In constructing kernel estimators of the spectral and cross-spectral densit -' of tationary time series, we would like to have a measure of their accuracy in terms of confidence intervals and confidence bands.It is relatively straightforward to develop asymptotic confidence intervals for estimating the value of the spectral density at a particular frequency (cf.[1],[4],[5]), and some asymptotic confidence bands (i.e. simultaneous confidence intervals for the whole curve) have However the applicability of such asymptotic confidence bands is debatable, because although the kernel spectral estimates at any two distinct fixed frequencies are asymptotically uncorrelated (cf.[4],p.456), in a finite sample setting they are not. In fact, one of the objectives in smoothing the periodogram (which also has asymptotically uncorrelated ordinates), is exactly to inject more correlation between estimates at neighboring frequencies.In this paper we focus on constructing nonparametric bootstrap confidence intervals and bands from kernel and lag-window spectral estimators that can be of use in a finite sample situation, especially in the case where we cannot assume the time series is Gaussian. It is conceivable that the bootstrap will provide a better penultimate (instead of purely asymptotic) approximation to the distribution of spectral estimates. Suppose X I , . . . , XN are observations from the 2nd order stationary process Xt with autocorrelation R(k) and spectral density f(w). Then the kernel estimator of f(w) using the Bartlett window is: been proposed [11,[21,[31. where R(s) is the sample autocorrelation sequence. The key idea which originally led Bartlett [6],[7] to propose this estimator is that, apart from some "end effects" corrections, this procedure is equivalent to splitting the observed series into [N/M] subseries each of length M, computing a periodogram for each subseries, and then averaging the [N/M] periodograms. By computing a tapered periodogram for each subseries, (with an appropriate choice of data window), and then averaging, we can obtain spectral estimates corresponding to an arbitrary spectral window ([4],p.585).So the kernel spectral estimator evaluated at discrete frequencies is nothing more than a sample mean of (tapered) periodograms and can be treated as a sample mean of [N/M] random vectors. The bootstrap procedure now amounts to resampling from these random vectors, which are in direct correspondence with the [N/M] subseries. A similar resampling scheme can be applied to a kernel estimator of the cross-spectral density, by replacing R(k) 88
A central problem in time series analysis is prediction of a future observation. The theory of optimal linear prediction has been well understood since the seminal work of A. Kolmogorov and N. Wiener during World War II. A simplifying assumption is to assume that one-step-ahead prediction is carried out based on observing the infinite past of the time series. In practice, however, only a finite stretch of the recent past is observed. In this context, Baxter's inequality is a fundamental tool for understanding how the coefficients in the finite-past predictor relate to those based on the infinite past. We prove a generalization of Baxter's inequality for triangular arrays of stationary random variables under the condition that the spectral density functions associated with the different rows converge. The motivating examples are statistical time series settings where the autoregressive coefficients are re-estimated as new data are acquired, producing new fitted processes -and new predictors -for each n.
Introduction. Nontraumatic splenic rupture is a rare event. On the other hand, splenic metastasis is also rare and usually found in the context of disseminated disease. Spontaneous splenic rupture caused by a metastatic deposit has been reported only as case reports. To the best of our knowledge, pancreatic cancer being the primary site has been reported in only a handful of cases. Case Presentation. A case of spontaneous splenic rupture in a 68-year-old male presenting with acute onset left shoulder pain, caused by metastatic pancreatic cancer to the spleen, is presented herein. During the investigation, the patient developed hypovolemic shock due to intra-abdominal hemorrhage, necessitating emergency splenectomy. The patient recovered well and was discharged from the hospital on the 8th postoperative day. Discussion. This study underlines the fact that the spleen is a rare site of metastasis, remaining mostly asymptomatic. However, it can nevertheless prove to be a life-threatening condition. The exact pathophysiological mechanism of splenic rupture due to metastasis still remains unknown, requiring further research. Emergency splenectomy remains the standard of care, and this clinical condition should be considered in the differential diagnosis of cases with acute abdomen and malignant neoplasm history.
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