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,siixo,in}..ni=m.,oxodinpe)nlT.uSiusnpiatpsv,oewscehtowerreeoofybipssreaerrdveiecrteaosrpcsoo.lnIlteseciss- tpThhaereatmupneenitneagrltsypianfrutanhmcetegitoeivnretnihnamt(oc)doenbltorauosnls∑dtsmjh=eth∣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(acttetoR)iaidngctoeronRdstuergcareiendstsr,iio∑dngβes)jr,ew≤grhetis,csihinonmot(ihsneeier- βˆlasso =argmβ in⎧⎪⎪⎨⎪⎪⎩∑i=n(yi−β−∑j=pxijβj)+λ∑j=p∣βj∣⎫⎪⎪⎬⎪⎪⎭. () words The lasso shrinks the OLS estimator toward zero and βˆridge =argβmin⎧⎪⎪⎨⎪⎪⎩∑iN=(yi−β−∑j=pxijβj)+λ∑j=pβj⎫⎪⎪⎬⎪⎪⎭, cdieepnetnsdtoinegxaocntlytzheero.vTaliubeshiorfanλi,(its)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+ shrinkagemethodperformsbetterwhenpislarge,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+isdefinedas 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=itij+(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 pointoutmorethanrelatedreferences. A simple ALT scenario is characterized by putting k λi =exp(−Zi−β−βXi), () 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−βXi)ri AbouttheAuthor i= FranciscoLouzada-NetoisanassociateprofessorofStatis- exp(−exp(−Zi−β−βXi)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 singleandjointauthorofmorethanpublicationsinsta- 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 lates,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 ≡ nthesizeofthesecondsampleandc(>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-,wassuggestedinbyDodge.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,POCIand 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 ageofandhisDoctor’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-)authoredoverbooksandarti- 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(onMay,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(minimizingtheLnorm)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 =βX+βX+...+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 =βxi+βxi+...+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. 