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| {{More footnotes|date=October 2010}}
| | They are typically a free website that are pre-designed for enabling businesses of every size in marking the presence on the internet and allows them in showcasing the product services and range through images, contents and various other elements. Online available for hiring are most qualified, well knowledgeable and talented Wordpress developer India from offshore Wordpress development services company. Step-4 Testing: It is the foremost important of your Plugin development process. If you need a special plugin for your website , there are thousands of plugins that can be used to meet those needs. By using this method one can see whether the theme has the potential to become popular or not and is their any scope of improvement in the theme. <br><br>The Internet is a vast open market where businesses and consumers congregate. Wordpress have every reason with it which promote wordpress development. When you loved this information and you would like to receive details about [http://undergroundsoles.com/2012/09/what-kicks-they-wear-jordan-brand-hq/ wordpress dropbox backup] i implore you to visit the web-page. It sorts the results of a search according to category, tags and comments. So if you want to create blogs or have a website for your business or for personal reasons, you can take advantage of free Word - Press installation to get started. Akismet is really a sophisticated junk e-mail blocker and it's also very useful thinking about I recieve many junk e-mail comments day-to-day across my various web-sites. <br><br>Saying that, despite the launch of Wordpress Express many months ago, there has still been no sign of a Wordpress video tutorial on offer UNTIL NOW. Note: at a first glance WP Mobile Pro themes do not appear to be glamorous or fancy. I've applied numerous Search engine optimization-ready Word - Press themes and I can say from knowledge that I consider the Genesis Search engine marketing panel one particular of the simplest to use. In crux the developer must have a detailed knowledge not only about the marketing tool but also about the ways in which it can be applied profitably. Search engine optimization pleasant picture and solution links suggest you will have a much better adjust at gaining considerable natural site visitors. <br><br>Additionally Word - Press add a default theme named Twenty Fourteen. And, that is all the opposition events with nationalistic agenda in favor of the individuals of Pakistan marching collectively in the battle in opposition to radicalism. Specialty about our themes are that they are easy to load, compatible with latest wordpress version and are also SEO friendly. The company gains commission from the customers' payment. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white. <br><br>He loves sharing information regarding wordpress, Majento, Drupal and Joomla development tips & tricks. As a website owner, you can easily manage CMS-based website in a pretty easy and convenient style. In simple words, this step can be interpreted as the planning phase of entire PSD to wordpress conversion process. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. Customers within a few seconds after visiting a site form their opinion about the site. |
| 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 in to determine the appropriate [[Insurance premium|premium]] for a group of insurance contracts. The model is named after Hans Bühlmann who first published a description in 1967.<ref>{{cite journal | last = Bühlmann | first = Hans | year = 1967 | title = Experience rating and credibility | url = http://www.casact.org/library/astin/vol4no3/199.pdf | publisher = ASTIN Bulletin | volume = 4 | issue = 3 | pages = 99–207}}</ref>
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| ==Model description==
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| 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
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| * ''X''<sub>ij</sub> for the ''j''-th claim on the ''i''-th risk (we assume that all claims for ''i''-th risk are [[independent and identically distributed]])
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| * <math>\scriptstyle \bar{X}_i=\frac{1}{m}\sum_{j=1}^{m}X_{ij}</math> for the average value.
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| * <math>\Theta_i</math> - the parameter for the distribution of the i-th risk
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| * <math>m(\vartheta)= \operatorname E\left [ X_{ij} |\Theta_i = \vartheta\right ]</math>
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| * <math>\Pi=\operatorname E(m(\vartheta)|X_{i1},X_{i2},...X_{im})</math> - premium for the i-th risk
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| * <math>\mu = \operatorname (m(\vartheta))</math>
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| * <math>s^2(\vartheta)=\operatorname{Var}\left [ X_{ij} |\Theta_i = \vartheta\right ]</math>
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| * <math>\sigma^2=\operatorname E\left [ s^2(\vartheta) \right ]</math>
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| * <math>v^2=\operatorname{Var}\left [ m(\vartheta) \right ]</math>
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| Note: <math>m(\vartheta)</math> and <math>s^2(\vartheta)</math> are functions of random parameter <math>\vartheta</math>
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| The Bühlmann model is the solution for the problem: | |
| :<math>\underset{a_{i0},a_{i1},...,a_{im}}{\operatorname{arg\,min}} \operatorname E\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi \right)^2\right ]</math>
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| where <math>a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}</math> is the estimator of premium <math>\Pi</math> and [[arg min]] represents the parameter values which minimize the expression. | |
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| ==Model solution==
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| The solution for the problem is:
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| <math>Z\bar{X}_i+(1-Z)\mu</math>
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| where:
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| <math>Z=\frac{1}{1+\frac{\sigma^2}{v^2m}}</math>
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| We can give this result the interpretation, that Z part of the premium is based on the information that we have about the specific risk, and (1-Z) part is based on the information that we have about the whole population.
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| ===Proof===
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| The following proof is slightly different from the one in the original paper. It is also more general, because it considers all linear estimators, while original proof considers only estimators based on average claim. Proof can be found on this site.<ref>url = http://www.math.ku.dk/~schmidli/rt.pdf</ref>
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| Lemma:
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| The problem can be stated alternatively as:
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| <math>f=\operatorname E\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-m(\vartheta)\right )^2\right ]\rightarrow max</math>
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| Proof:
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| <math>\operatorname E\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-m(\vartheta)\right )^2\right ]</math>
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| <math>=\operatorname E\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right )^2\right ]+\operatorname E\left [ \left ( m(\vartheta)-\Pi\right )^2\right ]+2E\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right ) \left ( m(\vartheta)-\Pi\right )\right ]</math>
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| <math>=\operatorname E\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right )^2\right ]+\operatorname E\left [ \left ( m(\vartheta)-\Pi\right )^2\right ]</math>
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| The last equation follows from the fact that
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| <math>\operatorname E\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right ) \left ( m(\vartheta)-\Pi\right )\right ]</math>
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| <math>\operatorname E_{\Theta}\left \{ E_X\left [ \left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right ) \left ( m(\vartheta)-\Pi\right ) | X_{i1},X_{i2},X_{im}\right ]\right \}</math>
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| <math>=\left ( a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-\Pi\right )\operatorname E_{\Theta}\left \{ \operatorname E_X\left [ \left ( m(\vartheta)-\Pi\right ) | X_{i1},X_{i2},X_{im}\right ]\right \}=0</math> | |
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| We are using here the law of total expectation and the fact, that <math>\Pi=\operatorname E(m(\vartheta)|X_{i1},X_{i2},...X_{im})</math>
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| In our previous equation, we decompose minimized function in the sum of two expressions. The second expression does not depend on parameters used in minimization. Therefore, minimizing the function is the same as minimizing the first part of the sum.
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| Let us find critical points of the function
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| <math>\frac{1}{2}\frac{\partial f}{\partial a_{01}}=</math>
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| <math>\operatorname E\left [ a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}-m(\vartheta)\right ]=a_{i0}+\sum_{j=1}^{m}a_{ij}\operatorname E(X_{ij})-\operatorname E(m(\vartheta))=a_{i0}-\left (\sum_{j=1}^{m}a_{ij}-1 \right )\mu</math>
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| <math>a_{i0}=\left (\sum_{j=1}^{m}a_{ij}-1 \right )\mu</math>
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| For <math>k\neq 0</math> we have:
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| <math>\frac{1}{2}\frac{\partial f}{\partial a_{ik}}=\operatorname E\left [ X_{ik}\left ( a_{i0} +\sum_{j=1}^{m}a_{ij}X_{ij}-m(\vartheta)\right ) \right ]</math>
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| <math>=\operatorname E\left [ X_{ik} \right ]a_{i0}+\sum_{j=1, j\neq k}^{m}a_{ij}\operatorname E[X_{ik}X_{ij}]+a_{ik}\operatorname E[X^2_{ik}]-\operatorname E[X_{ik}m(\vartheta)]=0</math>
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| We can simplify derivative, noting that:
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| <math>\operatorname E[X_{ij}X_{ik}]=\operatorname E[\operatorname E[X_{ij}X_{ik}|\vartheta]]=\operatorname E[cov(X_{ij}X_{ik}|\vartheta)+\operatorname E(X_{ij}|\vartheta)\operatorname E(X_{ik}|\vartheta)]=\operatorname E[(m(\vartheta))^2]=v^2+\mu^2</math>
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| and | |
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| <math>\operatorname E[X^2_{ik}]=\operatorname E[\operatorname E[X^2_{ik}|\vartheta]]=\operatorname E[s^2(\vartheta)+(m(\vartheta))^2]=\sigma^2+v^2+\mu^2</math>
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| and
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| <math>\operatorname E[X_{ik}m(\vartheta)]=\operatorname E[\operatorname E[X_{ik}m(\vartheta)|\Theta_i]=\operatorname E[(m(\vartheta))^2]=v^2+\mu^2</math>
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| Taking above equations and inserting into derivative, we have:
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| <math>\frac{1}{2}\frac{\partial f}{\partial a_{ik}}</math>
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| <math>=\left ( 1-\sum_{j=1}^{m}a_{ij} \right )\mu^2+\sum_{j=1,j\neq k}^{m}a_{ij}(v^2+\mu^2)+a_{ik}(\sigma^2+v^2+\mu^2)-(v^2+\mu^2)=a_{ik}\sigma^2-\left ( 1-\sum_{j=1}^{m}a_{ij} \right )v^2=0</math> | |
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| <math>\sigma^2a_{ik}=v^2\left (1- \sum_{j=1}^{m} a_{ij}\right)</math>
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| Right side doesn't depend on k. Therefore all <math>a_{ik}</math> are constant
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| <math>a_{i1}=a_{i2}=...=a_{im}=\frac{v^2}{\sigma^2+mv^2}</math>
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| From the solution for <math>a_{i0}</math> we have
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| <math>a_{i0}=(1-ma_{ik})\mu=\left ( 1-\frac{mv^2}{\sigma^2+mv^2} \right )\mu</math>
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| Finally, the best estimator is
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| <math>a_{i0}+\sum_{j=1}^{m}a_{ij}X_{ij}=\frac{mv^2}{\sigma^2+mv^2}\bar{X_i}+\left ( 1-\frac{mv^2}{\sigma^2+mv^2} \right )\mu=Z\bar{X_i}+(1-Z)\mu</math>
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| ==References==
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| <references />
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| *Frees, E.W., Young, V.R., Luo, Y., (1999) "A longitudinal data analysis interpretation of credibility models", ''Insurance: Mathematics and Economics'', 24 (3), 229–247. {{DOI| 10.1016/S0167-6687(98)00055-9}}
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| {{Stochastic processes}}
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| [[Category:Actuarial science]]
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| [[Category:Analysis of variance]]
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| {{DEFAULTSORT:Buhlmann Model}}
| |
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