers in these images are shown. Presentamos una metodología para la extracción de procesos de curvas en imágenes, mediante un resumen estadístico de la información direccional dado en medidas de localización, curvatura y dirección asociadas a los pixeles que componen cada curva. El propósito principal es obtener medidas que sirvan como insumo para la reconstrucción de los procesos de curvas que sean de interés, en formato de vector, de manera que las curvas extraídas puedan ser almacenadas fácilmente y reconstruidas en base a pocos parámetros conservando información representativa de su curvatura en cada pixel. Como punto de partida se usa la información direccional obtenida a partir de la metodología de detección consistente de curvas, la cual comprende la descomposición de la imagen en un dominio direccional contenido en R^(2^k), con k∈ N. Para este tipo de datos se proponen criterios básicos para las medidas de resumen y se muestra la aplicación a cuatro casos de imágenes satelitales para la extracción de tramos de río en dichas imágenes.

The Standard Quadratic Problem (StQP) is an NP-hard problem with many local minimizers (stationary points). In the literature, heuristics based on unconstrained continuous non-convex formulations have been proposed (Bomze & Palagi, 2005; Bomze, Grippo, & Palagi, 2012) but none dominates the other in terms of best value found. Following (Cassioli, DiLorenzo, Locatelli, Schoen, & Sciandrone, 2012) we propose to use Support Vector Machines (SVMs) to define a multistart global strategy which selects the “best” heuristic. We test our method on StQP arising from the Maximum Clique Problem on a graph which is a challenging combinatorial problem. We use as benchmark the clique problems in the DIMACS challenge.

Let h1, ...,hn be positive integers. We study new sums m(h1,...,hn = Σr1=0h1-1 ... Σrn=0hn-1 min {r1h1,..., rnhn} and M(h1,...,hn = Σr1=0h1-1 ... Σrn=0hn-1 max {r1h1,..., rnhn}, the first of which times h1 ... hn is the number of lattice points in a pyramid of dimension n + 1. We show that m(h1,...,hn)(h1 - 1)...(hn - 1) = 1 + Σ0 ≠ I ⊆ {1,...,n} (-1)|I| m({hi}i∈I)Πi∈I (hi - 1) if h1,...,hn > 1, and that M(h1,...,hn) - h1 ... hn + 1(h1 + 1)...(hn + 1) = Σ0≠I⊆{1,...,n} (-1)|I| M({hi}i∈I)Πi∈I (hi + 1). The sums m(h1, h2) and M(h1, h2) are connected with the reciprocity law for Dedekind sums. The values of m(h1, h2, h3, M(h1, h2, h3) and m(h1, h2, h3 h4) + M(h1, h2, h3 h4) are determined explicity in the paper.

Isotonic strength training remains one of the methods of strengthening muscles, as it requires inexpensive materials. The method requires the determination of maximal isotonic strength or one repetition maximum [1-RM] and/or sub-maximal isotonic strength (2-RM - 10-RM) of a muscle group in order to know the appropriate resistance to subject the muscle group to, and to monitor strength gains by the muscle group. Berger’s table contains percentages for predicting 1-RM - 10-RM, and it is intended to improve the efficiency of determining isotonic muscle strength. The aim of this study was to investigate the reliability of Berger’s table in predicting 1-RM and 10-RM of the right elbow flexor muscles.1-RM and 10-RM were estimated using Berger’s table in 100 young adults following the determination of the number of repetitions that could be carried out against randomly selected weights. Weight adjustments were made to obtain the actual 1-RM and 10-RM. Data were subjected to regression analysis. A significant linear relationship exists between the estimated and actual values of 1-RM and 10-RM. Results also showed that Berger’s table explains 93% of 1-RM and 98% of 10-RM. It was concluded that Berger’s table is reliable in determining 1-RM and 10-RM of the elbow flexors in young healthy adults. Regression equations that may improve the accuracy of estimation were derived.

In this paper the method devised by Fermat for the determination of the maximum and minimum of a function is exposed together with the method for the tangents. Moreover a problem of maximum geometrically solved by Fermat is studied.

We discuss tail behaviors, subexponentiality, and the extreme value distribution of logarithmic skew-normal random variables. With optimal normalized constants, the asymptotic expansion of the distribution of the normalized maximum of logarithmic skew-normal random variables is derived. We show that the convergence rate of the distribution of the normalized maximum to the Gumbel extreme value distribution is proportional to 1/(log n)1/2.

We present a new proof of the Burkholder-Davis-Gundy inequalities for $1\leq p, Comment: Published at http://dx.doi.org/10.3150/13-BEJ570 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Limit distributions of maxima of dependent Gaussian sequence are different according to the convergence rate of their correlations. For three different conditions on convergence rate of the correlations, in this paper, we establish the Piterbarg theorem for maxima of stationary Gaussian sequences.

WOS: 000310089700005, In this work we consider the problem of finding the distribution of linear functions of the minimum and the maximum of the bivariate log-normal distribution. We derive the distribution function, density function and moments of these statistics. This work will provide a generalization of the minimum and the maximum cases.

Šis magistro darbas yra skirtas stochastinių ekstremumų stabilumo uždavinių sprendimui. Šiame darbe yra atlikta N stabilumo analizė, kai N yra pasiskirstęs pagal geometrinį, Sibuya ir Harris dėsnius. Taip pat yra paskaičiuotas konvergavimo greitis maksimumams ir atlikta konvergavimo greičio kompiuterinė analizė. Gautais rezultatais mes patvirtinome jau žinomą faktą, kad geometrinis N maks stabilumas ir N min stabilumas yra vienas kitą sąlygojantys. Taip pat gavome ir įrodėme naujus rezultatus, jog Sibuya ir Harris N maks ir N min stabilumai nėra vienas kitą sąlygojantys. This master’s work is dedicated to solution of the stochastic extremums stability problems. In this work the analysis of N stability is done then N is distributed by geometrical, Sibuya and Harris laws. The convergence rate for maximums was constructed and also computerized analysis of results was done. In accordance with the main results, we confirmed the known fact that geometrical N max stability and N min stability are influencing each other. Also we obtained and proved new results, that Sibuya and Harris N max stability and N min stability are not influencing each other.

We analyze the tail behavior of the maximum N of Brownian motion minus a parabola and give an asymptotic expansion for P(N>x) as x tends to infinity. This extends a first order result on the tail behavior, which can be deduced from Huesler and Piterbarg (1999). We also point out the relation between certain results in Groeneboom (2010) and Janson, Louchard and Martin-L��f (2010)., 12 pages, submitted to the Electronic Communications in Probability

Given a stable L\'{e}vy process $X=(X_t)_{0\le t\le T}$ of index $\alpha\in(1,2)$ with no negative jumps, and letting $S_t=\sup_{0\le s\le t}X_s$ denote its running supremum for $t\in [0,T]$, we consider the optimal prediction problem \[V=\inf_{0\le\tau\le T}\mathsf{E}(S_T-X_{\tau})^p,\] where the infimum is taken over all stopping times $\tau$ of $X$, and the error parameter $p\in(1,\alpha)$ is given and fixed. Reducing the optimal prediction problem to a fractional free-boundary problem of Riemann--Liouville type, and finding an explicit solution to the latter, we show that there exists $\alpha_*\in(1,2)$ (equal to 1.57 approximately) and a strictly increasing function $p_*:(\alpha_*,2)\rightarrow(1,2)$ satisfying $p_*(\alpha_*+)=1$, $p_*(2-)=2$ and $p_*(\alpha), Comment: Published in at http://dx.doi.org/10.1214/10-AOP598 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The last decade has seen max-stable processes emerge as a common tool for the statistical modeling of spatial extremes. However, their application is complicated due to the unavailability of the multivariate density function, and so likelihood-based methods remain far from providing a complete and flexible framework for inference. In this article we develop inferentially practical, likelihood-based methods for fitting max-stable processes derived from a composite-likelihood approach. The procedure is sufficiently reliable and versatile to permit the simultaneous modeling of marginal and dependence parameters in the spatial context at a moderate computational cost. The utility of this methodology is examined via simulation, and illustrated by the analysis of United States precipitation extremes.