Change-Point Estimation

Transcription

1 Change-Point Estimation. Asymptotic Quasistationary Bias For notational convenience, we denote by {S n}, for n, an independent copy of {S n } and M = sup S k, k<inf as the maximum of {S n} when S = and σ M = arg sup S k k< as the corresponding maximum point. Conditional on N>ν, depending on whether ˆν >νor ˆν <ν, we can write the bias as ˆν ν =(ˆν ν)i [ˆν>ν] (ν ˆν)I [ˆν<ν], where I A denotes the indicator function for the event A. The notations are consistent with Chapter and, in addition, we shall denote by E θθ[.] the expectation when both P θ and P θ are involved. 5

2 6. Change-Point Estimation Pollak and Siegmund (986) shows that the quasistationary distribution of T ν converges to the stationary distribution of T ν as d. That means, lim lim P θ d ν (T ν <y N >ν) = lim lim P θ d ν (T ν <y) = P θ (M<y), as we note that T ν is asymptotically equivalent to M in distribution as ν. This implies that when the change occurs, T ν is asymptotically distributed as M. Thus, the event {ˆν >ν} is asymptotically equivalent to the event {τ M < }, i.e., the random walk S n eventually goes below to zero with initial starting point M. Given ˆν >ν, the bias ˆν ν is asymptotically equal to τ M plus the length, say γ m for a CUSUM process T n starting from zero until the last zero point time under P θ (.). Define E[X; A] =E[XI A ]. As d, ν,wehave E ν [ˆν ν;ˆν>ν] E θθ[τ M + γ m ; τ M < ] = E θθ[τ M ; τ M < ]+E θ [γ m ]P θθ(τ M < ). We can see that γ m is a geometric summation of i.i.d. random variables distributed as {τ ; τ < } with terminating probability P θ (τ = ). Thus, we have Lemma.: E θ γ m = E θ[τ ; τ < ]. P θ (τ = ) On the other hand, given ˆν <ν, by looking at T k backward in time starting from ν, we see that T νk behaves like a random walk {S k } for k with maximum value M and thus, ˆνν is asymptotically distributed as the maximum point σ M. Thus, as d, ν,wehave E ν [ν ˆν;ˆν<ν] E θθ[σ M ; τ M = ] = E θ [σ M P θ (τ M = )]. A similar argument is used in Srivastava and Wu (999) for the continuous-time analog. Summarizing the results, we get the following asymptotic first-order result. Theorem. : As ν, d, E ν [ˆν ν N >ν] E θθ[τ M ; τ M < ]+P θθ(τ M < ) E θ[τ ; τ < ] P θ (τ = ) E θθ[σ M ; τ M = ], E ν [ ˆν ν N >ν] E θθ[τ M ; τ M < ]+P θθ(τ M < ) E θ[τ ; τ < ] P θ (τ = ) +E θθ[σ M ; τ M = ].

3 .. Second-Order Approximation 7. Second-Order Approximation In this section, we shall derive the second-order approximation for the asymptotic bias given in Theorem. in order to investigate the bias numerically and also see some local properties by further assuming θ and θ approach zero at the same order. The main theoretical tool is the strong renewal theorem given in Section.3 and its applications to ladder variables. There are five terms in Theorem. which will be evaluated in a sequence of lemmas. Most of the results generalizes the ones given in Wu (999) in the fixed sample size case with normal distribution. However, the technique used here is much more general and can be used for any distribution of exponential family type and also raises more difficulties due to sequential sampling plan. The first lemma gives the approximation for the expected length τ if the random walk goes below zero. Lemma.: As θ, E θ [τ ; τ < ] = E S τ exp (θρ + θ µ ( ρ () ρ 5β )) (+o(θ ) ), E S τ where β is given in Lemma.. Proof: By using Wald s likelihood ratio identity by changing the measure P θ (.) top θ(.) and Lemma., we have E θ [τ ; τ < ] = E θ[τ e Sτ ] = µ E θs τ + E θ(τ S τ )+ E θ After some algebraic simplifications, we get the result. Corollary.: As θ, E θ γ m = µ e(β/esτ )θ ( + o(θ )). (τ S τ ) + o( ). To evaluate P θθ(τ M < ), we follow a similar technique used in Wu (999) and only the main steps are provided. First, by conditioning on whether M =orm>, we have From Lemma., we have P θθ(τ M < ) = P θ (τ < )P θ (τ + = ) + P θθ(τ M < ; M>). (.) P θ (τ < )P θ (τ + = ) = E S τ+ e θρ+ (+ E S τ+ )+o(θ ) = E S τ+ e θsτ + + o(θ ).

4 8. Change-Point Estimation For the second term in (.), by using the Wald s likelihood ratio identity by changing parameters θ to θ and θ to θ,wehave and From Corollary., we know P θ (τ x < ) = E θe Sτ x = e x E θe Rx, P θ (M>x) = P θ (τ x < ) = e x E θ e Rx. E R x ρ + = O ( e rx) and E R x ρ = O ( e rx), as x. Now, we write P θθ(τ M <,M >) = = = d P θ (τ x < )dp θ (M>x) E θe Sτ x deθ e Sτx e (xρ) de (x+ρ+) ( ( e (xρ) d e (x+ρ+) E θ e (Rxρ+)) ) ( ) e (xρ) E θ e (Rxρ) de (x+ρ+) ( ) e (xρ) E θ e (Rxρ) ( ( e (x+ρ+) E θ e (Rxρ+)) ). (.) The first term in (.) is e ρ ρ+. + The third term in (.) is approximately equal to (E R x ρ )dx + o(θ ). The fourth term in (.) is E (R x ρ )de (R x ρ + )+o(θ ).

5 .. Second-Order Approximation 9 The second term in (.), by integrating by parts, can be approximated as e ( ρ P θ (τ + < ) e ρ+) + (E R x ρ + )dx + o(θ ). Combining the above approximations, we have Lemma.3: As θ,θ at the same order, P θθ(τ M < ) = + e ρ ρ+ + e ρ ( e ρ+) + ( ρ E S τ + + (E R x ρ )d(e R x ρ + )+ (E R x ρ )dx + o(θ ). (E R x ρ + )dx) The evaluation of E θθ[τ M ; τ M < ] is similar. Lemma.4: As θ,θ at the same order, E θθ[τ M ; τ M < ] = ( µ ( + ) ρ ) + µ ( + ρ (E S τ+ ρ + ) (E R x ρ )d(e R x ρ + ) (E R x ρ + )dx)+o(). (E R x ρ )dx Proof: Again depending on whether {M =} or {M >}, wehave E θθ[τ M ; τ M < ] = E θ [τ ; τ < ]P θ (τ + = ) E θ [τ x ; τ x < ]dp θ (τ x < ). (.3) The first term in (.3) can be approximated by using Lemmas. and. as E θ [τ ; τ < ]P θ (τ + = ) = µ µ + o() = µ + o(). For the second term in (.3), by conditioning on the value of M given M> we use the similar techniques as in Lemma.3 and write it as

6 . Change-Point Estimation E θ [τ x ; τ x < ]dp θ (τ x < ) = E θ [ τ x e (xrx)] de θ e (x+rx) = E θ(τ x )e (xρ) de θ e (x+rx) + o() = µ [ (x + E R x )e (xρ) de (x+ρ+) (x + E R x )d(e R x ρ + )] + o(). (.4) The first term in (.4) is approximately equal to ( [e ρ ρ+ µ ( + ) ρ ) + The second term in (.4) is equal to µ [ (x ρ )d(e R x ρ + ) = µ [ρ (E S τ+ ρ + ) (E R x ρ )d(e R x ρ + )]. (E R x ρ + )dx Combining the above results, we complete the proof. Finally, we evaluate E θθ[σ M ; τ M < ]. We first write ] (E R x ρ )dx + o(). ] (E R x ρ )d(e R x ρ + ) E θθ[σ M ; τ M < ] =E θ σ M E θ [σ M E θe (M+R M ) ]. (.5) For the second term in (.5), we write E θ [σ M E θe (M+R M ) ] = E θ [ σm e M ] e ρ To evaluate E θ [σ M e M ], we note that under P θ (.) + E θ [ σm e M ( E θe R M e ρ)]. (σ M,M)= d (τ (K) +,S (K) τ ), where = d means equivalence in distribution, τ (k) + is the kth ladder epoch defined in Section.3, and K = inf{k >: τ (k) + < }. Note that K is a geometric random variable with + P (K = k) =p k ( p)

7 .. Second-Order Approximation for k, with terminating probability p = P θ (τ + = ). For given K = k, (τ (k) +,S (k) τ ) is, in distribution, equivalent to the sum of k i.i.d. + random variables distributed as (τ +,S τ+ ). Thus, [ E θ σm e M ] [ ] = E θ τ (K) + exp( S (K) τ ) = = = k= E θ [ τ (k) + + exp( S τ (k) + ] ); K = k [ ke θ τ+ exp( S τ+ ); τ + < ] k= ( [ E θ exp( Sτ+ ); τ + < ]) k Pθ (τ + = ) [ E θ τ+ exp( S τ+ ); τ + < ] ( [ Eθ exp( Sτ+ ); τ + < ]) P θ (τ + = ). The next two lemmas give the approximations for the related quantities. Lemma.5: As θ,θ at the same order, [ E θ exp( Sτ+ ); τ + < ] ( =( + )E S τ+ exp ( θ )ρ + + ( θ ) (ρ () + ρ +) θ ( + o(θ )). α E S τ+ ) Proof: Using Wald s likelihood ratio identity, we have E θ [ exp( Sτ+ ); τ + < ] = E θ e ( + )Sτ +. The Taylor series expansion following the lines of Lemma. will give the result after some algebraic simplification. In particular, by letting =, we have ( P θ (τ + = ) = E S τ+ exp θ ρ + + ( θ ρ () + ρ + α )) ( + o(θ E S )). τ+ The following lemma can be proved similarly as for Lemma., and its proof is omitted.

8 . Change-Point Estimation Lemma.6: As θ,θ at the same order, ] E θ [τ + e Sτ + ; τ+ < = E S τ+ exp(( θ )ρ + + µ ( θ ) (ρ () + ρ +) θ α α θ ( + ) )( + o(θ )). E S τ+ E S τ+ In particular, by letting =, we have E θ [τ + ; τ + < ] = E ( S τ+ exp θ ρ + + θ µ ( ρ () + ρ + 5α E S τ+ On the other hand, by conditioning on the value of M, wehave )) ( + o(θ )). E θ [ σm e M ( E θe R M e ρ)] (.6) = E θ [σ M (E R M ρ )]( + o()) = E θ [σ x M = x](e R x ρ )dp θ (M>x) = E θ [σ x M = x](e R x ρ )d(x + E R x )( + o()). Since E θ [S τ+ ; τ + < ] =E S τ+ ( + o()), as θ, thus, K = O p (x), where O p (.) means at the same order in probability. This implies E θ [σ x M = x] =O( x ). µ Thus, (.6) is at the the order of O(θ). By letting =, the first term of (.5) can be evaluated by combining Lemmas.5 and.6. Lemma.7: As θ, E θ σ M = E θ [τ + ; τ + < ] P θ (τ + = ) = e θ (α/esτ + ) ( + o(θ µ )). Finally, we have the following result: Lemma.8: As θ,θ, E θθ[σ M ; τ M = ] = e θ µ (α/esτ + )

9 .. Second-Order Approximation 3 () µ ( + ) eγ /3θ(θθ)(ρ + ρ + α/esτ + )(θθ) (α /E S τ+ ) ( + o(θ )). Combining Lemmas..8 and Lemmas.., we have the following local second-order expansion for the asymptotic bias of ˆν. Theorem.: As θ,θ at the same order, we have lim lim d ν Eν [ˆνν N >ν]= ( e ρ ρ+ + e ( ρ e ρ+) ) µ + µ ( ( + ) ρ ) e ρ ρ+ + + θ β + θ θ θ E S τ+ θ θ θ θ ρ ρ + ( ρ () + ρ + α E Sτ + ) + o(). A similar result can be obtained for the absolute bias E ν [ ˆν ν N >ν]. In particular, when θ = θ, we have the following result. Corollary.: As θ = θ, lim d lim ν E ν [ˆν ν N >ν] = 3 ( + ) γ ( + ) β 4 µ µ µ µ E S τ ( ρ () + ρ + α ) + o() E S τ+ = γ + 7γ 4θ 88 κ 6 β E S τ ( ρ () + ρ + α ) + o(). E S τ+ Proof: The first equation is a direct simplification of Theorem.. For the second equation, we note that as θ, µ = θ e θ(γ/)+θ (κ/6γ/8) ( + o(θ)), = θ e θ(γ/6)+θ (γ/4) ( + o(θ)), θ = θ e θ(γ/3)+θ (γ/8) ( + o(θ)), µ = θ e θ(γ/6)+θ (κ/67γ/7) ( + o(θ)). Some tedious simplifications give the expected results.

10 4. Change-Point Estimation Therefore, the local bias of ˆν is largely affected by the skewness γ. Ifγ>, the local bias becomes positive. If F (x) is symmetric, from Corollary., we have ρ () + ρ + α = κ E S τ+ 6, and thus, E ν [ˆν ν N >ν] 7 48 κ β + o(), E S τ which, surprisingly, is a nonzero constant, in contrast to the fixed sample size case as given in the normal case of the next section..3 Two Examples In this section, we present two cases: normal and exponential distributions..3. Normal Distribution From Example., the approximations for the related quantities are simplified as E θ (γ m ) = θ 4 + o(), P θθ(τ M < ) = θ e θ(θθ) + o(θ ), θ θ E θθ[τ M ; τ M < ] = θ θ(θ θ ) e(θθ) + o(), E θθ[σ M ; τ M < ] = θ (θ θ ) + o(). Summarizing the above results, we have the following corollary: Corollary.3: As θ,θ at the same order, lim lim d ν Eν [ˆν ν N >ν] = lim d lim ν Eν [ ˆν ν N >ν] = At θ = θ = θ, θ θ θ + 4(θ θ ) + o(), ( θ + ) θ (θ θ ) + lim lim d ν Eν [ˆν ν N >ν] = 8 + o(), lim lim 3 d ν Eν [ ˆν ν N >ν] = 8 + o(). 4θ θ 4(θ θ ) + o().

11 .3. Two Examples 5 Remark: Wu (999) considered the bias of the estimator in the large fixed sample size case, which corresponds to the maximum point of a two-sided random walk, and obtained the following result: E ν [ˆν ν] = θ θ + θ + θ + o(); 4 θ θ and at θ = θ, E ν [ ˆν ν ] = 3 4θ 4 + o(). Srivastava and Wu (999) also considered the continuous-time analog in the sequential sampling plan case, which gives lim lim d ν Eν [ˆν ν N >ν]= θ θ, which is zero at θ = θ and at θ = θ, lim lim d ν Eν [ˆν ν N >ν]= 3 4θ. We see that the sequential sampling plan has local effect at the second-order in the discrete-time case and is negative at θ = θ. To show the accuracy of the second-order approximations, we conduct a simple simulation study. For d = and θ =.5,.5, we let ν = 5 and. One thousand replications of the CUSUM charts are simulated for several values for θ. Only those runs with N>νare used for calculating ˆν. Table. gives the comparison between the simulated results and approximated values. The approximated values from Corollary.4 are given in the parentheses. We see that the approximations are generally good. The case ν = shows quite satisfactory results. Also, we see that approximations for the case θ =.5 perform better than those for the case θ =.5. The reason is that our results are given by first assuming d, ν and then letting θ,θ. The effect of ν is very little. However, as the local bias is at the order O(/θ)at θ = θ, which approaches as θ, there could be an error term at the order, say O(/(dθ)) for finitely large d. Thus, the approximation may perform better for θ = θ =.5. The case when θd approaches a constant, called moderate deviation as considered in Chang (99), is definitely worths a future study..3. Exponential Distribution Here, we are interested in quick detection of increment in the mean of an exponential distribution from the initial mean. From Example. and Theorem., we have the following result: Corollary.4. As θ, θ at the same order,

12 6. Change-Point Estimation Table.: Biases in the Normal Case ν θ θ E[ˆν ν N >ν] E[ ˆν ν N >ν] (.5) 9.737(.875).5 4.9(6.83) 7.9(8.39) (7.74) 6.376(7.86) (7.55) 6.88(7.8) (.5) 3.(.875).75.3(.).338(.49)..73(.583) 3.35(.97) (.5).78(.875) (6.83) 7.768(8.39) (7.74) 6.94(7.86). 6.5(7.55) 6.5(7.8).5.5.3(.5) 3.5(.875).75.9(.).8(.49)..564(.583).84(.97) lim d lim ν E ν [ˆν ν N >ν] At θ = θ, = + µ( + ) e(/) µ e / ( e ) µ( + ) e(/) + µ µ ( + ) e(/3) θ θ θ θ θ + 7 θ + o(). 8 θ θ lim d lim ν Eν [ˆν ν N >ν]= θ o(). We see that due to the asymmetry of F (x), the local bias becomes positive as θ is small..4 Case Study In this section, we conduct three classical case studies to illustrate the applications.

13 .4. Case Study 7 Table.: Nile River Flow from Year Flow Year Flow Year Flow Year Flow Nile River Data (Normal Case): The following Nile River flow data are reproduced from Cobb(978), and the data are read in columns. From a scatterplot, we see there is an obvious change around the year 9. A qq-normal plot shows the normality is roughly true. As in Cobb(978), we assume the independent normality. Assume the prechange mean is m = and the post-change mean is m = 85, with a change magnitude m m = 5 and standard deviation 5. To apply the CUSUM procedure, we first standardize the data by subtracting all the observations by m +(m m )/ = 975, the average of the pre-change and post-change means, then switch the sign in order to make the change positive, and then divide all the data by 5. After these transformations, we standardize the observations to x i for i =

14 8. Change-Point Estimation,,..., such that θ =, θ =, and σ =. Now we form the CUSUM process by letting T = and T i = max(,t i + x i ), for i =,...,. The calculated values are reported as follows: [] [6] [3] [46] [6] [76] [9] The reason we report the CUSUM process instead of drawing graphs is to inspect the path directly rather than guessing from the graph. By looking at the CUSUM process, we see that in the fixed sample size case, the maximum likelihood ratio estimator (the last zero point of the CUSUM process) is ˆν n =8 which is the year 898. The estimated pre-change mean is the average of the observations from 97 to 998, which is.98; the estimated post-change mean is.; and the prechange standard deviation is.8 and post-change standard deviation is.. We see that the assumption is roughly correct except for the slight discrepancy from the pre-change standard deviation. Let us look at the estimator from the sequential sampling plan point of view, which is more natural from the nature of monitoring. We see that as long as the threshold d is taken larger than. and less than 7, a change is always signaled and the maximum likelihood estimator ˆν =8 no matter what the value d is. Thus, we see that the estimator ˆν is stable to the selection of the threshold d..4. British Coal Mining Disaster (Exponential Case) The following table gives the intervals in days between successive coal mining disasters in Great Britain for the period A disaster is defined as involving the death of or more men. The data are taken from Maguire, Pearson, and Wynn(95) and appeared in many places; the most noticeable is Cox and Lewis (966). The data are read in columns.

15 .4. Case Study 9 Table.3: British Coal Mining Disaster Intervals

16 3. Change-Point Estimation We first explain the transformation on the data in the exponential case in order to fit them into the frame. Suppose the original data {Y i } follow exp(λ ) for i ν and exp(λ ) for i>ν where λ >λ are the corresponding hazard rates. Define λ =(λ λ )/ ln(λ /λ ), and make the following data transformation: X i = λ Y i, for i =,,... Denote for x, and then f (x) =e (x+), satisfies the standardized model with and f θ (x) = exp(θx c(θ))f (x) c(θ) =(θ + ln( θ)), θ i = λ i /λ for i =, such that c(θ )=c(θ ). A scatterplot or by looking at the data directly shows that there is a change around the observation 5. So we take the mean from observations to 5 as the pre-change mean and the mean from observations 5 to 9 as the post-change mean, which gives λ =/9 and λ =/335, and λ =.5. Now, after the data transformation by letting x i =.5y i for i =,...9, we can fit the data into the standardized model with θ =.55 and θ =.43. Next, we can formalize the CUSUM process by calculating T n s based on x i s, which are reported as follows: [] [6]..... [] [6] []..... [6]

17 .4. Case Study 3 [3] [36] [4] [46] [5] [56] [6] [66] [7] [76] [8] [86] [9] [96] [] [6] By inspecting the CUSUM process, we see that the estimator ˆν = 46 no matter what the threshold d is (at least 5 and at most 4). Again, we see that the CUSUM procedure is very reliable in terms of the change-point estimator..4.3 IBM Stock Price (Variance Change) Suppose the original independent observations {Y i } follow N(,σ ) for i ν and N(,σ ) for i>ν. For a reference value σ for σ, we define ( λ = σ σ ) / ln and make the following data transformation: ( ) σ σ X i = (λ Y i ). Let f (x) be the density function of (χ )/, where χ is the standard chi-square random variable with degree of freedom. Then from Example.3, we know c(θ) = θ ln( θ). Define and generally θ = ) ( /σ λ and θ = ( /σ θ = ( ) /σ λ. λ ),

18 3. Change-Point Estimation It can be verified that c (θ) = ( ) λ /σ and c(θ )=c(θ ). Thus, the standardized observations {X i } fit in our model. Remark: Generally, the original independent observations {Y i } follow ( + ɛ ) χ p for ı ν and ( + ɛ ) χ p for i>ν, where χ p is the standard chi-square random variable with degree of freedom p. Then, by using Example.3, we define ( λ = ( + ɛ ) ( + ɛ ) and make the following transformation: X i = p (λ Y i p). ) / ln[( + ɛ ) /( + ɛ ) ], For f (x) defined as in Example.3, let θ = p/ ( ) ( + ɛ) λ, and define θ and θ correspondingly. Then we can verify that c(θ )=c(θ ). The following data set is taken as the IBM stock daily closing prices from May 7 of 96 to Nov. of 96, [Box, Jenkins, and Reinsel(994), pp.54)] for a total of 369 observations. The data are read in rows. We use the geometric normal random walk model and find that it fits the data quite well with quite small autocorrelation. After taking the difference for the logarithm of the data, the plot of total 368 data shows an obvious increase in the variance roughly around the 5th observation. The standard deviation for the first 5 observations is found to be.978. So we divide all the data by.978, and denote the modified data as {Y i } s, which gives σ = and σ =7.39, where σ is the variance of the last 43 modified observations. Now, we calculate λ as ( λ = σ ) ( ) σ σ / ln σ =(/7.39)/ log(7.39) = The transformed data are calculated as X i = (λ Yi ), and the corresponding conjugate parameters are found to be θ = ( ) =

19 .4. Case Study 33 Table.4: IBM Stock Price

20 34. Change-Point Estimation and θ = ) ( = The CUSUM process formed by the standardized {X i } s are calculated as [] [3] [5] [37] [49] [6] [73] [85] [97] [9] [] [33] [45] [57] [69] [8] [93] [5] [7] [9] [4] [53] [65] [77] [89] [3] [33] [35] [337] [349] [36]

21 .4. Case Study 35 We see that no matter what the threshold d will be (at least 5 and at most 3), the change-point estimator is consistently ˆν = 34. The estimation for the post-change parameter is considered in Chapter 4.

22