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

Probability Inequalities for Likelihood Ratios and Convergence Rates of Sieve MLES

Wing Hung Wong and Xiaotong Shen
The Annals of Statistics
Vol. 23, No. 2 (Apr., 1995), pp. 339-362
Stable URL: http://www.jstor.org/stable/2242340
Page Count: 24
  • Read Online (Free)
  • Download ($19.00)
  • Subscribe ($19.50)
  • Cite this Item
If you need an accessible version of this item please contact JSTOR User Support
Probability Inequalities for Likelihood Ratios and Convergence Rates of Sieve MLES
Preview not available

Abstract

Let Y1,..., Yn be independent identically distributed with density p0 and let F be a space of densities. We show that the supremum of the likelihood ratios $\prod^n_{i=1} p(Y_i)/p_0(Y_i)$, where the supremum is over p ∈ F with |p1/2 - p1/2 0|2 ≥ ε, is exponentially small with probability exponentially close to 1. The exponent is proportional to nε2. The only condition required for this to hold is that ε exceeds a value determined by the bracketing Hellinger entropy of F. A similar inequality also holds if we replace F by Fn and p0 by qn, where qn is an approximation to p0 in a suitable sense. These results are applied to establish rates of convergence of sieve MLEs. Furthermore, weak conditions are given under which the "optimal" rate εn defined by H(εn, F) = nε2 n, where H(·, F) is the Hellinger entropy of F, is nearly achievable by sieve estimators.

Page Thumbnails

  • Thumbnail: Page 
339
    339
  • Thumbnail: Page 
340
    340
  • Thumbnail: Page 
341
    341
  • Thumbnail: Page 
342
    342
  • Thumbnail: Page 
343
    343
  • Thumbnail: Page 
344
    344
  • Thumbnail: Page 
345
    345
  • Thumbnail: Page 
346
    346
  • Thumbnail: Page 
347
    347
  • Thumbnail: Page 
348
    348
  • Thumbnail: Page 
349
    349
  • Thumbnail: Page 
350
    350
  • Thumbnail: Page 
351
    351
  • Thumbnail: Page 
352
    352
  • Thumbnail: Page 
353
    353
  • Thumbnail: Page 
354
    354
  • Thumbnail: Page 
355
    355
  • Thumbnail: Page 
356
    356
  • Thumbnail: Page 
357
    357
  • Thumbnail: Page 
358
    358
  • Thumbnail: Page 
359
    359
  • Thumbnail: Page 
360
    360
  • Thumbnail: Page 
361
    361
  • Thumbnail: Page 
362
    362