By Tang, Wan; He, Hua; Tu, Xin M
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5 Prove that V ar 1 ∂ f (Xi ,θ) ∂θ f (Xi , θ) = −E ∂2 ∂θ∂θ l (θ) . 6 A random variable X follows an exponential distribution with parameter λ takes positive values and Pr(X < t) = 1 − exp(λx). Suppose that Xi (i = 1, . . , n) is a random sample following an exponential distribution with parameter λ. Find the MLE of λ. 7 Let f (θ) be a n × 1 and g (θ) a 1 × m vector-valued function of θ. The ∂ ∂ derivatives of ∂θ f and ∂θ g are defined as follows: ∂f ∂g ∂g1 ∂fn 1 1 ∂θ1 · · · ∂θq ∂θ1 · · · ∂θ1 .
We can write the model as Yi = X i β + i, i ∼ N 0, σ 2 , 1 ≤ i ≤ n. 13) Thus, in regression models, interest lies in the relationship between Yi and X i , while accounting for random variation of Yi given the values of X i and the distribution of X i is of no interest. 13), the log-likelihood function is given by ln (θ) = − n 1 log 2πσ 2 − 2 2 2σ n 2 Yi − X i β . 8): ∂ 1 ln = 2 ∂β σ n X i Yi − X i X i β = 0. 14) 22 Applied Categorical and Count Data Analysis By solving for β, we obtain the MLE β = The second derivative of log f (Yi | X i , θ) is n i=1 −1 X iX i ( 1 ∂2 log f (yi | θ) = − 2 X i X i .
G1 · · · ∂θk ∂θ1 ∂gm ∂θ1 .. . ∂gm ∂θk . 9) k×m This delta method is quite useful for finding asymptotic distributions of functions of statistics. Similar to the relationship between CLT and LLN, we also have a version of the delta method for functions of consistent estimates. Let θ n be a vectorvalued consistent estimate of some vector-valued parameter θ. Let g (θ) = (g1 (θ) , . . , gm (θ)) be a continuous vector-valued function from Rk to Rm . Introduction 19 Then, the function g θ n is a consistent estimate of g (θ).