Estimation of the Mean and the ACVF
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1 Chapter 5 Estimatio of the Mea ad the ACVF A statioary process {X t } is characterized by its mea ad its autocovariace fuctio γ ), ad so by the autocorrelatio fuctio ρ ) I this chapter we preset the estimators of these statistics obtaied from observatios of X,,X ad discuss their properties 5 Estimatio of the Mea Deote by X = X,,X ) T, a -dimesioal radom vector each of whose compoets is a radom variable with expectatio µ i, that is E X = µ = µ,,µ ) T, ad whose variace-covariace matrix has the form varx ) covx, X 2 ) covx, X ) covx 2, X ) varx 2 ) covx 2, X ) V = covx, X ) covx, X 2 ) varx ) If {X t } is a statioary process the µ = µ,, µ) T = µ,, ) T 9
2 92 CHAPTER 5 ESTIMATION OF THE MEAN AND THE ACVF ad the variace covariace matrix simplifies to V = γ0) γ ) γ + ) γ) γ0) γ + 2) γ ) γ 2) γ0) The mea of a process is ot always zero ad its estimatio is importat for further iferece The momet estimator of the mea µ of a statioary process is the sample mea X = b T X, where b =,, ) T, X = X,,X ) T That is X = X i i= It is a ubiased estimator sice Eb T X) = b T EX) = b T µ =,, ) µ µ = µ The mea square error of X is E X µ) 2 = var X )
3 5 ESTIMATION OF THE MEAN 93 Usig the matrix otatio we ca write see Remark 32) var X ) = varb T X) = b T V b = 2,, ) = 2 = τ= + τ= + γτ) τ < γ0) γ ) γ + ) γ) γ0) γ + 2) γ ) γ 2) γ0) τ )γτ) τ ) γτ) Now, if γ) 0 as the the right had side coverges to zero It meas that X coverges to µ i the mea square sese If the series is Gaussia, the by the Remark 33, we have the ormality of X, X Nb T µ, b T V b) That is we ca write X N µ, v ), where v = τ < τ ) γτ) The a cofidece iterval for µ ca be obtaied so as P u α < X µ v/ < u α ) = α, what ca be rearraged to ) v P X u α < µ < X v + u α = α
4 94 CHAPTER 5 ESTIMATION OF THE MEAN AND THE ACVF Here u α is such that P U < u α ) = α ad U = X µ N0, ) For v/ α = 005 we have u α = 96 ad the cofidece iterval boudaries are ) v v X 96, X + 96, This results are obtaied assumig that v is kow I practice, usually it is ot the case ad we eed to estimate v To estimate v the covariace γτ) is replaced with γτ) ad ˆv is calculated as ˆv = τ ) ˆγτ) τ < Example 5 Let {X t } be a AR) process with mea µ, defied by X t µ = φx t µ) + Z t, where φ < ad Z t WN0, σ 2 ) For this process we have Hece, takig we obtai the result v = σ2 φ 2 τ < φ τ = γτ) = φ τ σ 2 φ 2 v = σ2 φ 2 τ < γτ) + 2 ) φ τ = τ=0 σ 2 φ) 2 I this case we eed to kow σ 2 ad φ to obtai v or their estimates to obtai ˆv 52 Estimatio of ACVF ad ACF Defiitio 43 gives the followig estimators for γτ) ad ρτ), respectively where γτ) = k τ X t k X k )X t+ τ X k ), k < τ < k 5) t= X k = k k X t t=
5 52 ESTIMATION OF ACVF AND ACF 95 ad ρτ) = γτ), k < τ < k 52) γ0) Both estimators are biased, however for for large k the bias is small The ACVF has the property that the k-dimesioal sample covariace matrix γ0) γ ) γk ) γ) γ0) γk 2) V k = γk ) γk 2) γ0) is oegative defiite To show it meas to show that a T Vk a 0 for ay k-dimesioal real vector a This ca be easily obtaied if we ca express the matrix V as the followig product V k = k CCT, for some matrix C Take vector rvs X = X,,X k ) T ad Y = X X,,X k X k ) T The Y Y 2 Y k 0 0 Y Y 2 Y k 0 C = 0 Y Y Y k 0 0 It is easy to see that multiplyig C by C T we obtai a matrix of sums of squares ad products of Y i which whe divided by k is the V k matrix Hece, a T Vk a = a T k CCT a = k at C)C T a) 0 Hece, due to the Theorem 4 γτ) is a autocovariace fuctio of a statioary process as it is oegative defiite ad eve We will be usig the forms 5 ad 52 as the estimators for the ACVF ad ACF The estimates of ρτ) are good if τ <<, where is the total umber of observatios For τ close to there are too few pairs X t, X t+τ ) for the estimate to be
6 96 CHAPTER 5 ESTIMATION OF THE MEAN AND THE ACVF reliable Box ad Jekis 976) suggest that should be at least 50 ad τ /4 For statistical iferece based o the ρτ) we eed to kow its distributio For large sample size it ca be approximated by a ormal distributio For liear models the vector ρ = ρ),, ρk)) T is approximately distributed as ρ N ρ, ) W, 53) approx where ρ = ρ),,ρk)) T ad W is the variace-covariace matrix where w ij is give by Bartlett s formula W = {w ij }, 54) w ij = [ρk + i) + ρk i) 2ρi)ρk)][ρk + j) + ρk j) 2ρj)ρk)] k= Example 52 Let {X t } IID0, σ 2 ) The ρτ) = 0 for all τ > 0 ad from 54 we obtai { if i = j w ij = 0 otherwise The by 53) the estimators ρτ) are approximately idepedet ad idetically distributed as ρτ) N 0, ) approx This gives us the cofidece bouds for ρτ) of a IID process, which are u α /, u α / ), where u α is such that P U < u α ) = α, where U N0, ) For α = 005 we have u α 96 ad the 95% cofidece iterval boudaries are 96/, 96/ )
7 52 ESTIMATION OF ACVF AND ACF 97 Example 53 Cosider MA) process X t = Z t + θz t, t = 0, ±, ±2,, where {Z t } WN0, σ 2 ) The from equatio 54) we obtai { 3ρ w ii = 2 ) + 4ρ 4 ), if i = + 2ρ 2 ), if i > The by 53) we get the cofidece iterval for ρ), amely ˆρ) u α 3ˆρ2 ) + 4ˆρ 4 )), ˆρ) + u α 3ˆρ2 ) + 4ˆρ 4 )) 55) We kow that for τ > the true value of the ACF is zero, hece as a kid of test we ca calculate the iterval i which the obtaied sample values of the ACF are ot sigificat This is ) u α + 2ˆρ2 )), u α + 2ˆρ2 )) ), 56) Now, take θ = 05 as i Example 43 The the theoretical value of ρ) is ρ) = From the simulatio we obtaied θ + θ = = 04 ρ) = ad so the 95% cofidece iterval is approximated be , ) = , ) which icludes the theoretical value of ρ) For lag τ > we have , ) = , ) I fact the bouds are ofte calculated accordig to the formula for IID oise, which depeds oly o Here = 00 ad we obtai u α /, u α / ) = 096, 096) Figure 44 shows such boudaries ad ideed all the sample autocorrelatios for lag τ > are withi these boudaries For lag τ = we obtaied the CI coverig the true value of ρ) These two facts support the compatibility of the simulated data with MA) model with θ = 05
8 98 CHAPTER 5 ESTIMATION OF THE MEAN AND THE ACVF Example 54 Cosider AR) process X t = φx t + Z t, where {Z t } is a iid oise ad φ < The the theoretical ACF is give by ρτ) = φ τ for ay τ = 0, ±, ±2, From the Bartlett s formula 54) for the variaces ad covariaces of ρ ad the form of ρτ) for AR) we obtai τ w ττ = φ 2τ φ k φ k ) 2 + φ 2k φ τ φ τ ) 2 k= k=τ+ = φ 2τ ) + φ 2 ) φ 2 ) 2τφ 2τ, 57) for τ =, 2, The, due to 53) the approximate cofidece bouds ca be computed as ) ˆρτ) u α wττ /, ˆρτ) + u α wττ / 58) Take φ = 05 as i the bottom plot of Figure 49 The sample ACF is give i the Figure 40 Is this sample ACF compatible with AR) for φ = 05? What coclusios ca you draw from the figure below?
9 52 ESTIMATION OF ACVF AND ACF Theoretical ACF Lower boud Upper boud Sample ACF Lag Figure 5: Theoretical ACF, sample ACF ad the CI bouds for the simulated AR) with φ = 05
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