Sample statistics

Definition

Simplerandomsample

Definition1LetX₁,X₂,...,X_nisasampleofcapacitynfromthepopulationX,ifX₁,X₂,...,X_naremutuallyindependentandeachisarandomvariablewiththesamedistributionasthepopulationX,thencalledX₁,X₂,...,X_nisasimplerandomsamplewithacapacityofninthepopulationX,referredtoasasimplesampleorsampleforshort.

Note:SampleX₁,X₂,...,X_ncanalsobeusedasann-dimensionalrandomvector(X₁,X₂,...,X_n)means.Letx_ibeanobservationvalueofX_i,andcallit(x₁,x₂,....,x_n)isanobservationofsampleX₁,X₂,...,X_nvalue.

Samplestatistics

Definition2SetX₁,X₂,...,X_nisthesampleofthepopulationX,Tisthen-dimensionalreal-valuedfunction,asthesampleX₁,X₂,...,X_{nT=T(X1,X2,...,Xn)(randomvariablewithoutunknownparameters),thevalueofTisrecordedast=T(x1,x2,...,xn),whichiscalledTorT(X1,X2,...,Xn)isthesamplestatistics,referredtoasStatistics.}

Note:1)Statisticsrefertothefunctionofthesampleanddonotcontainunknownparameters.Thefunctionofthesampleisequivalenttothefunctiondefinedonthesamplespace.

2)Oneobservationofagivensamplex=(x₁,x₂,...,x_n),thevalueofT(x₁,x₂,...,x_n)iscompletelydetermined.

Example

Example1SetX₁,X₂,...,X_nIsasampleofpopulationX,anditscapacityisn.Remember

thenandarebothstatistics,called

andarethemeanandvarianceofsamplesX₁,X₂,...,Xn,respectively.Theobservedvalues​​ofthesamplearex₁,x₂,...,x_n,and

Observationsarerecordedas

Extension

Orderstatistics

Definition3LetX₁,X₂,...,X_nbeasampleofthepopulationX,andnownfunctionsareestablishedfromthesample:

whereissuchastatistic,anditsobservationvalueis,

istheobservationvaluex₁ofsampleX₁,X₂,...,X_n>,x₂,...,x_nThekthindescendingorderisthevalue,theniscalledSequencestatistics.

Note:Itiseasytoseethatcallsthesmallestitemstatistic,andthelargestitemstatistic.

Extremelypoor

Definition4LetX₁,X₂,...,X_nisasampleofpopulationX,thenthestatisticiscalledtheextremedifferenceofthesample.

Note:Therangeisthedifferencebetweenthemaximumvalueandtheminimumvalueinthesample,whichreflectsthefluctuationrangeoftheobservedvalueofthesample.Likevariance,itisaquantitativeindicatorthatreflectsthedegreeofdispersionofobservations,anditiseasytocalculate.

Samplingdistribution

StatisticsarethemostimportantbasicconceptsforestimatingandinferringthedistributionfunctionornumericalcharacteristicsofthepopulationX.FindthestatisticT(x₁,x₂,...,x_n)Thedistributionfunctionisoneofthebasicproblemsofmathematicalstatistics.Thedistributionofstatisticsiscalledsamplingdistribution.