[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-399165-105":59,"doc-detail-399165-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","cas-exam-mas-ii-tables-06252018","CAS Exam MAS-II Tables - 06/25/2018","","Tables for CAS Exam MAS-II include statistical test tables and distribution function definitions used to write and calculate exam questions. The document standardizes AIC and BIC formulae for consistent scoring, using specified maximum-likelihood log-likelihood terms and parameter/observation counts. It also provides notation for linear mixed models, defining fixed and random effects, their design matrices, and assumed multivariate normal structures. Additional appendices outline continuous distribution formulas such as the incomplete gamma/beta functions and several transformed families.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/cas-exam-mas-ii-tables-06252018/399165/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/cas-exam-mas-ii-tables-06252018/399165.png","ImageObject",300,407,{"name":92,"@type":93},"Emma Wilson","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-02","2026-09-27",true,{"@type":102,"interactionType":103,"userInteractionCount":39},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What tables are provided to candidates for Exam MAS-II?","Question",{"text":112,"@type":113},"Candidates receive statistical test tables and a set of distribution functions definition tables used for writing exam questions and computations.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are AIC and BIC defined for calculations on the exam?",{"text":117,"@type":113},"Unless an alternative formula is explicitly given, the document provides specific AIC/BIC expressions using the observed-data log-likelihood at restricted maximum likelihood estimates, plus terms involving the number of parameters and observations.",{"name":119,"@type":110,"acceptedAnswer":120},"What notation does the exam use for mixed models with fixed and random effects?",{"text":121,"@type":113},"The document states that linear mixed model notation is used, with fixed-effect covariates in the design matrix, random-effect covariates collapsed into a random-effects design matrix, random effects assumed multivariate normal with covariance matrix, and residuals also assumed multivariate normal.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},399165,1791264602,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":39,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145,"preview_page_count":4},3848291630094,"https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","Tables for CAS Exam MAS-II  \nVersion date 06/25/2018  \nThe following tables will be provided to the candidate with the exam. The tables on pages 4 through 13 are reprinted with the permission of the Society of Actuaries; the tables on pages 14 through 18 are copyright material of the Casualty Actuarial Society.  \nWe are furnishing a set of tables for statistical tests as well as a set of distribution functions for Exam MAS-II. We do not have a single authoritative textbook for MAS-II. The nomenclature used to describe the distribution functions may vary from one textbook to the next. To avoid confusion on the part of the candidates we will use the formulae, tables and distribution functions definitions that follow when writing exam questions for Exam MAS-II.  \nAkaike Information Criterion (AIC) and Bayesian Information Criterion (BIC)  \nAIC and BIC are two common measures of relative model fit often used as criteria to select among competing models. These topics are covered in a number of the syllabus materials. Although the conceptual treatment is the same in each, the actual AIC and BIC formulae can vary by source.  \nCandidates are expected to understand AIC and BIC as presented in all of the texts. However, if an exam question requires a calculation of AIC and/or BIC, unless an alternative formula is explicitly given, the following will be used:  \n􀜣􀜫􀜥 = −2􀝈 (̂􀣂; 􀢟) + 2 ∗ 􀝌􀜤􀜫􀜥 = −2􀝈 (̂􀣂; 􀢟) + ln(􀝊) ∗ 􀝌  \nwhere:  \n􀝈 (̂􀣂; 􀢟) is the log-likelihood of the observed data, 􀢟 , under the model of interest at the (restricted) maximum likelihood estimates of the parameters, ̂􀣂 .  \n􀝌 is the number of parameters in the model (fixed effects + covariance parameters) .  \n􀝊 is the number of observations in the modeled dataset.  \nMixed Model Notation  \nTo specify models that may include both fixed and random effects, notation consistent with Linear Mixed Models: A Practical Guide Using Statistical Software will be employed.  \nSpecifically, for the 􀝅􀯧ℎsubject with 􀝊􀯜 repeated observations in a model with 􀝌 fixed effect covariates and 􀝍 random effects the model will be specified as follows:  \n􀢅􀢏 = 􀢄􀢏 􀪺 + 􀜈􀜑 􀢛 􀢏 + 􀣕􀢏  \n􀢛 􀢏 ~ N(􀫙, 􀡰)  \n􀣕􀢏 ~ N(􀫙, 􀡾 􀢏)  \nWhere  \n􀜻1􀯜  \n􀢅􀢏 = (  \n􀜻2􀯜  \n⋮  \n) ; 􀢄􀢏 =  \n􀜻􀯡 􀳔􀯜  \n􀜺1(1􀯜) 􀜺1(2􀯜)  \n􀜺2(1􀯜) 􀜺2(2􀯜)(􀜺􀯡1􀳔􀯜) 􀜺􀯡2􀳔􀯜)  \n⋯ 􀜺1(􀯣􀯜)  \n⋯ 􀜺2(􀯣􀯜)  \n 􀜺􀯡􀯣􀳔􀯜))  \n; 􀢼 = (􀟚􀟚􀟚⋮12􀯣)  \n􀢆􀢏 =  \n􀜼1( 1􀯜) 􀜼1(2􀯜) ⋯ 􀜼1(􀯤􀯜)  \n􀜼2( 1􀯜) 􀜼2(2􀯜) ⋯ 􀜼2(􀯤􀯜)(􀜼􀯡1􀳔􀯜) 􀜺􀯡2􀳔􀯜)  􀜼􀯡􀯤􀳔􀯜))  \n; 􀢛 􀢏 = (􀝑􀝑􀝑􀯜􀯜􀯜 )  \n; 􀣕􀢏 = (􀟳􀟳􀟳12􀯡⋮ 􀯜􀯜􀳔􀯜 )  \nWith the notation above, the fixed effect covariates for the 􀝅 􀯧ℎ subject are contained in the 􀢄􀢏 design matrix, the fixed effects are represented by the 􀢼 vector, the random effect covariates for the 􀝅 􀯧ℎ subject are collapsed into the 􀢆􀢏 design matrix, the random effects are stored in the 􀢛 􀢏 vector which is assumed to be multivariate normal with covariance matrix 􀡰 , and the 􀝊 􀯜 residuals for the 􀝅 􀯧ℎ subject are represented by 􀣕􀢏 vector which is also assumed multivariate normal with covariance matrix 􀡾􀢏 .  \nExcerpts from the Appendices to Loss Models: From Data to  \nDecisions, 2nd edition  \nApril 21, 2005  \nAppendix A  \nAn Inventory of Continuous Distributions  \nA.1 Introduction  \nThe incomplete gamma function is given by  \nΓ (α; x) = Γ~~ ~~1(α) Z0 x tα−1e−t dt, α > 0, x > 0 with Γ (α) = Z0 ∞ tα−1e−t dt, α > 0.  \nAlso, define  \nG (α; x) = Zx∞ tα−1e −t dt, x > 0.  \nIntegration by parts produces the relationship  \nG (α; x) = − xαeα−x + 1αG (α +1; x)  \nFor negative α, this can repeated until the first argument is postive, say at α + k. Then the incomplete gamma function can be evaluated from  \nG (α + k; x) = Γ(α + k)[1 − Γ(α + k; x)] .  \nThe incomplete beta function is given by  \nβ (a, b; x) = ΓΓ((aa)(bb)) Z0 x ta−1(1 − t)b−1 dt, a > 0, b > 0 , 0 \u003C x \u003C 1.  \nAPPENDIX A. AN INVENTORY OF CONTINUOUS DISTRIBUTIONS  \nA.2 Transformed beta family  \nA.2.3 Three-parameter distributions  \nA.2.3.1 Generalized Pareto (beta of the second kind)–α, θ , τ  \nΓ (α + τ) θα xτ −1   x   \nf (x) = F (x) = β(τ α; u), u =Γ (α)Γ (τ) (x + θ)α+τ , x + θ  \n","cbCaiaOOofq6eK1f","https://ap.wps.com/l/cbCaiaOOofq6eK1f","pdf",627276,18,"English","# Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC)\n# Mixed Model Notation\n# Excerpts from Appendices to Loss Models: From Data to Decisions\n## An Inventory of Continuous Distributions\n## Transformed beta family\n## Burr (Burr Type XII, Singh-Maddala)\n## Inverse Burr (Dagum)\n## Two-parameter distributions\n## Pareto and Inverse Pareto","[{\"question\":\"What tables are provided to candidates for Exam MAS-II?\",\"answer\":\"Candidates receive statistical test tables and a set of distribution functions definition tables used for writing exam questions and computations.\"},{\"question\":\"How are AIC and BIC defined for calculations on the exam?\",\"answer\":\"Unless an alternative formula is explicitly given, the document provides specific AIC/BIC expressions using the observed-data log-likelihood at restricted maximum likelihood estimates, plus terms involving the number of parameters and observations.\"},{\"question\":\"What notation does the exam use for mixed models with fixed and random effects?\",\"answer\":\"The document states that linear mixed model notation is used, with fixed-effect covariates in the design matrix, random-effect covariates collapsed into a random-effects design matrix, random effects assumed multivariate normal with covariance matrix, and residuals also assumed multivariate normal.\"}]","CAS Exam MAS-II Tables - 06/25/2018 | PDF",1790494103,45]