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- W339987254 abstract "Under the usual regularity conditions an estimator θ n of a real valued parameter θ is asymptotically efficient if $$ {hat{theta }_n} - theta = {{{(frac{1}{n}sumlimits_{{j = 1}}^n {frac{{partial log f({X_j};theta )}}{{partial theta }}} )}} left/ {{I(theta ) + {varepsilon_n}}} right.} $$(4.1)where ( sqrt {n} {varepsilon_n} to 0 ) in P θ -probability as n → ∞, and I(θ) is Fisher’s information$$ I(theta ) = {E_theta}{(frac{{partial log f({X_1};theta )}}{{partial theta }})^2} = - {E_theta}(frac{{{partial^2}log f({X_1};theta )}}{{partial {theta^2}}}) $$(4.2). Here X1, X2,... are i.i.d. observations with values in some measure space (χ, B, μ) and, for each θ in the parameter space Θ = (a, b) (-∞ ≤ a < b ≤ ∞) f(x; θ) is the density of the common distribution of the observations with respect to the sigmafinite measure μ. As before, for each θ ∈ Θ, (Ω, F, P θ ) is a probability space on which X′ j s are defined. In particular, any consistent solution of the likelihood equation$$ sumlimits_{{j = 1}}^n {frac{{partial log f({X_j};theta ')}}{{partial theta '}} = 0} $$(4.3)is asymptotically efficient. If there is a unique solution to (4.3) this solution is the maximum likelihood estimator. But even in this case of a unique solution there are other estimators which arise naturally and satisfy (4.1). For example, in attempting to solve (4.3) a common procedure is to take an initial estimator ({tilde theta _n}) and use the Newton-Raphson method to get a first approximation (theta _n^ * ) to the solution of (4.3): $$ theta_n^{ star }: = {tilde{theta }_n} - {(frac{{sumlimits_{{j = 1}}^n {{{{partial log f({X_j};theta ')}} left/ {{partial theta '}} right.}} }}{{sumlimits_{{j = 1}}^n {{{{{partial^2}log f({X_j};theta ')}} left/ {{partial {{theta '}^2}}} right.}} }})_{{theta ' = {{tilde{theta }}_n}}}} $$(4.3). If, in addition to the usual regularity conditions, ( sqrt {n} ({tilde{theta }_n} - theta ) ) is stochastically bounded under P θ (i.e. if for any given e > 0 there exists a constant -A e such that ( sqrt {n} ({tilde{theta }_n} - theta ) ) for all n), then θ n * is asymptotically efficient." @default.
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- W339987254 date "1990-01-01" @default.
- W339987254 modified "2023-09-24" @default.
- W339987254 title "Second Order Efficiency" @default.
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