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Bühlmann Model
In credibility theory, a branch of study in actuarial science, the Bühlmann model is a random effects model (or "variance components model" or hierarchical linear model) used to determine the appropriate premium for a group of insurance contracts. The model is named after Hans Bühlmann who first published a description in 1967. Model description Consider ''i'' risks which generate random losses for which historical data of ''m'' recent claims are available (indexed by ''j''). A premium for the ''i''th risk is to be determined based on the expected value of claims. A linear estimator which minimizes the mean square error is sought. Write * ''X''ij for the ''j''-th claim on the ''i''-th risk (we assume that all claims for ''i''-th risk are independent and identically distributed) * \scriptstyle =\frac\sum_^X_ for the average value. * \Theta_i - the parameter for the distribution of the i-th risk * m(\vartheta)= \operatorname E\left \Theta_i = \vartheta\right /math> * \Pi=\opera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Credibility Theory
Credibility theory is a form of statistical inference used to forecast an uncertain future event developed by Thomas Bayes. It is employed to combine multiple estimates into a summary estimate that takes into account information on the accuracy of the initial estimates. This is typically used by actuaries working for insurance companies when determining the premium values. For example, in group health insurance an insurer is interested in calculating the risk premium, RP, (i.e. the theoretical expected claims amount) for a particular employer in the coming year. The insurer will likely have an estimate of historical overall claims experience, x, as well as a more specific estimate for the employer in question, y. Assigning a credibility factor, z, to the overall claims experience (and the reciprocal to employer experience) allows the insurer to get a more accurate estimate of the risk premium in the following manner: RP = xz + y(1-z).The credibility factor is derived by calculati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Effects Model
In statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are drawn from a hierarchy of different populations whose differences relate to that hierarchy. A random effects model is a special case of a mixed model. Contrast this to the biostatistics definitions, as biostatisticians use "fixed" and "random" effects to respectively refer to the population-average and subject-specific effects (and where the latter are generally assumed to be unknown, latent variables). Qualitative description Random effect models assist in controlling for unobserved heterogeneity when the heterogeneity is constant over time and not correlated with independent variables. This constant can be removed from longitudinal data through differencing, since taking a first difference will remove any time invariant components of the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hierarchical Linear Model
Multilevel models (also known as hierarchical linear models, linear mixed-effect model, mixed models, nested data models, random coefficient, random-effects models, random parameter models, or split-plot designs) are statistical models of parameters that vary at more than one level. An example could be a model of student performance that contains measures for individual students as well as measures for classrooms within which the students are grouped. These models can be seen as generalizations of linear models (in particular, linear regression), although they can also extend to non-linear models. These models became much more popular after sufficient computing power and software became available. Multilevel models are particularly appropriate for research designs where data for participants are organized at more than one level (i.e., nested data). The units of analysis are usually individuals (at a lower level) who are nested within contextual/aggregate units (at a higher level ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Insurance Premium
Insurance is a means of protection from financial loss in which, in exchange for a fee, a party agrees to compensate another party in the event of a certain loss, damage, or injury. It is a form of risk management, primarily used to hedge against the risk of a contingent or uncertain loss. An entity which provides insurance is known as an insurer, insurance company, insurance carrier, or underwriter. A person or entity who buys insurance is known as a policyholder, while a person or entity covered under the policy is called an insured. The insurance transaction involves the policyholder assuming a guaranteed, known, and relatively small loss in the form of a payment to the insurer (a premium) in exchange for the insurer's promise to compensate the insured in the event of a covered loss. The loss may or may not be financial, but it must be reducible to financial terms. Furthermore, it usually involves something in which the insured has an insurable interest established by o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Independent And Identically Distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usually abbreviated as ''i.i.d.'', ''iid'', or ''IID''. IID was first defined in statistics and finds application in different fields such as data mining and signal processing. Introduction In statistics, we commonly deal with random samples. A random sample can be thought of as a set of objects that are chosen randomly. Or, more formally, it’s “a sequence of independent, identically distributed (IID) random variables”. In other words, the terms ''random sample'' and ''IID'' are basically one and the same. In statistics, we usually say “random sample,” but in probability it’s more common to say “IID.” * Identically Distributed means that there are no overall trends–the distribution doesn’t fluctuate and all items in the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arg Min
In mathematics, the arguments of the maxima (abbreviated arg max or argmax) are the points, or elements, of the domain of some function at which the function values are maximized.For clarity, we refer to the input (''x'') as ''points'' and the output (''y'') as ''values;'' compare critical point and critical value. In contrast to global maxima, which refers to the largest ''outputs'' of a function, arg max refers to the ''inputs'', or arguments, at which the function outputs are as large as possible. Definition Given an arbitrary set a totally ordered set and a function, the \operatorname over some subset S of X is defined by :\operatorname_S f := \underset\, f(x) := \. If S = X or S is clear from the context, then S is often left out, as in \underset\, f(x) := \. In other words, \operatorname is the set of points x for which f(x) attains the function's largest value (if it exists). \operatorname may be the empty set, a singleton, or contain multiple elements. In th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
