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This theorem follows from the fact that if X n converges in distribution to X and Y n converges in probability to a constant c, then the joint vector (X n, Y n) converges in distribution to (X, c) . Next we apply the continuous mapping theorem , recognizing the functions g ( x , y ) = x + y , g ( x , y ) = xy , and g ( x , y ) = x y −1 are ...
However, according to Scheffé’s theorem, convergence of the probability density functions implies convergence in distribution. [4] The portmanteau lemma provides several equivalent definitions of convergence in distribution. Although these definitions are less intuitive, they are used to prove a number of statistical theorems.
Proof of the theorem: Recall that in order to prove convergence in distribution, one must show that the sequence of cumulative distribution functions converges to the F X at every point where F X is continuous. Let a be such a point. For every ε > 0, due to the preceding lemma, we have:
Download as PDF; Printable version; In other projects ... Law of large numbers; ... Slutsky's theorem; Stein's lemma; V.
In statistics, a consistent estimator or asymptotically consistent estimator is an estimator—a rule for computing estimates of a parameter θ 0 —having the property that as the number of data points used increases indefinitely, the resulting sequence of estimates converges in probability to θ 0.
where n is the number of observations and Σ is a (symmetric positive semi-definite) covariance matrix. Suppose we want to estimate the variance of a scalar-valued function h of the estimator B . Keeping only the first two terms of the Taylor series , and using vector notation for the gradient , we can estimate h(B) as
Pentagonal number theorem (number theory) Perelman's Geometrization theorem (3-manifolds) Perfect graph theorem (graph theory) Perlis theorem (graph theory) Perpendicular axis theorem ; Perron–Frobenius theorem (matrix theory) Peter–Weyl theorem (representation theory) Phragmén–Lindelöf theorem (complex analysis) Picard theorem (complex ...
The normal equations can be derived directly from a matrix representation of the problem as follows. The objective is to minimize = ‖ ‖ = () = +.Here () = has the dimension 1x1 (the number of columns of ), so it is a scalar and equal to its own transpose, hence = and the quantity to minimize becomes