Distribution of $8n\operatorname{ln}(\sqrt{0.2\hat{p}} + \sqrt{0.8(1-\hat{p})} )$

Question

Let $X_1, ..., X_n$ be a random sample taken from a Bernoulli$(p)$ distribution. Consider $\hat{p} = (1/n)\sum_i X_i = \bar{X}_n$ as the estimator of $p$. We wish to test $H_0: p=0.2$ against $H_1: p\neq 0.2$ using the statistic $$ 8nB = - 8n\operatorname{ln}(\sqrt{0.2\hat{p}} + \sqrt{0.8(1-\hat{p})} ) $$ where $B = -\operatorname{ln}(\sqrt{0.2\hat{p}} + \sqrt{0.8(1-\hat{p})} )$ is the Bhattacharya's distance.

The Problem: Determine the asymptotic distribution of $8nB$.

I haven't been able to create any analytical solution to the problem. I have verified that $\hat{p}$ satisfies the WLLN, that is, $\hat{p} \stackrel{P}{\longrightarrow} p$. Simulation (using R) suggests that $8nB$ converges in distribution to a $\chi^2_1$:

N <- 10^4
n <- 10^4

B_distr <- c()

for(i in 1:N){
  X <- sample(0:1, n, prob=c(0.8, 0.2),replace=TRUE)
  p_hat <- mean(X)

  B <- -8*n*log(sqrt(0.2*p_hat) + sqrt(0.8*(1 - p_hat)))
  B_distr <- c(B_distr, B)
}

hist(B_distr)
psych::describe(B_distr)
ks.test(B_distr, p="pchisq", 1)

Answer 1

Assuming $H_0$ is true we can say $$\sqrt{n}\Big(\hat{P}-\frac{1}{5}\Big) \longrightarrow N\Big(0,\frac{4}{25}\Big)$$ where $\longrightarrow$ denotes convergence in distribution. Take $$g(x)=-8\ln\Big(\sqrt{0.2x}+\sqrt{0.8(1-x)}\Big)$$ Note $g'(1/5)=0$ so we cannot use the delta$-$method. But we can, however, use the second order delta method and say that $$n \bigg(g\big(\hat{P}\big)-g(1/5)\bigg) \longrightarrow \frac{4}{25}\cdot \frac{g''(1/5)}{2} \cdot \chi^2_1$$ Since $g(1/5)=0$ and $g''(1/5)=\frac{25}{2}$ we have $$ng\big(\hat{P}\big) \longrightarrow \chi^2_1$$ This is what you're looking for.