formula sheet

formula sheet - Statistical Inference for FE. Mathematical...

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Statistical Inference for FE. Mathematical Formula Sheet. F -distribution F n 1 ,n 2 = χ 2 n 1 /n 1 χ 2 n 2 /n 2 , where χ 2 n 1 and χ 2 n 2 are two independent χ 2 distribution with d.f. n 1 and n 2 , respectively. F n 1 , = χ 2 n 1 /n 1 , where χ 2 n 1 is a χ 2 distribution with d.f. n 1 . Convergence in probability: X n converges to X in probability, if, for any ε > 0 , P ( | X n X | > ε ) 0 , as n →∞ . Law of Large Numbers and Central Limit Theorem. Suppose X 1 , X 2 , X 2 ,.... are i.i.d. random variables with mean μ = E [ X ] and variance σ 2 .Th en P n i =1 X i /n E [ X ] , a.s., P n i =1 X i σ n N (0 , 1) in distribution. Suppose that we try to f nd the root of f ( x )= a. The Newton algorithm is x m +1 = x m + a f ( x m ) f 0 ( x m ) . A standard one dimensional Brownian motion W ( t ) , t 0 ,start ingat 0 , is a continuous-time process with the properties that (1) W (0) = 0 , and the sample path of W ( t ) is continuous, almost surely; (2) W ( t 1 ) , W ( t 2 ) W ( t 1 ) , W ( t 3 ) W ( t 2 ) ,..., W ( t m ) W ( t m 1 ) are independent for every m 1 and 0 <t 1 <t 2 < ··· <t m < ;(3) W ( t ) W ( s ) is normally distributed with mean zero and variance t s . A Brownian motion W μ, σ ( t ) with drift μ and variance σ 2 is de f ned as W μ, σ ( t ):= μt + σ W t . Geometric Brownian motion: e at + σ W t , t 0 . Simpson’s paradox occurs when some overall conclusion concerning a set of objects fails to hold within each of a collection of subsets of those subjects. If the estimator ˆ θ has an asymptotic normal distribution ˆ θ N ( θ ,v ( ˆ θ )) , where v ( ˆ θ ) is the asymp- totic variance of ˆ θ ,then 95% con f dence interval for the unknown parameter θ is ( ˆ θ 1 . 96 q v ( ˆ θ ) , ˆ θ + 1 . 96 q v ( ˆ θ )) . An estimator ˆ θ of an unknown parameter θ is unbiased if E [ ˆ θ ]= θ . The bias of an estimator ˆ θ is de f ned to be E [ ˆ θ θ ] . Suppose we have two unbiased estimators ˆ θ 1 and ˆ θ 2 .W esay ˆ θ 1 is more e cient than ˆ θ 2 if Var ( ˆ θ 1 ) <Var ( ˆ θ 2 ) . The mean squared error of an estimator ˆ θ is MSE ( ˆ θ )= E [( ˆ θ θ ) 2 ] . Note that if θ is a scalar, then MSE ( ˆ θ )= Var ( ˆ θ )+( Bias ( ˆ θ ))
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formula sheet - Statistical Inference for FE. Mathematical...

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