统计学要点摘要英文版 Statistic Review

Chapter 2 Statistic ReviewA. Random variables; 1. expected value:Define : X is a discrete random variable, “ the mean (or expected value) of X ” is theweighted average of the possible outcomes, where the probabilities of the outcome serve as the appropriate weight.∑++===n n i i X X p X p X p X p X E ......)(2211μ p i is ith of prob., i=1,2, ……nInterpretation : The random variable is a variable that have a probability associated witheach outcome. Outcome is not controlled.Discrete random V ar. : has finite outcome, or outcome is countable infinite.Continuous random V ar.: uncountable infinite outcome, the probability of each outcome issmall because of too many numbers.For normal random V ar., probability density function is used to calculate the probabilitybetween the are.E( ): the expectations operator, ∑=1i p→ NX obs of N XX XX X Ni iN∑==+++=`1321.)(....… “ sample mean ”, used to estimate X μThe Xis changed from sample to sample. is not a fixed on time, the outcome selected should not be the same. There is prob. associated with each X . X is also a random variable, we can calculate E(X ).2. variance: measure the dispersion(分散), the range of the value∑-=-=-==2222)()]([})]([{)(Xi i i iXX E X E X E X E X pX Var μσ “population variance ” constant)(X Var X=σ…………... “population standard deviation ”→ 1)(ˆ222--==∑N X XS iX Xσ “sample variance ” used to estimate 2X σ2ˆXX X S S ==σ……….. “sample standard deviation ”3. joint distribution: (linear relation of X and Y bi-variance random variance.))])([(),(Y X Y X E Y X Cov μμ--=Covariance, measuring the linear relationship between X and Y.),(Y X Cov , depends on the units of X and Y; different unit-> different),(Y X Cov >0: the best-fitting line has a positive slope, positive relationship between X and Y . ),(Y X Cov <0: the best-fitting line has a negative slope, negative relationship between X and Y . ),(Y X Cov =0: there is no linear relationship between X and Y , but may be have nonlinearrelationship.YX XY Y X σσρ),cov(=“population correlation coefficient ” is scale free.XY ρ>0, a positive correction, XY ρ<0, a negative correction XY ρ=0, no linear relationship between X and Y11-≥≥XY ρ, XY ρ=1: regression line is a straight line with positive slope, XY ρ=-1: regressionline is a straight line with negative slope.1))((),(ˆ1---=∑=N Y Y X XY X vCo Ni i i,YX XYS S Y X ),(v ˆco =γ=221)()())((∑∑∑----=Y Y X XY Y X Xi iNi i i“sample correlation coefficient ”)int (,1,1)(,1)(11prob jo p Y p X p ijN i Nj ij ===∑∑∑∑==E(X) =0.263×1+0.403×2+0.334×3=2.071 V ar(X) =4.881-(2.071)2=0.591959 E(Y) =0.298×6+0.385×5+0317×4=4.981 V ar(Y)= 25.425-(4.981)2=0.614639Cov(X,Y)=)()(Y E X E Y X p ijj i j i -∑∑=10.001-2.071×4.981= -0.3174. formulaE(b)=b , V ar(b)=0；E(aX)=aE(X), V ar(aX)=a 2 V ar(X )； E(aX+b)=aE(X)+b, Var(aX+b)= a 2 V ar(X ))()]([)]}([{])([)]()[()(2222222X Var a X E X E a X E X a E b X aE b ax E b aX E b aX E b aX Var =-=-=+-+=+-+=+E(X+Y)=E(X)+E(Y), V ar(X+Y)=V ar(X)+V ar(Y)+2Cov(X, Y)\),(2)()()]}()][([2)]([)]({[)]}([)]({[)]()([)]()[()(22222Y X Cov Y Var X Var Y E Y X E X Y E Y X E X E Y E Y X E X E Y E X E Y X E Y X E Y X E Y X Var ++=----+-=-+-=--+=+-+=+ If X and Y is independent (linear uncorrelated), than E(X+Y)=E(X)+E(Y) → Cov(X,Y)=0, XY ρ=0, → V ar(X+Y)=Var(X)+V ar(Y))(][)]()()()([)]()][([),(=-=-=+--=+--=--=Y X Y X Y X Y X Y X XY E X Y XY E Y E X E Y XE X XE XY E Y E Y X E X E Y X Cov μμμμμμμμμμ,0),(=Y X Cov can ’t define the X and Y are independent.2))(()(X E X X E ≠∙ X is not independent of itself.B. (probability) distributions:1. the normal distribution:),(`~2XX N X σμ2. the standard normal distribution:)1,0(~,N Z X Z XXσμ-=3. the Chi-square distribution:∑=+++==Ni NiNZ Z Z Z12222122 χZ i : N independently normal distribution random variables with 0 mean and variance 1. → As N gets larger, the χ2 distribution because an approximation of normal distribution. the rang of χ2 : 0 → ∞ 4. the t distribution:If (1) Z is normal distribution with mean 0 and variance 1, (2) χ2 is distribution as Chi-square with N degrees of freedom, (3) Z and χ2 are independent Thennnt NZ~2χ→As N gets large, the t distribution will tend to approximately be normal distribution. 5. the F distribution:if X and Y are independent and distribution as Chi-square will N1 and N2 degrees of freedom, respectively thus21,21~N N F N YN Xassume Xi ’s are independent each other,),(~2NX XX σμ regardless of the distribution of XE(X)=)(1)(1][21Ni iXX X E NX E NNX E +++==∑∑=X XXXX N N XXNμμμ=∙=+++)(1NN NXNXNX XXiiX22222)(1)var(1)1var()var(σσσ=∙====∑∑22121)]()[()var(N N i X X X E X X X E X +++-+++=∑=22211)]}([)([)]({[N N X E X X E X X E X E -++-+- =∑∑--+-)]()][([2)]([2j j i i i i X E X X E X E X E X =)var()var()var()var(21N i X X X X +++=∑=2222X X X X N σσσσ=+++If ),(~2X X N X σμ, then i.i.d.(independent with other variables, identically distributed over time) If ),(~2XX N X σμ, then ),(~2NN X XX σμ＊ Central limit theorem:If ),(~2XX X σμ (at any distribution), then ),(~2NN X XX σμ as N(sample size) increase.1)(ˆ2--==∑N X XS iX σ, what is the distribution ofNS X 0μ- ?C. hypothesis testing:ex. H 0: μ=100… null hypothesis, H 1: μ≠100 …alternative hypothesis α=0.05 … level of significanceto get the value of X , the test statistic, NX Z 2100σ-=if σ2 is know.To check the critical value, Zc ( whether or not to accept H 0)If accept H 0:“we can not reject H 0 at 95% confidence level based on the data we have ”If reject H 0:“we can reject H 0 at 5% level of confidence.”Type I error: reject H 0 when H 0 is true, the probability of making type I error is α.Type II error: accept H 0 when H 0 is false, the probability of making type II error is difficult to determine.When the confidence interval increase, then the type I error will reduce and type II error will increase.※ 22000)1()1()()(σσμμμ---=-=-N S N NX SNX NS X(1) if ),(~2XX N X σμ, then ),(~2NN X XX σμ, the numerator(分子) (NS X 0μ-) is the standardnormal variable Z, and22)1(σSN - is 21-N χ (21222222)()1()(11-=-=-⇒--=∑∑N i i XX SN XX NSχσσσσ)121~)1(~----∴N N t N ZNS X χμ, if numerator and denominator(分母) are independent.(2) if ),(~2X X N X σμ, the we will appeal to the central limit theorem.D. point and interval estimate:1. point estimate: ex X =40 , but we don ’t know whether the true value approximates it ornot.2. interval estimate:N Z X Xσα2±, α=0.05 , if X σ is know.NS t X Xt 1,2-±α,α=0.05 , if X σ is unknown and N is small (<31).95% interval will include the true value.E. properties of estimator: unbiasedness, efficiency , consistency1. unbiasedness:βˆ is an unbiased estimator of β, if E(βˆ) =β(true value)→ ex. X is an unbiased estimator of X μ；ex. 2X S is an unbiased estimator of 2X σ]})()([11{]1)([)(2222∑∑----=--=X X X N X N E N X XE S E μμ=222)(11XXXNNN N σσσ=--∑∑∑----+-=---=-)])((2)()[()]()[()(2222X X XXXXX X X X X X X X μμμμμμ =∑∑----+-)()(2)()(22XXXXX X X N X μμμμ =222)(2)()(XXXX N X N X μμμ---+-∑ =22)()(XXX N X μμ---∑→ bias= E(βˆ) －β2. efficiency:define: (a) minimum mean square error , MSE(βˆ)=E(βˆ-β)2 =[bias(βˆ)]2+Var(βˆ).Pove : 22]})([)](ˆ{[)ˆ(ββββββ-+-=-E E E E=]})([)](ˆ{[2])([)](ˆ[22ββββββββ--+-+-E E E E E E =)}ˆ()]ˆ([ˆ)ˆ(ˆ{2)]ˆ([)ˆvar(22βββββββββE E E E bias +--++ =0)]ˆ([)ˆvar(2++ββbias→ we check X , it is mostly around μ.→ when bias(βˆ)=0, then MSE(βˆ)=V ar(βˆ)→ If we have an unbiased estimator with large dispersion of true value (ie. High var.) and abiased estimator with low var. we might prefer biased estimator than unbiased estimator to maximize the precision of prediction.Def. (b) If for a given sample size, (1) βˆis unbiased, (2) V ar(βˆ))~var(β≤, when β~is anyother unbiased estimator of β. Then βˆ is an efficient estimator of β.Cramer-Rao lower bounds: gives a lower limit for the variance of any unbiased estimator. EX. The lower var(X )=N2σ; the lower variance for N422σσ=( between the all unbiased estimators, the min. is kN -=422)ˆvar(σσ)3. consistency:The probability limit of βˆ, plim (βˆ)=β as N →∞ ;,1)ˆ(lim =<-∞→δββprob N for any δ>0.0)ˆ(,1)ˆ(:=>-==∞→∞→δββββN N prob prob ie2. criteria to consistency:(a) sufficient condition for consistency: not necessary for consistency βˆ is a consistent estimator of β if plim (βˆ)=β as N →∞ → consistency ⇒⨯ plim (βˆ)=β (b) mean square error consistency:If βˆ is an estimator of β and if ,0)ˆ(lim =∞→βMSE N then βˆ is a consistent estimatorof β→ ie X is a consistent estimator of X μ0lim ,0)var()]([)(222==+=+=∞→MSE NNX X bias X MSE N XXσσSlutsky ’s theorem: If plim (βˆ)=β and g(βˆ) is a continuous function of βˆ,then plim g(βˆ)=g[ plim (βˆ)]( This is “ consistency carries over ” However, consistency is not always carries over. ) Biasedness doesn ’t carry over.4. asymptotic unbiasedness: as N becomes large and large. “βˆ” is an asymptotically unbiased estimator of β if ,)ˆ(lim ββ=∞→E N→ if estimator is unbiased ,⇐⨯⇒ it will be asym. unbiased.→ Ex. If ∑-=22)(1~X XNiσ, it is asym. unbias221)~(σσNN E -= , 22)~(lim σσ=∞→E N5. asymptotic efficiency: N →∞“βˆ” is an asymptotically efficient estimator of β if all of the following conditions are satisfied:(a) βˆ is an asymptotic distribution with finite mean and finite variance. (b) βˆ is consistent ;(c) no other consistent estimator of β has small asymptotic variance than βˆ. → if estimator is efficient, ⇐⨯⇒it will be asym. efficient.unbiaedness efficiency consistency asym. unbiasedness asym. efficiencybiaedness inefficiency consistency asym. unbiasedness asym. efficiencybiaedness inefficiency inconsistency asym. unbiasedness asym. inefficiencyReview of linear algebra1. a mrtrix A is idempotent iff AA=A.2. If the inner product of the 2 vectors vanishes (ie., the scalar is 0), then the vector areorthogonal. [An inner product (or scalar, or dot product) of 2 vectors is a row vector times a column vector, yielding a scalar. An outer product of 2 vectors is a column vector times a new vector, yielding a matrix ]3. The rank of any matrix A, ρ(A), is the size of the largest non -vanishing determinantcontained in A;Or, the rank is the maximum number of linearly independent rows or columns of A, wherea 1, a 2, … a N is a set of linearly independent vectors, iff∑==Nj j ja k10 ,necessarily implies021====N k k k →Ex.AA ,6832⎥⎦⎤⎢⎣⎡==12-24=-12≠ 0, =∴)(A ρ 2BB ,4263⎥⎦⎤⎢⎣⎡==12-12=0 , =∴)(B ρ 1→ properties:a. )(0A ρ≤=intege r ≦min(M,N) where A is an M ×N matrix.b. )(A ρ=N, 0)0(=ρc. )(A ρ=)'()'()'(AA A A A ρρρ==d. if A and B are of the same order, )()()(B A B A ρρρ+≤+.e. if AB is defined, )](),([min )(B A AB ρρρ≤f. If A is diagonal, )(A ρ= number of nonzero elements.g. If A is idempotent, )(A ρ=trace(A.)h. The ranks of a matrix is not changed if one row ( or column) is multiplied by a nonzero constant, or if such a multiple of one row (column) is added to another row (column).4. A aquare matrix of order N is nonsingular iff it is of full rank, )(A ρ=N (or ≠A 0);otherwise, it is aingular. The rank of matrix is unchanged by premultiplying or postmultiplying by a nonsingularmatrix. 5. differentiation in matrix notation (rules):a. If ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⨯⨯MMM MM a a a X X X 111, where a i =(i=1,2,…M) are constant, then=∂∂XX a )'( a.b. IF A is a symmetric matrix of order M×M where the typical element is a constant a ij , then=∂∂XAX X )'(2AX,if not a symmetric matrix=∂∂XAX X )'((A ’+A)X.c. IF A is a symmetric matrix of order M×M where the typical element is a constant a ij , then=∂∂∂')'(X X AX X 2A,=∂∂AAX X )'(XX ’If Y and Z are vectors, B is a matrix, then=∂∂YBZ Y )'(BZ ,=∂∂ZBZ Y )'(B ’Y ,=∂∂BBZ Y )'(YZ ’Formula for Matrix(A ’)’=NM A ⨯ =⨯⨯)'(KN N M B A B ’A ’A B ≠BA in general'')'(C A C A NM ±=±⨯111)(---⨯⨯=DE E DMM M M , iff det D≠0 and E≠0 ( ie. Iff D and E nonsingular matrices)(D -1)’= (D’)-1 Det D=det D ’Trace (D ± E )= trace (D) ±trace (E) Trace )()(NN MN N M FA trace F A ⨯⨯⨯=Trace (scalar)=saclar E[trace (D)]=trace (E(D))。