International Encyclopedia of Statistical Sciences PDF
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A of λ, the greater the amount of shrinkage. The quadratic AbsolutePenaltyEstimation penalty term makes βˆridge a linear function of y. Frank EjazS.Ahmed,EnayeturRaheem,Shakhawat and Friedman () introduced bridge regression, a Hossain generalized version of penalty (or absolute penalty type) ProfessorandDepartmentHeadofMathematicsand estimation,whichincludesridgeregressionwhenγ=.For agivenpenaltyfunctionπ(⋅)andregularizationparameter Statistics λ,thegeneralformcanbewrittenas UniversityofWindsor,Windsor,ON,Canada UniversityofWindsor,Windsor,ON,Canada ϕ(β)=(y−Xβ)T(y−Xβ)+λπ(β), In statistics, the technique of (cid:55)least squares is used for wherethepenaltyfunctionisoftheform estimating the unknown parameters in a linear regres- sionmodel(see(cid:55)LinearRegressionModels).Thismethod p minimizes the sum of squared distances between the π(β)=∑∣βj∣γ, γ>. () j= observedresponsesinasetofdata,andthefittedresponses aftrinoodnmxotifhd=eart(aexgi{r,yeisx,siixo,in}..ni=m.,oxodinpe)nlT.uSiusnpiatpsv,oewscehtowerreeoofybipssreaerrdveiecrteaosrpcsoo.lnIlteseciss- tpThhaereatmupneenitneagrltsypianfrutanhmcetegitoeivnretnihnamt(oc)doenbltorauosnls∑dtsmjh=eth∣aβemj∣Lγoγu≤nntto,orwmfhsehorrfeintthkies- convenienttowritethemodelinmatrixnotation,as, age. We see that for γ = , we obtain ridge regression. y= Xβ+ε, () However,ifγ ≠ ,thepenaltyfunctionwillnotberota- tionally invariant. Interestingly, for γ < , it shrinks the where y is n× vector of responses, X is n×p matrix, coefficienttowardzero,anddependingonthevalueofλ,it knownasthedesignmatrix, β = (β,β,...,βp)T isthe setssomeofthemtobeexactlyzero.Thus,theprocedure unknownparametervectorandεisthevectorofrandom combinesvariableselectionandshrinkageofcoefficientsof errors.Inordinaryleastsquares(OLS)regression,weesti- penalizedregression.Animportantmemberofthepenal- mateβbyminimizingtheresidualsumofsquares,RSS = ized least squares (PLS) family is the L penalized least (y−Xβ)T(y− Xβ),givingβˆ =(XTX)−XTy.Thisesti- squaresestimatororthelasso[leastabsoluteshrinkageand OLS matorissimpleandhassomegoodstatisticalproperties. selectionoperator,Tibshirani()].Inotherwords,the However, the estimator suffers from lack of uniqueness absolutepenaltyestimator(APE)ariseswhentheabsolute if the design matrix X is less than full rank, and if the valueofpenaltytermisconsidered,i.e.,γ=in().Similar columnsofXare(nearly)collinear.Toachievebetterpre- totheridgeregression,thelassoestimatesareobtainedas dictionandtoalleviateillconditioningproblemofXTX, H(cid:55)mRoizeiedrslgaethnaednRKdSeSrSnusarurrbdojge(acttetoR)iaidngctoeronRdstuergcareiendstsr,iio∑dngβes)jr,ew≤grhetis,csihinonmot(ihsneeier- βˆlasso =argmβ in⎧⎪⎪⎨⎪⎪⎩∑i=n(yi−β−∑j=pxijβj)+λ∑j=p∣βj∣⎫⎪⎪⎬⎪⎪⎭. () words The lasso shrinks the OLS estimator toward zero and βˆridge =argβmin⎧⎪⎪⎨⎪⎪⎩∑iN=(yi−β−∑j=pxijβj)+λ∑j=pβj⎫⎪⎪⎬⎪⎪⎭, cdieepnetnsdtoinegxaocntlytzheero.vTaliubeshiorfanλi,(its)eutssedsoamqeuacdoreaffitic- () programming method to solve () for βˆlasso. Later, where λ≥ is known as the complexity parameter that Efron et al. () proposed least angle regression controls the amount of shrinkage. The larger the value (LAR), a type of stepwise regression, with which the MiodragLovric(ed.),InternationalEncyclopediaofStatisticalScience,DOI./----, ©Springer-VerlagBerlinHeidelberg A AbsolutePenaltyEstimation lasso estimates can be obtained at the same compu- wherez+=max(,z).ThePSEisparticularlyimportantto tational cost as that of an ordinary least squares esti- controltheover-shrinkinginherentin βˆS.Theshrinkage mation Hastie et al. (). Further, the lasso esti- estimators can be viewed as a competitor to the APE mator remains numerically feasible for dimensions m approach. Ahmed et al. () finds that, when p is that are much higher than the sample size n. Zou and relatively small with respect to p, APE performs bet- Hastie () introduced a hybrid PLS regression with ter than the shrinkage method. On the other hand, the the so called elastic net penalty defined as λ∑pj=(αβj+ shrinkagemethodperformsbetterwhenpislarge,which (−α) ∣βj∣). Here the penalty function is a linear com- is consistent with the performance of the APE in linear bination of the ridge regression penalty function and models.Importantly,theshrinkageapproachisfreefrom lasso penalty function. A different type of PLS, called anytuningparameters,easytocomputeandcalculations garotte is due to Breiman (). Further, PLS estima- arenotiterative.Theshrinkageestimationstrategycanbe tionprovidesageneralizationofbothnonparametricleast extendedinvariousdirectionstomorecomplexproblems. squaresandweightedprojectionestimators,andapopu- Itmaybeworthmentioningthatthisisoneofthetwoareas larversionofthePLSisgivenbyTikhonovregularization BradleyEfronpredictedfortheearlytwenty-firstcentury (Tikhonov ). Generally speaking, the ridge regres- (RSSNews,January).Shrinkageandlikelihood-based sion is highly efficient and stable when there are many methodscontinuetobeextremelyusefultoolsforefficient small coefficients. The performance of lasso is superior estimation. whenthereareasmall-to-mediumnumberofmoderate- sized coefficients. On the other hand, shrinkage esti- mators perform well when there are large known zero AbouttheAuthor coefficients. TheauthorS.EjazAhmedisProfessorandHeadDepart- Ahmed et al. () proposed an APE for partially ment of Mathematics and Statistics. For biography, see linearmodels.Further,theyreappraisedthepropertiesof entry(cid:55)OptimalShrinkageEstimation. shrinkageestimatorsbasedonStein-ruleestimation.There exists a whole family of estimators that are better than OLSestimatorsinregressionmodelswhenthenumberof CrossReferences predictors is large. A partially linear regression model is (cid:55)Estimation definedas (cid:55)Estimation:AnOverview yi =xTi β+g(ti)+εi, i=,...,n, () (cid:55)James-SteinEstimator (cid:55)LinearRegressionModels where ti ∈ [,] are design points, g(⋅) is an unknown (cid:55)OptimalShrinkageEstimation real-valuedfunctiondefinedon[,],andyi,x,β,andεi’s (cid:55)Residuals are as defined in the context of (). We consider experi- (cid:55)RidgeandSurrogateRidgeRegressions mentswherethevectorofcoefficientsβinthelinearpart (cid:55)SemiparametricRegressionModels of () can be partitioned as (βT,βT)T, where β is the coefficientvectoroforderp ×formaineffects(e.g.,treat- ment effects, genetic effects) and β is a vector of order ReferencesandFurtherReading p × for “nuisance” effects (e.g., age, laboratory). Our AhmedSE,DoksumKA,HossainS,YouJ()Shrinkage,pretest relevant hypothesis is H : β = . Let βˆ be a semi- andabsolutepenaltyestimatorsinpartiallylinearmodels.Aust parametric least squares estimator of β , and we let β˜ NZJStat():– denotetherestrictedsemiparametricleastsquaresestima- Breiman L () Better subset selection using the non-negative tor of β . Then the semiparametric Stein-type estimator garotte.Technicalreport,UniversityofCalifornia,Berkeley Efron B, Hastie T, Johnstone I, Tibshirani R () Least angle (see(cid:55)James-SteinEstimatorandSemiparametricRegres- regression(withdiscussion).AnnStat():– sionModels),βˆS,ofβ is FrankIE,FriedmanJH()Astatisticalviewofsomechemomet- ricsregressiontools.Technometrics:– βˆS =β˜ +{−(p −)T−}(βˆ −β˜ ), p ≥ () HastieT,TibshiraniR,FriedmanJ()Theelementsofstatisti- callearning:datamining,inference,andprediction,ndedn. where T is an appropriate test statistic for the H . Springer,NewYork Apositive-ruleshrinkageestimator(PSE)βˆS+isdefinedas HoerlAE,KennardRW()Ridgeregression:biasedestimation fornonorthogonalproblems.Technometrics:– TibshiraniR()Regressionshrinkageandselectionviathelasso. βˆS+=β˜+{−(p−)T−}+(βˆ−β˜), p ≥ () JRStatSocB:– A AcceleratedLifetimeTesting Tikhonov An () Solution of incorrectly formulated problems by a lifetime distribution, such as exponential, Weibull, A and the regularization method. Soviet Math Dokl :– log-normal, log-logistic, among others. The other is a , English translation of Dokl Akad Nauk SSSR , , stress-responserelationship(SRR),whichrelatesthemean – lifetime (or a function of this parameter) with the stress ZouH,HastieT()Regularizationandvariableselctionviathe levels. Common SRRs are the power law, Eyring and elasticnet.JRStatSocB():– Arrhenius models (Meeker and Escobar ) or even a generallog-linearorlog-non-linearSRRwhichencompass the formers. For sake of illustration, we shall assume an exponentialdistributionasthelifetimemodelandagen- AcceleratedLifetimeTesting eral log-linear SRR. Here, the mean lifetime under the usual working conditions shall represent our device reli- FranciscoLouzada-Neto abilitymeasureofinteresting. AssociateProfessor Let T > be the lifetime random variable with an UniversidadeFederaldeSãoCarlos,SaoPaulo,Brazil exponentialdensity f(t,λi)=λiexp{−λit}, () Acceleratedlifetests(ALT)areefficientindustrialexperi- where λi > isanunknownparameterrepresentingthe mentsforobtainingmeasuresofadevicereliabilityunder constant failure rate for i = ,...,k (number of stress theusualworkingconditions. levels).Themeanlifetimeisgivenbyθi =/λi. Apracticalproblemforindustriesofdifferentareasis The likelihood function for λi, under the i-th stress to obtain measures of a device reliability under its usual levelXi,isgivenby working conditions. Typically, the time and cost of such ceixepnetrfiomrehnatnatdiloinngarseuclohnsgituanatdioenx,pseinnscieveth.TheinefAorLmTaatrioeneffion- Li(λi)=⎛⎝∏ri f(tij,λi)⎞⎠(S(tiri,λi))ni−ri =λriiexp{−λiAi}, j= the device performance under the usual working condi- tionsareobtainedbyconsideringatimeandcost-reduced where S(tiri,λi) is the survival function at tiri and Ai = experimental scheme. The ALT are performed by test- ∑rj=itij+(ni−ri)tiri denotesthetotaltimeontestforthe ing items at higher stress covariate levels than the usual i-thstresslevel. working conditions, such as temperature, pressure and Considering data under the k random stress levels, voltage. the likelihood function for the parameter vector λ = ThereisalargeliteratureonALTandinterestedread- (λ,λ,...,λk)isgivenby erscanrefertoMannetal.(),Nelson(), Meeker k and Escobar () which are excellent sources for ALT. L(λ)=∏λriiexp{−λiAi}. () Nelson(a,b)providesabriefbackgroundonacceler- i= atedtestingandtestplansandsurveystherelatedliterature Weconsideragenerallog-linearSRRdefinedas pointoutmorethanrelatedreferences. A simple ALT scenario is characterized by putting k λi =exp(−Zi−β−βXi), () groups of ni items each under constant and fixed stress covariatelevels,Xi (hereafterstresslevel),fori = ,...,k, where X is the covariate, Z = g(X) and β and β are wherei=generallydenotestheusualstresslevel,thatis, unknownparameterssuchthat−∞<β ,β <∞. theusualworkingconditions.Theexperimentendsaftera TheSRR()hasseveralmodelsasparticularcases.The certainpre-fixednumberri <nioffailures,ti,ti,...,tiri, ArrheniusmodelisobtainedifZi = ,Xi = /Vi,β=−α at each stress level, characterizing a type II censoring and β = α , where Vi denotes a level of the tempera- scheme (Lawless ; see also (cid:55)Censoring Methodol- turevariable.IfZi = ,Xi = −log(Vi), β = log(α)and ogy).Otherstressschemes,suchasstep(see(cid:55)Step-Stress β = α ,whereVidenotesalevelofthevoltagevariable AcceleratedLifeTests)andprogressiveones,arealsocom- weobtainthepowermodel.FollowingLouzada-Netoand moninpracticebutwillnotbeconsideredhere.Examples Pardo-Fernandéz (), the Eyring model is obtained if ofthosemoresophisticatedstressschemescanbefoundin Zi = −logVi, Xi = /Vi, β = −α and β = α, where Nelson(). Videnotesa level ofthe temperaturevariable. Interested The ALT models are composed by two components. readerscanrefertoMeekerandEscobar()formore One is a probabilistic component, which is represented informationaboutthephysicalmodelsconsideredhere. A AcceleratedLifetimeTesting From()and(),thelikelihoodfunctionforβ andβ Two types of software for ALT are provided by isgivenby Meeker and Escobar () and ReliaSoft Corporation (). k L(β,β)=∏{exp(−Zi−β−βXi)ri AbouttheAuthor i= FranciscoLouzada-NetoisanassociateprofessorofStatis- exp(−exp(−Zi−β−βXi)Ai)}. () tics at Universidade Federal de São Carlos (UFSCar), Brazil.HereceivedhisPh.DinStatisticsfromUniversityof Themaximumlikelihoodestimates(MLEs)ofβ and Oxford(England).HeisDirectoroftheCentreforHazard β can be obtained by direct maximization of (), or by Studies(–,UFSCar,Brazil)andEditorinChiefof solvingthesystemofnonlinearequations,∂logL/∂θ=, the Brazilian Journal of Statistics (–, Brazil). He whereθ′ =(β,β).Obtainingthescorefunctioniscon- isapast-DirectorforUndergraduateStudies(–, ceptuallysimpleandtheexpressionsarenotgivenexplic- UFSCar, Brazil) and was Director for Graduate Studies itly.TheMLEsofθicanbeobtained,inprinciple,straight- inStatistics(–,UFSCar,Brazil).Louzada-Netois forwardly by considering the invariance property of the singleandjointauthorofmorethanpublicationsinsta- MLEs. tisticalpeerreviewedjournals,booksandbookchapters, Large-sample inference for the parameters can be He has supervised more than assistant researches, basedontheMLEsandtheirestimatedvariances,obtained Ph.Ds,mastersandundergraduates. by inverting the expected information matrix (Cox and Hinkley).Forsmallormoderate-sizedsampleshow- CrossReferences everwemayconsidersimulationapproaches,suchasthe (cid:55)DegradationModelsinReliabilityandSurvivalAnalysis bootstrapconfidenceintervals(see(cid:55)BootstrapMethods) (cid:55)ModelingSurvivalData thatarebasedontheempiricalevidenceandaretherefore (cid:55)Step-StressAcceleratedLifeTests preferred(DavisonandHinkley).Formalgoodness- (cid:55)SurvivalData of-fittestsarealsofeasiblesince,from(),wecanusethe likelihoodratiostatistics(LRS)fortestinggoodness-of-fit ofhypothesessuchasH :β =. ReferencesandFurtherReading Although we considered only an exponential dis- BaiDS,ChaMS,ChungSW()Optimumsimpleramptestsfor tribution as our lifetime model, more general lifetime the Weibull distribution and type-I censoring. IEEE T Reliab :– distributions,suchastheWeibull(see(cid:55)WeibullDistribu- Cox DR, Hinkley DV () Theoretical statistics. Chapman and tionandGeneralizedWeibullDistributions),log-normal, Hall,London log-logistic, among others, could be considered in prin- DavisonAC,HinkleyDV()Bootstrapmethodsandtheirappli- ciple. However, the degree of difficulty in the calcula- cation.CambridgeUniversityPress,Cambridge tionsincreaseconsiderably.Alsoweconsideredonlyone Khamis IH () Comparison between constant- and step-stress testsforWeibullmodels.IntJQualReliabManag:– stress covariate, however this is not critical for the over- LawlessJF()Statisticalmodelsandmethodsforlifetimedata, allapproachtoholdandthemultiplecovariatecasecanbe ndend.Wiley,NewYork handlestraightforwardly. Louzada-Neto F, Pardo-Fernandéz JC () The effect of Astudyontheeffectofdifferentreparametrizationson reparametrizationontheaccuracyofinferencesforaccelerated theaccuracyofinferencesforALTisdiscussedinLouzada- lifetimetests.JApplStat:– Mann NR, Schaffer RE, Singpurwalla ND () Methods for Neto and Pardo-Fernandéz ). Modeling ALT with a statistical analysis of reliability and life test data. Wiley, log-non-linearSRRcanbefoundinPerdonáetal.(). NewYork Modeling ALT with a threshold stress, below which the Meeker WQ, Escobar LA () Statistical methods for reliability lifetime of a product can be considered to be infinity or data.Wiley,NewYork muchhigherthanthatforwhichithasbeendevelopedis Meeker WQ, Escobar LA () SPLIDA (S-PLUS Life Data Analysis)software–graphicaluserinterface.http://www.public. proposedbyTojeiroetal.(). iastate.edu/~splida WeonlyconsideredALTinpresenceofconstantstress MillerR,NelsonWB()Optimumsimplestep-stressplansfor loading,howevernon-constantstressloading,suchasstep acceleratedlifetesting.IEEETReliab:– stressandlinearlyincreasingstressareprovidedbyMiller NelsonW()Acceleratedtesting–statisticalmodels,testplans, andNelson()and Bai,ChaandChung(),respec- anddataanalyses.Wiley,NewYork NelsonW(a)Abibliographyofacceleratedtestplans.IEEET tively.Acomparisonbetweenconstantandstepstresstests Reliab:– is provided by Khamis (). A log-logistic step stress NelsonW(b)AbibliographyofacceleratedtestplanspartII– modelisprovidedbySrivastavaandShukla(). references.IEEETReliab:– A AcceptanceSampling PerdonáGSC,LouzadaNetoF,TojeiroCAV()Bayesianmod- Regarding the decision on the batches, we distin- A ellingoflog-non-linearstress-responserelationshipsinaccel- guish three different approaches: () acceptance without eratedlifetimetests.JStatTheoryAppl():– inspection, applied when the supplier is highly reliable; Reliasoft Corporation () Optimum allocations of stress lev- ()%inspection,whichisexpensiveandcanleadtoa els and test units in accelerated tests. Reliab EDGE :–. sloppy attitude towards quality; () an intermediate deci- http://www.reliasoft.com Srivastava PW, Shukla R () A log-logistic step-stress model. sion, i.e., an acceptance sampling program. This increases IEEETReliab:– the interest on quality and leads to the lemma: make TojeiroCAV,LouzadaNetoF,BolfarineH()ABayesiananalysis things right in the first place. The type of inspection that foracceleratedlifetimetestsunderanexponentialpowerlaw shouldbeapplieddependsonthequalityofthelastbatches modelwiththresholdstress.JApplStat():– inspected.Atthebeginningofinspection,aso-callednor- mal inspection is used, but there are two other types of inspection,atightenedinspection(forahistoryoflowqual- ity),andareducedinspection(forahistoryofhighquality). AcceptanceSampling There are special and empirical switching rules between thethreetypesofinspection,aswellasfordiscontinuation M.IvetteGomes ofinspection. Professor UniversidadedeLisboa,DEIOandCEAUL,Lisboa, Portugal FactorsforClassificationsofSampling Introduction Plans Acceptance sampling (AS) is one of the oldest statisti- Samplingplansbyattributesversussamplingplansbyvari- cal techniques in the area of (cid:55)statistical quality control. ables. Iftheiteminspectionleadstoabinaryresult(con- It is performed out of the line production, most com- formingornonconforming),wearedealingwithsampling monlybeforeit,fordecidingonincomingbatches,butalso byattributes,detailedlateron.Iftheiteminspectionleads afterit,forevaluatingthefinalproduct(seeDuncan; toacontinuousmeasurementX,wearesamplingbyvari- Stephens ; Pandey ; Montgomery ; and ables.Then,wegenerallyusesamplingplansbasedonthe Schilling and Neubauer , among others). Accepted sample mean and standard deviation, the so-called vari- batches go into the production line or are sold to ablesamplingplans.IfXisnormal,itiseasytocomputethe consumers; the rejected ones are usually submitted to a numberofitemstobecollectedandthecriteriathatleads rectificationprocess.Asamplingplanisdefinedbythesize totherejectionofthebatch,withchosenrisksαandβ.For of the sample (samples) taken from the batch and by the different sampling plans by variables, see Duncan (), associatedacceptance–rejectioncriterion.Themostwidely amongothers. usedplansaregivenbytheMilitaryStandardtables,devel- Incoming versus outgoing inspection. If the batches are oped during the World War II, and first issued in . inspectedbeforetheproductissenttotheconsumer,itis We mention MIL STD E () and the civil version calledoutgoinginspection.Iftheinspectionisdonebythe ANSI/ASQC Z. () of the American National Stan- consumer (producer), after they were received from the dards Institution and the American Society for Quality supplier,itiscalledincominginspection. Control. At the beginning, all items and products were Rectifyingversusnon-rectifyingsamplingplans. Alldepends inspectedfortheidentificationofnonconformities.Atthe on what is done with nonconforming items that were lates,DodgeandRomig(seeDodgeandRomig), found during the inspection. When the cost of replac- in the Bell Laboratories, developed the area of AS, as an ing faulty items with new ones, or reworking them is alternativeto%inspection.TheaimofASistoleadpro- accountedfor,thesamplingplanisrectifying. ducers to a decision (acceptance or rejection of a batch) Single, double, multiple and sequential sampling and not to the estimation or improvement of the qual- plans. ityofabatch.Consequently,ASdoesnotprovideadirect form of quality control, but its indirect effects in quality ● Singlesampling.Thisisthemostcommonsampling are important: if a batch is rejected, either the supplier plan: we draw a random sample of n items from the triesimprovingitsproductionmethodsortheconsumer batch,andcountthenumberofnonconformingitems (producer)looksforabettersupplier,indirectlyincreasing (orthenumberofnonconformities,ifmorethanone quality. nonconformity is possible on a single item). Such a A AcceptanceSampling planisdefinedbynandbyanassociatedacceptance- with a % inspection. When a pre-specified num- rejectioncriterion,usuallyavaluec,theso-calledaccep- beriofconsecutivenonconformingitemsisachieved, tance number, the number of nonconforming items the plan changes into sampling inspection, with the that cannot be exceeded. If the number of noncon- inspection of f items, randomly selected, along the forming items is greater than c, the batch is rejected; continuousproduction.Ifonenonconformingitemis otherwise,thebatchisaccepted.Thenumberr,defined detected(thereasonfortheterminologyCSP-),% as the minimum number of nonconforming items inspectioncomesagain,andthenonconformingitem leading to the rejection of the batch, is the so-called isreplaced.Forpropertiesofthisplananditsgeneral- rejection number. In the most simple case, as above, izationsseeDuncan(). r=c+,butwecanhaver>c+. ● Doublesampling. A double sampling plan is charac- terizedbyfourparameters:n <<n,thesizeofthefirst AFewCharacteristicsofaSamplingPlan sample,c theacceptancenumberforthefirstsample, OCC.Theoperationalcharacteristiccurve(OCC)isPa ≡ nthesizeofthesecondsampleandc(>c)theaccep- Pa(p) = P(acceptanceofthebatch ∣ p), where p is the tancenumberforthejointsample.Themainadvantage probabilityofanonconformingiteminthebatch. ofadoublesamplingplanisthereductionofthetotal AQL and LTPD (or RQL). The sampling plans are built inspectionandassociatedcost,particularlyifwepro- taken into account the wishes of both the supplier and ceedtoacurtailmentinthesecondsample,i.e.westop the consumer, defining two quality levels for the judg- theinspectionwheneverc isexceeded.Another(psy- ment of the batches: the acceptance quality level (AQL), chological)advantageoftheseplansisthewaytheygive theworstoperatingqualityoftheprocesswhichleadsto asecondopportunitytothebatch. ahighprobabilityofacceptanceofthebatch,usually% ● Multiple sampling. In the multiple plans a pre- –fortheprotectionofthesupplierregardinghighquality determined number of samples are drawn before batches,andthelottolerancepercentdefective(LTPD)or takingadecision. rejectablequalitylevel(RQL),thequalitylevelbelowwhich ● (cid:55)Sequentialsampling.Thesequentialplansareagen- an item cannot be considered acceptable. This leads to a eralizationofmultipleplans.Themaindifferenceisthat smallacceptanceofthebatch,usually%–forthepro- thenumberofsamplesisnotpre-determined.If,ateach tectionoftheconsumeragainstlowqualitybatches.There step,wedrawasampleofsizeone,theplan,basedon existtwotypesofdecision,acceptanceorrejectionofthe Wald’stest,iscalledsequentialitem-to-item;otherwise, batch,andtwotypesofrisks,torejecta“good"(highqual- it is sequential by groups. For a full study of multiple ity)batch,andtoaccepta“bad"(lowquality)batch.The andsequentialplanssee,forinstance,Duncan() probabilitiesofoccurrenceoftheserisksaretheso-called (seealsotheentry(cid:55)SequentialSampling). supplier risk and consumer risk, respectively. In a single Specialsamplingplans.Amongthegreatvarietyofspecial sampling plan, the supplier risk is α = −Pa(AQL) and plans,wedistinguish: the consumer risk is β = Pa(LTPD). The sampling plans shouldtakeintoaccountthespecificationsAQLandLTPD, ● Chainsampling.Whentheinspectionproceduresare i.e.wearesupposedtofindasingleplanwithanOCCthat destructiveorveryexpensive,asmallnisrecommend- passesthroughthepoints(AQL,-α)and(LTPD,β).The able.Wearethenledtoacceptancenumbersequalto constructionofdoubleplanswhichprotectboththesup- zero.Thisisdangerousforthesupplierandifrectifying plierandtheconsumeraremuchmoredifficult,anditis inspectionisused,itisexpensivefortheconsumer.In no longer sufficient to provide indication on two points ,Dodgesuggestedaprocedurealternativetothis of the OCC. There exist the so-called Grubbs’ tables (see typeofplans,whichusesalsotheinformationofpre- Montgomery)providing(c ,c ,n ,n ),forn =n , ceding batches, the so-called chain sampling method as an example, α = ., β = . and several rates (seeDogdgeandRomig). RQL/AQL. ● Continuoussamplingplans(CSP).Therearecontinu- ousproductionprocesses,wheretherawmaterialisnot AOQ, AOQL and ATI. If there is a rectifying inspection naturallyprovidedinbatches.Forthistypeofproduc- program–acorrectiveprogram,basedona%inspec- tionitiscommontoalternatesequencesofsampling tion and replacement of nonconforming by conforming inspectionwith%inspection–theyareinacertain items, after the rejection of a batch by an AS plan –, senserectifyingplans.Thesimplestplanofthistype, the most relevant characteristics are the average outgoing theCSP-,wassuggestedinbyDodge.Itbegins quality(AOQ),AOQ(p) = p(−n/N)Pa,whichattains A ActuarialMethods a maximum at the so-called average output quality limit The broad range of existing and applicable actuarial A (AOQL), the worst average quality of a product after a calculationsrequireuseofvariousmethodsandinevitably rectifyinginspectionprogram,aswellastheaveragetotal predetermines a necessity of their alteration depending inspection(ATI),theamountofitemssubjecttoinspection, onconcretecasesofcomparisonanalysisandselectionof equaltonifthereisnorectification,butgivenbyATI(p)= mostefficientofthem. nPa+N(−Pa),otherwise. Theconditionofsuccessisatypologyofactuarialcal- culationsmethods,basedonexistingtypologyfieldsand Acknowledgments objectsoftheirapplications,aswellasknowledgeofrule ResearchpartiallysupportedbyFCT/OE,POCIand forselectionofmostefficientmethods,whichwouldpro- PTDC/FEDER. videselectionoftargetresultswithminimumcostsorhigh accuracy. AbouttheAuthor Regardingthecontinuouscharacteroffinancialtrans- ForbiographyofM.IvetteGomesseetheentry(cid:55)Statistical actions, the actuarial calculations are carried out QualityControl. permanently. The aim of actuarial calculations in every particularcaseisprobabilisticdeterminationofprofitshar- CrossReferences ing (transaction return) either in the form of financial liabilities (interest, margin, agio, etc.) or as commission (cid:55)IndustrialStatistics charges(suchasroyalty). (cid:55)SequentialSampling The subject of actuarial calculations can be distin- (cid:55)StatisticalQualityControl guishedinthenarrowandinthebroadsenses. (cid:55)StatisticalQualityControl:RecentAdvances The given subject in the broad sense covers financial andactuarialaccounts,budgeting,balance,audit,assess- ReferencesandFurtherReading ment of financial conditions and financial provision for all categories and types of borrowing institutions, basis DodgeHF,RomigHG()Samplinginspectiontables,singleand doublesampling,ndedn.Wiley,NewYork fortheirpreferentialfinancialdecisionsandtransactions, DuncanAJ()Qualitycontrolandindustrialstatistics,thedn. conditionsandresultsofworkfordifferentfinancialand Irwin,Homehood credit institutions; financial management of cash flows, MontgomeryDC()Statisticalqualitycontrol:amodernintro- resources,indicators,mechanisms,instruments,aswellas duction,thedn.Wiley,Hoboken,NJ financialanalysisandauditoffinancialactivityofcompa- PandeyBN()Statisticaltechniquesinlife-testing,reliability, samplingtheoryandqualitycontrol.Narosa,NewDelhi nies, countries, nations their groups and unions, includ- SchillingEG,NeubauerDV()Acceptancesamplinginquality ingnationalsystemoffinancialaccount,financialcontrol, control,ndedn.ChapmanandHall/CRC,NewYork engineering, and forecast. In other words, the subject of StephensKS()Thehandbookofappliedacceptancesampling: actuarialcalculationsisaprocessofdeterminationofany plans,principles,andprocedures.ASQQuality,Milwaukee expendituresandincomesfromanytypeoftransactionsin theshortestway. In the narrow sense it is a process of determination, inthesameway,offutureliabilitiesandtheircomparison with present assets in order to estimate their sufficiency, ActuarialMethods deficitofsurplus. Wecandefinegeneralandefficientactuarialcalcula- VassiliySimchera tions,theprincipalsofwhicharegivenbelow. Director Efficient actuarial calculations imply calculations of Rosstat’sStatisticalResearchInstitute,Moscow,Russia any derivative indicators, which are carried out through conjugation (comparison) of two or more dissimilar ini- tial indicators, the results of which are presented as dif- A specific (and relatively new) type of financial calcula- ferent relative numbers (coefficients, norms, percents, tions are actuarial operations, which represent a special shares,indices,rates,tariffs,etc.),characterizingdifferen- (inmajorityofcountriestheyareusuallylicensed)sphere tial (effect) of anticipatory increment of one indicator in ofactivityrelatedtoidentificationsofrisksoutcomesand comparisonwithanotherone. market assessment of future (temporary) borrowed cur- In some cases similar values are called gradients, rentassetsandliabilitiescostsfortheirredemption. derivatives (of different orders), elasticity coefficients, or A ActuarialMethods anticipatory coefficients and can be determined by ref- ofdocumentaryadoption,whichincludeconstructionof erence to more complex statistical and mathematical actuarialbalancesandpreparationofactuarialreportsand methodsincludinggeometrical,differential,integral,and conclusions, are called actuarial estimation; the organi- correlationandregressionmultivariatecalculations. zations that are carrying out such procedures are called Herewithincaseofapplicationofnominalcomparison actuarialorganizations. scalesfortwoormoresimplevalues(socalledscaleofsim- Hence,thereisanecessitytolearntheorganizationand pleinterests,whicharecalculatedandrepresentedinterms techniqueofactuarialmethods(estimations)inaggregate; ofcurrentprices)theyaredeterminedandoperatedasitwas aswellastointroducetheknowledgeofactuarialsubjects mentioned by current nominal financial indicators, but in to any expert who is involved in direct actuarial estima- case of real scales application, i.e. scales of so called com- tionsoffutureassetsandliabilitiescostsofvariousfunds, poundinterests,theyarecalculatedandrepresentedinterms credit,insurance,andsimilarlyfinancialcompanies.This of future or current prices, that is real efficient financial istrueforassetsandliabilitiesofanycountry. indicators. The knowledge of these actuarial assessments and Incaseofinsuranceschemethecalculationofefficient practicaluseisasignificantreserveforincreasingnotonly financialindicatorssignifythespecialtypeoffinancialcal- efficiencybut(moreimportanttoday)legitimate,transpar- culationsi.e.actuarialcalculations,whichimplyadditional ent,andprotectedfuturesforbothborrowingandlending profit (discounts) or demanding compensation of loss companies. (loss,damageorlossofprofit)inconnectionwithoccur- rence of contingency and risks (risk of legislation alter- KeyTerms ation,exchangerates,devaluationorrevaluation,inflation Actuary(actuarius–Latin)–profession,appraiserofrisks, ordeflation,changesinefficiencycoefficients). certifiedexpertonassessmentofdocumentaryinsurance Actuarial calculations represent special branch of (and wider – financial) risks; in insurance – insurer; in activity (usually licensed activity) dealing with market realty agencies – appraiser; in accounting – auditor; in assessment of compliance of current assets of insurance, financialmarkets–broker(orbookmaker);inthepastreg- joint-stock, investment, pension, credit and other finan- istrar and holder of insurance documents; in England – cialcompanies(i.e.companiesengagedincreditrelations) adjusterorunderwriter. withfutureliabilitiestotherepaymentofcreditinorder Actuarialtransactions–specialfieldofactivityrelated topreventinsolvencyofadebtorandtoprovideefficient todeterminationofinsuranceoutcomesincircumstances protectionforinvestors-creditors. ofuncertaintythatrequireknowledgeofprobabilitytheory Actuarialcalculationsassumethecomparisonofassets andactuarialstatisticsmethodsandmathematics,includ- (waysofuseorallocationofobtainedfunds)withliabili- ingmoderncomputerprograms. ties(sourcesofgainedfunds)forborrowingcompaniesof Actuarial assessment – type of practical activity, alltypesandforms,whicharecarriedoutinaggregateby licensed in the majority of countries, related to prepara- particularitemsoftheirexpensesundercircumstancesof tion of actuarial balances, market assessment of current mutualrisksinordertoexposethedegreeofcomplianceor and future costs of assets and liabilities of insurer (in incompliance(surplusordeficit)ofborrowedassetswith case of pension insurance assets and liabilities of non- futureliabilitiesintermofrepayment,inotherwordsto governmental pension funds, insurances companies and checkthesolvencyofborrowingcompanies. specializedmutualtrustfunds);completedwithprepara- Borrowingcompanies–insurance,stock,brokerand tionofactuarialreportaccordingtostandardmethodolo- auditorfirms,banks,mutual,pension,andotherspecial- gies and procedures approved, as a rule in conventional ized investment funds whose accounts payable two or (sometimesinlegislative)order. more times exceeds their own assets and appear to be Actuarial estimations – documentary estimations of a source of high risk, which in turn affects interests of chanceoutcomes(betting)ofanyrisk(gambling)actions broad groups of business society as well as population – (games) with participation of two or more parties with areconsideredascompaniesthataresubjectstoobligatory fixed(registered)ratesofrepaymentofinsurancepremium insuranceandactuarialassessment. andcompensationspremiumforpossiblelosses.Theydif- Actuarialcalculationsassumetheconstructionofbal- ferbycriteriaofcomplexity–thatiselementary(simple ancesforfutureassetsandliabilities,probabilisticassess- or initial) and complex. The most widespread cases of mentoffutureliabilitiesrepayment(debts)attheexpense elementary actuarial estimations are bookmaker estima- of disposable assets with regard to risks of changes of tions of profit and loss from different types of gambling theiramountonhandandmarketprices.Theprocedures includingplayingcards,lottery,andcasinos,aswellasrisk A ActuarialMethods takingonmodernstockexchange,foreignexchangemar- FacultyofActuaries”),ChartedInsuranceInstitute,Inter- A kets,commodityexchanges,etc.Thecomplexestimations nationalAssociationofActuaries,InternationalForumof assume determination of profit from second and conse- ActuariesAssociations,InternationalCongressofActuar- quent derived risks (outcomes over outcomes, insurance ies,andGroupeConsultatifActuarielEuropéen. over insurance, repayment on repayment, transactions withderivatives,etc.).Alloftheseestimationsarecarried out with the help of various method of high mathemat- AbouttheAuthor ics(firstofall,numericmethodsofprobabilitytheoryand Professor Vassiliy M. Simchera received his PhD at the mathematicalstatistics).Theyarealsooftenrepresentedas ageofandhisDoctor’sdegreewhenhewas.Hehas methodsofhighactuarialestimations. beenVice-presidentoftheRussianAcademyofEconom- Generally due to ignorance about such estimations, icalSciences(RAES),ChairmanoftheAcademicCouncil current world debt (in approximately trillion and Counsel of PhDs dissertations of RAES, Director of USD, including trillion USD in the USA) has dras- RussianStateScientificandResearchStatisticalInstituteof tically exceeded real assets, which account for about Rosstat(Moscow,from).HewasalsoHeadofChair trillion USD, which is actually causing the enormous ofstatisticsintheAll-RussianDistantFinancialandStatis- financialcrisiseverywhereintheworld. ticalInstitute(–),DirectorofComputerStatistics Usuallysuchestimationsarebeingundertakentowards DepartmentintheStateCommitteeonstatisticsandtech- futureinsuranceoperations,profitsandlosses,andthatis niques of the USSR (–), and Head of Section of whytheyareclassifiedasstrictlyapproximateandrepre- StatisticalResearchesintheScienceAcademyoftheUSSR sentedincategoriesofprobabilisticexpectations. (–). He has supervised Doctors and over The fundamental methods of actuarial estimations are PhD’s.Hehas(co-)authoredoverbooksandarti- the following: methods for valuing investments, select- cles,includingthefollowingbooks:EncyclopediaofStatis- ing portfolios, pricing insurance contracts, estimating ticalPublications(,p.,inco-authorship),Financial reserves, valuing portfolios, controlling pension scheme, and Actuarial Calculations (), Organization of State finances,assetmanagement,timedelaysandunderwriting Statistics in Russian Federation () and Development cycle, stochastic approach to life insurance mathematics, of Russia’s Economy for Years, – (). pensionfundingandfeedback,multiplestateanddisabil- Professor Simchera was founder and executive director ityinsurance,andmethodsofactuarialbalances. (–)ofRussianStatisticalAssociation,memberof The most popular range of application for actuarial various domestic and foreign academies, as well as sci- methods are: ) investments, (actuarial estimations) of entific councils and societies. He has received numerous investments assets and liabilities, internal and external, honorsandawardsforhiswork,includingHonoredScien- realandportfoliotypestheirmathematicalmethodsand tistofRussianFederation()(DecreeofthePresident models,investmentsrisksandmanagement;)lifeinsur- oftheRussianFederation)andSaintNicolayChudotvoretz ance (various types and methods, insurance bonuses, honor of III degree (). He is a full member of the insurance companies and risks, role of the actuarial InternationalStatisticalInstitute(from). methods in management of insurance companies and reductionofinsurancerisks);)generalinsurance(insur- anceschemes,premiumrating,reinsurance,reserving);) CrossReferences actuarialprovisionofpensioninsurance(pensioninvest- (cid:55)CareersinStatistics ments – investment policy, actuarial databases, meeting (cid:55)Insurance,Statisticsin thecost,actuarialresearches). (cid:55)Kaplan-MeierEstimator Scientistwhohavegreatlycontributedtoactuarialprac- (cid:55)LifeTable tices: William Morgan, Jacob Bernoulli, A. A. Markov, (cid:55)PopulationProjections V. Y. Bunyakovsky, M. E. Atkinson, M. H. Amsler, (cid:55)Probability,Historyof B.Benjamin, G. Clark, C. Haberman, S. M. Hoem, (cid:55)QuantitativeRiskManagement W.F.Scott,andH.R.Watson. (cid:55)RiskAnalysis World’s famous actuary’s schools and institutes: The (cid:55)StatisticalAspectsofHurricaneModelingand Institute of Actuaries in London, Faculty of Actuaries in Forecasting Edinburgh(onMay,followingaballotofFellows (cid:55)Statistical Estimation of Actuarial Risk Measures for ofbothinstitutions,itwasannouncedthattheInstituteand Heavy-TailedClaimAmounts Facultywouldmergetoformonebody–the“Instituteand (cid:55)SurvivalData A AdaptiveLinearRegression ReferencesandFurtherReading (cid:55)Computational Statistics and (cid:55)Statistical Software: An BenjaminB,PollardJH()Theanalysisofmortalityandother Overview). On the other hand, this progress has put an actuarialstatistics,ndedn.Heinemann,London applied statistician into a difficult situation: If one needs BlackK,SkipperHD()Lifeinsurance.PrenticeHall,Englewood to fit the data with a regression hyperplane, he (she) is Cliffs,NewJersey hesitatingwhichproceduretouse.Ifthereismoreinfor- BoothP,ChadburnR,CooperD,HabermanSandJamesD() mationonthemodel,thentheestimationprocedurecan Modernactuarialtheoryandpractice.ChapmanandHall/CHC, London,NewYork be chosen accordingly. If the data are automatically col- SimcheraVM()Introductiontofinancialandactuarialcalcu- lectedbyacomputerandthestatisticianisnotabletomake lations.FinancyandStatistikaPublishingHouse,Moscow anydiagnostics,thenhe(she)mightuseoneofthehigh TeugelsJL,SundtB()Theencyclopediaofactuarialscience, breakdown-pointestimators.However,manydeclinethis vols.Wiley,Hoboken,NJ ideaduetothedifficultcomputation.Then,attheend,the TransactionsofInternationalCongressofActuaries,vol.–;JInst Actuar,vol.– statisticiancanpreferthesimplicitytotheoptimalityand useseithertheclassicalleastsquares(LS),LAD-methodor otherreasonablysimplemethod. Insteadoftofixourselvesononefixedmethod,onecan trytocombinetwoconvenientestimationmethods,andin AdaptiveLinearRegression thiswaydiminisheventualshortagesofboth.Taylor() suggestedtocombinetheLAD(minimizingtheL norm) JanaJurecˇková andtheleastsquares(minimizingtheLnorm)methods. Professor Arthanari and Dodge () considered a convex com- CharlesUniversityinPrague,Prague,CzechRepublic bination of LAD- and LS-methods. Simulation study by DodgeandLindstrom()showedthatthisprocedure is robust to small deviations from the normal distribu- Consider a set of data consisting of n observations of a tion(see(cid:55)NormalDistribution,Univariate).Dodge() response variable Y and of vector of p explanatory vari- extendedthismethodtoaconvexcombinationofLADand ablesX=(X,X,...,Xp)⊺.Theirrelationshipisdescribed Huber’sM-estimationmethods(see(cid:55)RobustStatisticsand by the linear regression model (see (cid:55)Linear Regression RobustStatisticalMethods).DodgeandJurecˇková() Models) observed that the convex combination of two methods Y =βX+βX+...+βpXp+e. could be adapted in such a way that the resulted esti- mator has the minimal asymptotic variance in the class Intermsoftheobserveddata,themodelis of estimators of a similar kind, no matter what is the unknown distribution. The first numerical study of this Yi =βxi+βxi+...+βpxip+ei, i=,,...,n. procedure was made by Dodge et al. (). Dodge and The variables e,...,en are unobservable model errors, Jurecˇková(,)thenextendedtheadaptiveproce- whichareassumedbeingindependentandidenticallydis- duretothecombinationsofLAD-withM-estimationand tributedrandomvariableswithadistributionfunctionF withthetrimmedleastsquaresestimation.Theresultsand anddensityf.Thedensityisunknown,weonlyassumethat examples are summarized in monograph of Dodge and itissymmetricaround.Thevectorβ=(β,β,...,βp)⊺ Jurecˇková(),wherearemanyreferencesadded. isanunknownparameter,andtheproblemofinterestis Letusdescribethegeneralidea,leadingtoaconstruc- to estimate β based on observations Y,...,Yn and xi = tionofanadaptiveconvexcombinationoftwoestimation (xi,...,xip)⊺, i=,...,n. methods: We consider a family of symmetric densities Besides the classical (cid:55)least squares estimator, there indexedbyansuitablemeasureofscales: existsabigvarietyofrobustestimatorsofβ.Somearedis- tributionallyrobust(lesssensitivetodeviationsfromthe assumed shape of f), others are resistant to the leverage F ={f :f(z)=s−f (z/s), s>}. points in the design matrix and have a high breakdown point[introducedoriginallybyHampel(),thefinite sampleversionisstudiedinDonohoandHuber()]. Theshapeoff isgenerallyunknown;itonlysatisfiessome The last years brought a host of statistical pro- regularity conditions and the unit element f ∈ F has cedures, many of them enjoying excellent properties the scale s = . We take s = /f() when we combine and being equipped with a computational software (see L -estimatorwithotherclassofestimators.
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