1442 d n n n n x x f x fx part 11 asymptotic

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™    14/42 d n n n n x  F (x ) F(x) →∞ → →
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Part 11: Asymptotic Distribution  Theory Limiting Distribution ™    15/42 ( 29 ( 29 2 1 2 0 2 2 1 1 1 , ,..., = a random sample from N[ , ] For purpose of testing H : 0, the usual test statistic is t , where s 1 / The exact density of the random variable t is t with n-1 n n i n n i n n n n x x x x x x n s n = - - μ σ μ = - = = - /2 2 1 1 1 1 degrees of freedom. The density varies with n; ( / 2) 1 f(t ) 1 [( 1) / 2] 1 ( 1) The cdf, F (t) = ( ) . The distribution has mean zero and variance (n-1)/(n-3). As n , t n n n t n n t n n n n f x dx - - - - - -∞ Γ = + Γ - - - π → ∞ 1 he distribution and the random variable converge to standard normal, which is written t N[0,1]. d n - →
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Part 11: Asymptotic Distribution  Theory A Slutsky Theorem for Random Variables (Continuous Mapping) ™    16/42 d n n d n n If x x,  and if g(x ) is a continuous function with continuous derivatives and does not involve n, then g(x ) g(x). Example :  t  =  random variable with t distribution with                       n deg → → 2 n d n d 2 2 n rees of freedom.                t  =  exactly, an F random variable with [1,n]                        degrees of freedom.                t N(0,1),                  t [N(0,1)]  =  chi-squared[1]. → →
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Part 11: Asymptotic Distribution  Theory An Extension of the Slutsky Theorem ™    17/42 d n n d n n n n n If x x (x  has a limiting distribution) and  is some relevant constant, and g(x , ) g (g  has a limiting distribution that is                         some function of  ) and plim y ,  then g(x , y → θ θ → θ = θ d ) g (replacing   with a consistent estimator leads to the same limiting distribution). → θ
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Part 11: Asymptotic Distribution  Theory Application of the Slutsky Theorem ™    18/42 χ → σ = → σ →χ 2 d 2 p 2 d 2 Large sample behavior of the F statistic for testing restrictions ( ) [J] ( )/J J J F= /(n-K) /(n-K) 1 Therefore, JF [J] Establishing the numerator requires a c e * 'e *  - e'e e * 'e *  - e'e e'e e'e entral limit theorem.
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Part 11: Asymptotic Distribution  Theory Central Limit Theorems Central Limit Theorems describe the large sample behavior of random variables that involve sums of variables. “Tendency toward normality.” Generality: When you find sums of random variables, the CLT shows up eventually. The CLT does not state that means of samples have normal distributions. ™    19/42
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Part 11: Asymptotic Distribution  Theory A Central Limit Theorem ™    20/42 μ σ - μ → σ 1 n 2 d Lindeberg-Levy CLT (the simplest version of the CLT) If x ,..., x  are a random sample from a population with finite mean   and finite variance  ,  then n(x )               N(0,1)
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