Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If you need an accessible version of this item please contact JSTOR User Support

Variable Length Markov Chains

Peter Buhlmann and Abraham J. Wyner
The Annals of Statistics
Vol. 27, No. 2 (Apr., 1999), pp. 480-513
Stable URL: http://www.jstor.org/stable/120101
Page Count: 34
  • Read Online (Free)
  • Download ($19.00)
  • Cite this Item
If you need an accessible version of this item please contact JSTOR User Support
Variable Length Markov Chains
Preview not available

Abstract

We study estimation in the class of stationary variable length Markov chains (VLMC) on a finite space. The processes in this class are still Markovian of high order, but with memory of variable length yielding a much bigger and structurally richer class of models than ordinary high-order Markov chains. From an algorithmic view, the VLMC model class has attracted interest in information theory and machine learning, but statistical properties have not yet been explored. Provided that good estimation is available, the additional structural richness of the model class enhances predictive power by finding a better trade-off between model bias and variance and allowing better structural description which can be of specific interest. The latter is exemplified with some DNA data. A version of the tree-structured context algorithm, proposed by Rissanen in an information theoretical set-up is shown to have new good asymptotic properties for estimation in the class of VLMCs. This remains true even when the underlying model increases in dimensionality. Furthermore, consistent estimation of minimal state spaces and mixing properties of fitted models are given. We also propose a new bootstrap scheme based on fitted VLMCs. We show its validity for quite general stationary categorical time series and for a broad range of statistical procedures.

Page Thumbnails

  • Thumbnail: Page 
480
    480
  • Thumbnail: Page 
481
    481
  • Thumbnail: Page 
482
    482
  • Thumbnail: Page 
483
    483
  • Thumbnail: Page 
484
    484
  • Thumbnail: Page 
485
    485
  • Thumbnail: Page 
486
    486
  • Thumbnail: Page 
487
    487
  • Thumbnail: Page 
488
    488
  • Thumbnail: Page 
489
    489
  • Thumbnail: Page 
490
    490
  • Thumbnail: Page 
491
    491
  • Thumbnail: Page 
492
    492
  • Thumbnail: Page 
493
    493
  • Thumbnail: Page 
494
    494
  • Thumbnail: Page 
495
    495
  • Thumbnail: Page 
496
    496
  • Thumbnail: Page 
497
    497
  • Thumbnail: Page 
498
    498
  • Thumbnail: Page 
499
    499
  • Thumbnail: Page 
500
    500
  • Thumbnail: Page 
501
    501
  • Thumbnail: Page 
502
    502
  • Thumbnail: Page 
503
    503
  • Thumbnail: Page 
504
    504
  • Thumbnail: Page 
505
    505
  • Thumbnail: Page 
506
    506
  • Thumbnail: Page 
507
    507
  • Thumbnail: Page 
508
    508
  • Thumbnail: Page 
509
    509
  • Thumbnail: Page 
510
    510
  • Thumbnail: Page 
511
    511
  • Thumbnail: Page 
512
    512
  • Thumbnail: Page 
513
    513