Asymptotic consistency and inconsistency of the chain ladder
The distribution-free chain ladder reserving method belongs to the most frequently used approaches in general insurance. It is well known, see Mack (1993), that the estimators f̂j of the development factors are unbiased and mutually uncorrelated under some mild conditions on the mean structure and u...
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| Veröffentlicht in: | Insurance, mathematics & economics mathematics & economics, 2012-09, Vol.51 (2), p.472-479 |
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| creator | Pesta, Michal Hudecová, Sárka |
| description | The distribution-free chain ladder reserving method belongs to the most frequently used approaches in general insurance. It is well known, see Mack (1993), that the estimators f̂j of the development factors are unbiased and mutually uncorrelated under some mild conditions on the mean structure and under the assumption of independence of the claims in different accident years. In this article we deal with some asymptotic properties of f̂j. Necessary and sufficient conditions for asymptotic consistency of the estimators of true development factors fj are provided. A rate of convergence for the consistency is derived. Possible violation of these conditions and its consequences are discussed, and some practical recommendations are given. Numerical simulations and a real data example are provided as well.
► Consistency of the distribution-free chain ladder method is examined. ► Unbiasedness of the development factors can be of less importance than consistency. ► A necessary and sufficient condition for the conditional consistency is derived. ► Mean square error of the development factors’ estimates is calculated. ► Real data example and numerical simulations are shown. |
| doi_str_mv | 10.1016/j.insmatheco.2012.07.004 |
| format | Article |
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► Consistency of the distribution-free chain ladder method is examined. ► Unbiasedness of the development factors can be of less importance than consistency. ► A necessary and sufficient condition for the conditional consistency is derived. ► Mean square error of the development factors’ estimates is calculated. ► Real data example and numerical simulations are shown.</description><identifier>ISSN: 0167-6687</identifier><identifier>EISSN: 1873-5959</identifier><identifier>DOI: 10.1016/j.insmatheco.2012.07.004</identifier><identifier>CODEN: IMECDX</identifier><language>eng</language><publisher>Amsterdam: Elsevier B.V</publisher><subject>Asymptotic methods ; Asymptotic properties ; Chain ladder ; Claims reserving ; Development factors ; Development financing ; Estimating techniques ; Estimation ; IM10 ; IM20 ; Insurance ; Insurance claims ; Mathematical economics ; Simulation ; Studies</subject><ispartof>Insurance, mathematics & economics, 2012-09, Vol.51 (2), p.472-479</ispartof><rights>2012 Elsevier B.V.</rights><rights>Copyright Elsevier Sequoia S.A. Sep 2012</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c443t-5d3b0db79459d0d7f1782c8175f53fd0cd866ecfdd4366fee5f7439ca3f15a263</citedby><cites>FETCH-LOGICAL-c443t-5d3b0db79459d0d7f1782c8175f53fd0cd866ecfdd4366fee5f7439ca3f15a263</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.insmatheco.2012.07.004$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids></links><search><creatorcontrib>Pesta, Michal</creatorcontrib><creatorcontrib>Hudecová, Sárka</creatorcontrib><title>Asymptotic consistency and inconsistency of the chain ladder</title><title>Insurance, mathematics & economics</title><description>The distribution-free chain ladder reserving method belongs to the most frequently used approaches in general insurance. It is well known, see Mack (1993), that the estimators f̂j of the development factors are unbiased and mutually uncorrelated under some mild conditions on the mean structure and under the assumption of independence of the claims in different accident years. In this article we deal with some asymptotic properties of f̂j. Necessary and sufficient conditions for asymptotic consistency of the estimators of true development factors fj are provided. A rate of convergence for the consistency is derived. Possible violation of these conditions and its consequences are discussed, and some practical recommendations are given. Numerical simulations and a real data example are provided as well.
► Consistency of the distribution-free chain ladder method is examined. ► Unbiasedness of the development factors can be of less importance than consistency. ► A necessary and sufficient condition for the conditional consistency is derived. ► Mean square error of the development factors’ estimates is calculated. ► Real data example and numerical simulations are shown.</description><subject>Asymptotic methods</subject><subject>Asymptotic properties</subject><subject>Chain ladder</subject><subject>Claims reserving</subject><subject>Development factors</subject><subject>Development financing</subject><subject>Estimating techniques</subject><subject>Estimation</subject><subject>IM10</subject><subject>IM20</subject><subject>Insurance</subject><subject>Insurance claims</subject><subject>Mathematical economics</subject><subject>Simulation</subject><subject>Studies</subject><issn>0167-6687</issn><issn>1873-5959</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2012</creationdate><recordtype>article</recordtype><recordid>eNqFkMtKAzEUhoMoWKvvMODGzYzJ5DrgphZvUHCj65DmQjPMJDWZCn17UyooblwdOHz_uXwAVAg2CCJ22zc-5FFNG6tj00LUNpA3EJITMEOC45p2tDsFs4LymjHBz8FFzj2EEHWMz8DdIu_H7RQnrysdQ_Z5skHvKxVM5cPvTnRVWVLpjfKhGpQxNl2CM6eGbK--6xy8Pz68LZ_r1evTy3KxqjUheKqpwWto1rwjtDPQcIe4aLVAnDqKnYHaCMasdsYQzJizljpOcKcVdoiqluE5uDnO3ab4sbN5kqPP2g6DCjbuskQQiw4iQWhBr_-gfdylUK47UIRSBvGBEkdKp5hzsk5ukx9V2hdIHrTKXv5olQetEnJZtJbo_TFqy8Of3iaZtS-CrPHJ6kma6P8f8gUCn4Xf</recordid><startdate>20120901</startdate><enddate>20120901</enddate><creator>Pesta, Michal</creator><creator>Hudecová, Sárka</creator><general>Elsevier B.V</general><general>Elsevier Sequoia S.A</general><scope>AAYXX</scope><scope>CITATION</scope><scope>8BJ</scope><scope>FQK</scope><scope>JBE</scope><scope>JQ2</scope></search><sort><creationdate>20120901</creationdate><title>Asymptotic consistency and inconsistency of the chain ladder</title><author>Pesta, Michal ; Hudecová, Sárka</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c443t-5d3b0db79459d0d7f1782c8175f53fd0cd866ecfdd4366fee5f7439ca3f15a263</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2012</creationdate><topic>Asymptotic methods</topic><topic>Asymptotic properties</topic><topic>Chain ladder</topic><topic>Claims reserving</topic><topic>Development factors</topic><topic>Development financing</topic><topic>Estimating techniques</topic><topic>Estimation</topic><topic>IM10</topic><topic>IM20</topic><topic>Insurance</topic><topic>Insurance claims</topic><topic>Mathematical economics</topic><topic>Simulation</topic><topic>Studies</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Pesta, Michal</creatorcontrib><creatorcontrib>Hudecová, Sárka</creatorcontrib><collection>CrossRef</collection><collection>International Bibliography of the Social Sciences (IBSS)</collection><collection>International Bibliography of the Social Sciences</collection><collection>International Bibliography of the Social Sciences</collection><collection>ProQuest Computer Science Collection</collection><jtitle>Insurance, mathematics & economics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Pesta, Michal</au><au>Hudecová, Sárka</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Asymptotic consistency and inconsistency of the chain ladder</atitle><jtitle>Insurance, mathematics & economics</jtitle><date>2012-09-01</date><risdate>2012</risdate><volume>51</volume><issue>2</issue><spage>472</spage><epage>479</epage><pages>472-479</pages><issn>0167-6687</issn><eissn>1873-5959</eissn><coden>IMECDX</coden><abstract>The distribution-free chain ladder reserving method belongs to the most frequently used approaches in general insurance. It is well known, see Mack (1993), that the estimators f̂j of the development factors are unbiased and mutually uncorrelated under some mild conditions on the mean structure and under the assumption of independence of the claims in different accident years. In this article we deal with some asymptotic properties of f̂j. Necessary and sufficient conditions for asymptotic consistency of the estimators of true development factors fj are provided. A rate of convergence for the consistency is derived. Possible violation of these conditions and its consequences are discussed, and some practical recommendations are given. Numerical simulations and a real data example are provided as well.
► Consistency of the distribution-free chain ladder method is examined. ► Unbiasedness of the development factors can be of less importance than consistency. ► A necessary and sufficient condition for the conditional consistency is derived. ► Mean square error of the development factors’ estimates is calculated. ► Real data example and numerical simulations are shown.</abstract><cop>Amsterdam</cop><pub>Elsevier B.V</pub><doi>10.1016/j.insmatheco.2012.07.004</doi><tpages>8</tpages></addata></record> |
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| subjects | Asymptotic methods Asymptotic properties Chain ladder Claims reserving Development factors Development financing Estimating techniques Estimation IM10 IM20 Insurance Insurance claims Mathematical economics Simulation Studies |
| title | Asymptotic consistency and inconsistency of the chain ladder |
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