Originanddevelopment
Mathematicalstatisticsisabranchofmathematicsdevelopedwiththedevelopmentofprobabilitytheory.Itstudieshowtoeffectivelycollect,organizeandanalyzedataaffectedbyrandomfactors,andMakeinferencesorpredictionsontheissuesunderconsideration,andprovidebasisorsuggestionsfortakingcertaindecisionsandactions.
Mathematicalstatisticsoriginatedfromvariousdescriptivestatisticalactivitiessuchasdemographicstatisticsandsocialsurveys.In2250BC,Dayu'swatercontrolwasdividedintoKyushuinaccordancewiththequalityofmountainsandrivers,humanandmaterialresources;YinZhouIntheera,theminefieldsystemwasimplemented,andthelandandhouseholdregistrationwerecalculatedaccordingtothepopulation.IntheSpringandAutumnPeriod,thestrengthoftheprinceswasoftenjudgedbythenumberofmilitaryvehicles.Itcanbeseenthatmilitarysurveysandcomparisonshavebeencarriedout.ThestatisticsofthenationalhouseholdregistrationandageintheHanDynastyarewelldocumented;TheYellowBookandtheFishScaleBookwereinitiallycompiled.TheYellowBookisanationalhouseholdregister,andtheFishScaleBookisanationallandmap,withtopographydrawn,andithasthenatureofmodernstatisticalcharts.Itcanbeseenthatourcountryhasattachedgreatimportancetostatisticalworkinthepastdynasties,butlackssystematicresearchandhasnotformedspecialworks.
InWesterncountries,statisticalworkbeganin3050BC.Egyptbuiltpyramidstocollectconstructioncostsandconductcensusesandstatisticsonthenationalpopulation.IntheAristotleera,statisticalworkbegantobecomerational.Evolution.Atthistime,theapplicationofstatisticsinhealth,insurance,domesticandforeigntrade,militaryandadministrativemanagementwaswelldocumented.ThetermstatisticsevolvedgraduallyfromthetermItaly.
Thedevelopmentofmathematicalstatisticscanberoughlydividedintothreestages:theclassicalperiod,themodernperiodandthemodernperiod.
Classicalperiod
(Beforethe19thcentury)
Thisistheformationanddevelopmentstageofdescriptivestatistics,andtheembryonicperiodofmathematicalstatistics.Duringthisperiod,theSwissmathematicianBernoulli(1654-1705)systematicallydemonstratedthelawoflargenumbersearlier.In1763,BritishmathematicianBayesproposedatheoryofinductivereasoning,whichwaslaterdevelopedintoamethodofstatisticalinference-Bayesianmethod,whichcreatedaprecedentformathematicalstatistics.TheFrenchmathematicianDiMofo(1667-1754)firstdiscoveredthedensityfunctionofthenormaldistributionin1733,andcalculatedtheprobabilityofthecurveinvariousintervals,whichlaidthefoundationfortheentiretheoryoflargesamples.In1809,theGermanmathematicianGauss(1777-1855)andFrenchmathematicianLegendre(1752-1833)independentlydiscoveredtheleastsquaresmethodandappliedittotheerroranalysisofobservationaldata,inthetheoryandapplicationofmathematicalstatisticsTheyhavemadeimportantcontributions.Henotonlyappliedmathematicalstatisticstobiology,butalsoappliedtothestudyofeducationandpsychology,anddemonstratedindetailthewiderangeofapplicationsofmathematicalstatistics.Heoncepredicted:"Statisticalmethodscanbeapplied.Invariousdepartmentsofvariousdisciplines".
Modernperiod
(fromtheendofthe19thcenturyto1945)
Themainbranchofmathematicalstatisticswasestablished,whichwastheformationperiodofmathematicalstatistics.Atthebeginningofthelastcentury,duetoThedevelopmentofprobabilitytheoryisclosetocompletetheoretically,coupledwiththeurgentneedofindustrialandagriculturalproduction,whichpromotesthevigorousdevelopmentofthissubject.
In1889,theBritishmathematicianPearson(1857-1936)proposedthemethodofmomentestimation,andthefollowingyearproposedthetheoryoffrequencycurve.In1900,theGermanmathematicianHelmetdiscoveredThec2testisproposedonthebasisofthec2distribution,whichisthefirstsmallsampledistributioninthehistoryofmathematicalstatistics.
In1908,theBritishstatisticianGossett(1876-1937)createdthetheoryandmethodofsmallsampletestinsteadoflargesampletest(ietdistributionandttestmethod),whichismathematicaltheoryAnotherbranchofstatistics-multivariateanalysislaysthetheoreticalfoundation.
In1912,theBritishstatisticianFischer(1890-1962)promotedthet-testmethodanddevelopednewbranchesofmathematicalstatisticssuchassignificancetestingandestimationandanalysisofvariance.
Inthisway,someimportantbranchesofmathematicalstatistics,suchashypothesistesting,regressionanalysis,analysisofvariance,orthogonaldesign,etc.,havecontentandtheoriesthatdeterminetheirappearance.Mathematicalstatisticshasbecomeasubjectofmathematicswithawiderangeofapplicationsanduniquemethods.
Modernperiod
(after1945)
RomanianAmericanMathematicalStatisticianVarade(1902-1950)devotedhimselftousingmathematicalmethodstomakestatisticsHehasmademanyimportantachievementsinlearningprecisionandrigor.Hedevelopeddecision-makingtheory,raisedgeneraldiscriminantproblems,createdsequentialanalysistheory,andproposedthefamoussequentialprobabilityratiotest.Wald’stwobooks"SequentialAnalysisandStatisticalDecisionFunctionTheoryareconsideredclassicsinthehistoryofmathematicaldevelopment.
Duetotheapplicationofcomputers,thetheoreticalresearchandapplicationofmathematicalstatisticshavebeencontinuouslydevelopedindepth,andsomenewbranchesandmarginalnewdisciplineshavebeengenerated,suchasoptimaldesignandnon-parametricstatisticalinferenceWait.
Currently,theapplicationrangeofmathematicalstatisticsisbecomingmoreandmoreextensive.Ithaspenetratedintomanyscientificfields,appliedtovarioussectorsofthenationaleconomy,andhasbecomeanindispensabletoolforscientificresearch.
Introduction
Definition
Mathematicalstatisticsisamethodbasedonprobabilitytheorytostudythebasiclawsofalargenumberofrandomphenomenainsocietyandnature.Itsmaincontentincludesparameterestimation,hypothesistesting,correlationanalysis,experimentaldesign,non-parametricstatistics,processstatistics,etc.
Features
Ittakestheobservationandexperimentofrandomphenomenaasastartingpoint,usesprobabilitytheoryasthetheoreticalbasistostudyrandomphenomena,selectsmathematicalmodelsforrandomphenomenabasedonthedata,andusesmathematicsDatatoverifywhetherthemathematicalmodelisappropriate,andthenstudyitscharacteristics,natureandregularityonanappropriatebasis.
Forexample,whenabulbfactoryproducesbulbs,afewoftheproductsonacertaindayareselectedfortesting.Beforethetest,itisnotknownhowlongthelifeofthebulbswillbeonthatday,theirprobability,andtheirdistribution.Afterthetest,thelifespanoftheseseveralbulbsisobtainedasdata,fromwhichtheservicelifeandqualificationrateofthebulbsproducedinthewholebatchcanbeinferred.Inordertostudyitsdistribution,usethemathematicalmodelprovidedbyprobabilitytheorytocarryouttheexponentialdistribution,findthevalue,andthenuseafewdaysofsamplingtesttodeterminetheappropriatenessoftheexponentialdistribution.
Alllinksofstatisticalwork
Whenusingmathematicalstatisticstosolveapracticalproblem,therearegenerallythefollowingsteps:establishamathematicalmodel,collectandorganizedata,performstatisticalinference,predictionanddecisionmaking.Theselinkscannotbecompletelyseparated,andtheyarenotnecessarilyintheaboveorder,andsometimestheyareinterleavedwitheachother.
①Selectionandestablishmentofmodel.Inmathematicalstatistics,amodelreferstoacertainassumptionaboutthepopulationunderstudy,andgenerallyspecifiesacertaintypeofpopulationdistribution.Theestablishmentofamodelshouldbebasedontheknowledgeofprobability,theprofessionalknowledgeoftheresearchproblem,pastexperienceandsamples(data)drawnfromthepopulation.
②Datacollection.Therearethreewaysofcomprehensiveobservation,samplingobservationandarrangingspecificexperiments.Comprehensiveobservationisalsocalledcensus,whichmeansthateveryindividualinthepopulationisobservedtodeterminetherequiredindicators.Samplingobservation,alsoknownasspotcheck,referstotakingapartfromthepopulationanddeterminingitsrelatedindexvalues.Theresearchcontentinthisareaconstitutesabranchofmathematicalstatistics.Calledasamplesurvey.
③Arrangespecificexperimentstocollectdata.Thesespecificexperimentsmustberepresentativeandmakethedataeasytoanalyze.Themathematicalproblemscontainedthereinconstituteanotherbranchofmathematicalstatistics,thatis,thecontentofexperimentaldesign.
④Dataorganization.Thepurposeistoextracttheusefulinformationcontainedinthedata.Oneformistodevelopappropriatecharts,suchasscatterplots,toreflecttheroughregularitiesorgeneraltrendsimplicitinthedata.Anotherformistocalculateseveralnumericalfeaturestocharacterizecertainaspectsofthesample'sproperties,suchassimpledescriptivestatisticssuchassamplemeanandsamplevariance.
⑤Statisticalinference.Referstomakingcertainconclusionsaboutthedistributionofthepopulationbasedonthepopulationmodelandthesamplesdrawnfromthepopulation.Datacollectionandorganizationarenecessarypreparationsforstatisticalinference,andstatisticalinferenceisthemaintaskofmathematicalstatistics.
⑥Statisticalforecast.Theobjectofstatisticalpredictionisthevaluethatarandomvariablewilltakeatacertaintimeinthefuture,orthevaluethatitwilltakewhenitisassumedtobeobservedundercertainconditions.Forexample,predictthemarketsalesvolumeofaproductinthenext3years,andtheheightandweightofa10-year-oldboyin3years.
⑦Statisticaldecision-making.Anactionplanbasedonstatisticalinferencesorpredictionsmadeandtakingintoaccounttheconsequencesoftheaction(expressedintheformofeconomiclosses).Thegoalistomakethelossassmallaspossible,or,conversely,tomakethegainaslargeaspossible.Forexample,astorewantstodeterminethepurchasequantityofacertainproductthisyear.Basedonasamplesurvey,thestore’sstatisticianpredictsthatthestore’ssalesofthisproductwillbe1,000piecesthisyear.Assumingalossof20yuanforeachbacklogofaproduct,andalossof10yuanforamissingproduct,adecisionaboutthequantityofpurchasesmustbemadeaccordingly.
Disciplinaryapplication
Mathematicalstatisticshavebeenwidelyandprofoundlyappliedinnaturalsciences,engineeringtechnology,managementsciences,andhumanitiesandsocialsciences.Technologyandpolitics,economyandsocietycontinuetodevelopandgraduallyexpand,butingeneraltermscanbedividedintotwocategories:⑴experimentdesignandresearch,thatis,tostudyhowtoobtainobservationdatamorerationallyandeffectively;⑵statisticalinference,Thatistostudyhowtousecertaindatatomakeasaccurateandreliableconclusionsaspossibleontheissuesofconcern.Ofcourse,thetwopartsarecloselyrelatedandshouldbegivenconsiderationinpracticalapplications.Butaccordingtotheoveralldesignofthismajor,ourmathematicalstatisticscourseonlydiscussesstatisticalinference.Mathematicalstatisticsisadisciplinethatstudiesthestatisticalregularityofrandomphenomenabasedonthetheoryofprobabilityanddataobtainedfromexperimentsorobservations.Thepurposeofthiscourseistoenablestudentstounderstandstatisticalinferencetestingandothermethodsandbeabletoapplythesemethodstomakevariousreasonableestimatesandjudgmentsontheobjectiveregularityoftheresearchobject.Masterthepointestimationandintervalestimationofoverallparameters.Masterthebasicmethodsandtechniquesofhypothesistesting.Understandtheprinciplesofsquareerroranalysisandregressionanalysis,andbeabletouseitsmethodsandtechniquestomakestatisticalinferences.