Applied categorical and count data analysis by Tang, Wan; He, Hua; Tu, Xin M

By Tang, Wan; He, Hua; Tu, Xin M

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Example text

1) where Pr (·) denotes probability and “→” denotes convergence. For convenience, the asymptotic distribution is often expressed in several different forms. 3) where “→d ” denotes convergence in distribution as is often called in statistics asymptotics and “∼a ” indicates an approximate rather than exact relationship. 3) simply by: Xn − µ ∼ N 0, σ2 n , or X n ∼ N µ, σ2 n . , Pr [Xi = 1] = p, Let pn = 1 n n i=1 Pr [Xi = 0] = 1 − p = q. Xi . Then, by CLT, √ n (pn − p) →d N (0, pq) , So, for large n, pn − p ∼ N 0, pq n n → ∞.

The solution is µ = X n . The Fisher information is I (θ) = −E ∂2 ∂θ∂θ log f (Xi , θ) = − σ12 . Thus, the asymptotic variance of 2 µ is σn . In the special case, µ has an exact rather than asymptotic normal distribution. In regression analysis, we have one outcome designated as the response or dependent variable Y and a set of other variables specified as explanatory variables, or independent variables, predictors, covariates, X. We are interested in the change of the response as a function of the explanatory variables.

In this special case, the 2 distribution of X n is a normal with mean µ and variance σn . However, if Xi is not a normal variate, the sampling distribution of X n is generally nonnormal. For example, if Xi ∼ Bernoulli (p), a Bernoulli variable with the probability of success p, X n no longer follows a normal distribution. In fact, the distribution of X n is not even a continuous function. For large n, however, we can approximate the distribution of X n of any random variate Xi using a normal distribution according to the CLT.

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