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^ 2.3.2.2 Properties of the Estimator By construction, a is unbiased under ID1a and ID2, and if, in addition, we assume ID1b its variance is simply Var ^ a s2 w2 : j j 2:24



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Use an application-level heartbeat to learn whether the application has returned to its main loop or listening state. It if gets caught processing a request, the heartbeat stops and the equivalent of a failover management system can step in and kill it with the intent of restarting it, or can try to send it secret input to grab control of the application. Make sure you understand the deadlock detection and back-out procedures for whatever scheduling facilities you use. If you need to back out or reschedule a task, don t just blindly go down the same path; execute with a delay or out-of-order to prevent future deadlocks. Don t mix and match locking mechanisms. If two applications are locking the same pieces of data in different orders, or using different granularities of locks, eventually they re going to deadlock and get stuck in a deadly embrace, where each one waits for the other.





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Note that the variance is a function of the amount of data, the nj , but is independent both of the data itself, Eq. (2.22), and of the actual (unknown) value of a. It is also independent of the precise choice of mother wavelet, except indirectly through the choice of N , the number of vanishing moments. A quantitative study of this dependence is given by Delbeke and Abry [27].

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It has been shown [75], in the limit of nj large for each j in j1 ; j2 , that the ^ Cramer Rao bound for the full problem is attained, showing that a is asymptotically the minimum variance unbiased estimator under ID1 ID2. The decrease in variance ^ of a as a function of the size of the data is then explicitly seen to be 1=n, a remarkable result, being the rate of decay typical of SRD problems, yet appearing in a dif cult scaling context. Numerical comparisons [75] show that away from the ^ limit the variance of a remains extremely close to the Cramer Rao bound. This is not surprising as the assumption that nj is large is a very good one, except possibly for the nj corresponding to the largest j, since nj 1 % nj =2. Examining the limit in more detail, for large nj we have [75] log2 e nj 2 log2 e 2 : nj

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The rst of these relations indicates that for nj large, yj can be identi ed with log2 mj . It has moreover been shown [3] that, under ID1 and ID2, log2 mj is asymptotically normally distributed: 3 2 log2 e 2 log2 mj $ N ja log2 cf C ; : nj

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^ Since a consists of a sum of the yj , most of which are approximately Gaussian and ^ weighted according to their (known) variances, a, can be considered as approxi^ mately Gaussian distributed. Con dence intervals for the yj and a have been calculated using these arguments. 2.3.2.3 Robustness with Respect to ID1 and ID2 Simulation studies show [5, 75] that the above properties hold to an excellent approximation, even for small size data, upon the mild departures from ID1 characteristic of the FGN series used. Numerical simulations presented by Abry and Veitch [5], as well as those described below, show that the above properties also hold to an excellent approximation when the Gaussian hypothesis ID2, as well as ID1, is dropped. The robustness with respect to ID2 can be justi ed using the following asymptotic arguments. Let Y f X and s2 and s2 be the variances of X and Y, respectively. X Y Standard approximation formulas for a change of variable [54] are Ef X 9 f EX f HH EX s2 =2 and s2 9 j f H EX j2 s2 . Because Var mj decreases X Y X

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