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Panjer Recursion
The Panjer recursion is an algorithm to compute the probability distribution approximation of a compound random variable S = \sum_^N X_i\, where both N\, and X_i\, are random variables and of special types. In more general cases the distribution of ''S'' is a compound distribution. The recursion for the special cases considered was introduced in a paper by Harry Panjer (Distinguished Emeritus Professor, University of Waterloo). It is heavily used in actuarial science (see also systemic risk). Preliminaries We are interested in the compound random variable S = \sum_^N X_i\, where N\, and X_i\, fulfill the following preconditions. Claim size distribution We assume the X_i\, to be i.i.d. and independent of N\,. Furthermore the X_i\, have to be distributed on a lattice h \mathbb_0\, with latticewidth h>0\,. : f_k = P[X_i = hk].\, In actuarial practice, X_i\, is obtained by discretisation of the claim density function (upper, lower...). Claim number distribution The number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm can be expressed within a finite amount of spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Systemic Risk
In finance, systemic risk is the risk of collapse of an entire financial system or entire market, as opposed to the risk associated with any one individual entity, group or component of a system, that can be contained therein without harming the entire system.Banking and currency crises and systemic risk George G. Kaufman (World Bank), It can be defined as "financial ''system'' instability, potentially catastrophic, caused or exacerbated by idiosyncratic events or conditions in financial intermediaries". It refers to the risks imposed by ''interlink ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fréchet Distribution
The Fréchet distribution, also known as inverse Weibull distribution, is a special case of the generalized extreme value distribution. It has the cumulative distribution function :\Pr(X \le x)=e^ \text x>0. where ''α'' > 0 is a shape parameter. It can be generalised to include a location parameter ''m'' (the minimum) and a scale parameter ''s'' > 0 with the cumulative distribution function :\Pr(X \le x)=e^ \text x>m. Named for Maurice Fréchet who wrote a related paper in 1927, further work was done by Fisher and Tippett in 1928 and by Gumbel in 1958. Characteristics The single parameter Fréchet with parameter \alpha has standardized moment :\mu_k=\int_0^\infty x^k f(x)dx=\int_0^\infty t^e^ \, dt, (with t=x^) defined only for k1 the expectation is E \Gamma(1-\tfrac) * For \alpha>2 the variance is \text(X)=\Gamma(1-\tfrac)-\big(\Gamma(1-\tfrac)\big)^2. The quantile q_y of order y can be expressed through the inverse of the distribution, :q_y= ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Variables
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads H and tails T) in a sample space (e.g., the set \) to a measurable space, often the real numbers (e.g., \ in which 1 corresponding to H and -1 corresponding to T). Informally, randomness typically represents some fundamental element of chance, such as in the roll of a dice; it may also represent uncertainty, such as measurement error. However, the interpretation of probability is philosophically complicated, and even in specific cases is not always straightforward. The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup. In the formal mathematical language of measure theory, a rando ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Generating Function
In probability theory, the probability generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random variable. Probability generating functions are often employed for their succinct description of the sequence of probabilities Pr(''X'' = ''i'') in the probability mass function for a random variable ''X'', and to make available the well-developed theory of power series with non-negative coefficients. Definition Univariate case If ''X'' is a discrete random variable taking values in the non-negative integers , then the ''probability generating function'' of ''X'' is defined as http://www.am.qub.ac.uk/users/g.gribakin/sor/Chap3.pdf :G(z) = \operatorname (z^X) = \sum_^p(x)z^x, where ''p'' is the probability mass function of ''X''. Note that the subscripted notations ''G''''X'' and ''pX'' are often used to emphasize that these pertain to a particular random variable ''X'', and to its distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
(a,b,0) Class Of Distributions
In probability theory, a member of the (''a'', ''b'', 0) class of distributions is any distribution of a discrete random variable ''N'' whose values are nonnegative integers whose probability mass function satisfies the recurrence formula : \frac = a + \frac, \qquad k = 1, 2, 3, \dots for some real numbers ''a'' and ''b'', where p_k = P(N = k). Only the Poisson, binomial and negative binomial distributions satisfy the full form of this relationship. These are also the three discrete distributions among the six members of the natural exponential family with quadratic variance functions (NEF–QVF). More general distributions can be defined by fixing some initial values of ''pj'' and applying the recursion to define subsequent values. This can be of use in fitting distributions to empirical data. However, some further well-known distributions are available if the recursion above need only hold for a restricted range of values of ''k'': for example the logarithmic distribu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Waterloo
The University of Waterloo (UWaterloo, UW, or Waterloo) is a public research university with a main campus in Waterloo, Ontario, Canada. The main campus is on of land adjacent to "Uptown" Waterloo and Waterloo Park. The university also operates three satellite campuses and four affiliated university colleges. The university offers academic programs administered by six faculties and thirteen faculty-based schools. Waterloo operates the largest post-secondary co-operative education program in the world, with over 20,000 undergraduate students enrolled in the university's co-op program. Waterloo is a member of the U15, a group of research-intensive universities in Canada. The institution originates from the Waterloo College Associate Faculties, established on 4 April 1956; a semi-autonomous entity of Waterloo College, which was an affiliate of the University of Western Ontario. This entity formally separated from Waterloo College and was incorporated as a university with the pass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distinguished Emeritus Professor
The ruling made by the judge or panel of judges must be based on the evidence at hand and the standard binding precedents covering the subject-matter (they must be ''followed''). Definition In law, to distinguish a case means a court decides the holding or legal reasoning of a precedent case will not apply due to materially different facts between the two cases. Two formal constraints constrain the later court: the expressed relevant factors (also known as considerations, tests, questions or determinants) in the '' ratio'' (legal reasoning) of the earlier case must be recited or their equivalent recited or the earlier case makes an exception for their application in the circumstances otherwise it envisages, and the ruling in the later case must not expressly doubt (criticise) the result reached in the precedent case.Lamond, Grant"Precedent and Analogy in Legal Reasoning: 2.1 Precedents as laying down rules:2.1.2 The practice of distinguishing". ''Stanford Encyclopedia of Phi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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