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D.A.S. Fraser 24322.4 Inference for regular models: Frequency(i) Normal, exponential, and regular models. Much of contemporary inferencetheory is organized around Normal statistical models with side concernsfor departures from Normality, thus neglecting more general structures. Recentlikelihood methods show, however, that statistical inference is easy anddirect for exponential models and more generally for regular models using anappropriate exponential approximation. Accordingly, let us briefly overviewinference for exponential models.(ii) Exponential statistical model. The exponential family of models iswidely useful both for model building and for model-data analysis. The full exponentialmodel with canonical parameter ϕ and canonical variable u(y) bothof dimension p is f(y; ϕ) =exp{ϕ ′ u(y)+k(ϕ)}h(y). Let y 0 with u 0 = u(y 0 )be observed data for which statistical inference is wanted. For most purposeswe can work with the model in terms of the canonical statistic u:g(u; ϕ) =exp{l 0 (ϕ)+(ϕ − ˆϕ 0 ) ′ (u − u 0 )}g(u),where l 0 (ϕ) =a +lnf(y 0 ; ϕ) istheobservedlog-likelihoodfunctionwiththeusual arbitrary constant chosen conveniently to subtract the maximum loglikelihoodln f(y 0 ;ˆϕ 0 ), using ˆϕ 0 as the observed maximum likelihood value.This representative l 0 (ϕ) has value 0 at ˆϕ 0 , and −l 0 (ϕ) relative to ˆϕ 0 is thecumulant generating function of u − u 0 , and g(u) isaprobabilitydensityfunction. The saddle point then gives a third-order inversion of the cumulantgenerating function −l 0 (ϕ) leadingtothethird-orderrewriteg(u; ϕ) =ek/n(2π) p/2 exp{−r2 (ϕ; u)/2}|j ϕϕ (ˆϕ)| −1/2 ,where ˆϕ =ˆϕ(u) isthemaximumlikelihoodvalueforthetiltedlikelihoodl(ϕ; u) =l 0 (ϕ)+ϕ ′ (u − u 0 ),r 2 (ϕ; u)/2 =l(ˆϕ; u) − l(ϕ; u) istherelatedlog-likelihoodratioquantity,j ϕϕ (ˆϕ) =−∂∂ϕ∂ϕ ′ l(ϕ; u)| ˆϕ(u)is the information matrix at u, and finally k/n is constant to third order. Thedensity approximation g(u; ϕ 0 ) gives an essentially unique third-order nulldistribution (Fraser and Reid, 2013) for testing the parameter value ϕ = ϕ 0 .Then if the parameter ϕ is scalar, we can use standard r ∗ -technology tocalculate the p-value p(ϕ 0 )forassessingϕ = ϕ 0 ;see,e.g.,Brazzaleetal.(2007). For a vector ϕ, adirectedr ∗ departure is available; see, e.g., Davisonet al. (2014). Thus p-values are widely available with high third-order accuracy,