Question

Let Xi ∼iid Ber(p). Define Y = Xbar. Approximate the mean of W = g(Y ) = Y (1 − Y ) using the first and second order delta methods. Also, estimate the variance of W using the first order method.

Answer 1

we will use delta method of first order which is univariate method as well. i am sharing the image containing the solution of the questing. i am using the method of central limit theorem also .

the first order delta method is ,

x₁, x₂....x_n is sequence of random variables,

if for n tends to infinite,

sqrt(n)[x_n-p]--d--->N(0,sigma²)

where, p = mean of x_1,x₂, ....... ,x_n

and singma² = variance of x₁,x₂, ...., x_n

then for n tends to infinite,

sqrt(n)[g(x_n)-g(p)]--d--->N(0,sigma².[g^' (p)]²)

where g^'(p) exist and non zero.

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