Asymptotic Techniques for Use in StatisticsThe use in statistical theory of approximate arguments based on such methods as local linearization (the delta method) and approxi mate normality has a long history. Such ideas play at least three roles. First they may give simple approximate answers to distributional problems where an exact solution is known in principle but difficult to implement. The second role is to yield higher-order expansions from which the accuracy of simple approximations may be assessed and where necessary improved. Thirdly the systematic development of a theoretical approach to statistical inference that will apply to quite general families of statistical models demands an asymptotic formulation, as far as possible one that will recover 'exact' results where these are available. The approximate arguments are developed by supposing that some defining quantity, often a sample size but more generally an amount of information, becomes large: it must be stressed that this is a technical device for generating approximations whose adequacy always needs assessing, rather than a 'physical' limiting notion. Of the three roles outlined above, the first two are quite close to the traditional roles of asymptotic expansions in applied mathematics and much ofthe very extensive literature on the asymptotic expansion of integrals and of the special functions of mathematical physics is quite directly relevant, although the recasting of these methods into a probability mould is quite often enlightening. |
Contents
Some basic limiting procedures | 25 |
7 | 38 |
Asymptotic expansions | 71 |
Copyright | |
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Common terms and phrases
a₁ apply argument asymptotic expansion asymptotically normal Barndorff-Nielsen behaviour bivariate calculation characteristic function chi-squared distribution coefficients conditional distribution constant convergence correction term corresponding covariance matrix cumulant generating function cumulative distribution function defined degrees of freedom denote derivatives discussion distributed random variables Edgeworth series Edgo equation exact Example exponential family exponential model exponential tilt finite formula further gamma distribution Hermite polynomials identically distributed random independent and identically integral inverse leading term Legendre transform Let Y₁ limit theorem linear Math maximum likelihood estimate moment generating function moments normal approximation notation Note obtain parameter particular partitions Poisson distribution probability density quantiles random variables random vectors S₁ saddlepoint sample Section 4.2 sequence Show stochastic sums of independent Suppose tensor theory tilted approximation univariate write X₁ and X2 Y₂ zero mean