```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

# Part 1: Wald, Score, and likelihood ratio test statistics

```{r, echo = F}
#https://www.cs.ubc.ca/~arnaud/stat461/lecture_stat461_WaldRaoLRtests_handouts_2008.pdf
```

Write out the likelihood function, and derive the test statistics of the Wald, Score, and likelihood ratio test. 

1. $X_{i} \stackrel{\text { i.i.d. }}{\sim} f(x \mid \theta)$ 
 
$$f(x \mid \theta)=\theta \exp (-x \theta) \mathbb{I}\{x>0\}$$ 

## Solution

The likelihood function, $L(\theta | \mathbf{x})$, is the joint density of the observed data $\mathbf{x} = (x_1, ..., x_n)$. Due to the i.i.d. assumption, it is the product of the individual PDFs:
$$ L(\theta | \mathbf{x}) = \prod_{i=1}^{n} f(x_i | \theta) = \prod_{i=1}^{n} \theta e^{-x_i \theta} =  \theta^n \exp\left(-\theta \sum_{i=1}^{n} x_i\right) $$
The log-likelihood function, $\ell(\theta | \mathbf{x})$ is:
$$ \ell(\theta | \mathbf{x}) = \log L(\theta | \mathbf{x}) = n \log(\theta) - \theta \sum_{i=1}^{n} x_i $$
To derive the test statistics, we first need the Maximum Likelihood Estimator (MLE), the Score function, and the Fisher Information.


The first derivative of the log-likelihood.
$$ U(\theta) = \frac{\partial \ell}{\partial \theta} = \frac{n}{\theta} - \sum_{i=1}^{n} x_i $$

We find the MLE by setting the score function to zero and solving for $\theta$.
$$ \frac{n}{\hat{\theta}} - \sum_{i=1}^{n} x_i = 0 \implies \hat{\theta} = \frac{n}{\sum_{i=1}^{n} x_i} = \frac{1}{\bar{x}} $$

The negative expected value of the second derivative of the log-likelihood.
$$ \frac{\partial^2 \ell}{\partial \theta^2} = -\frac{n}{\theta^2} $$
Since the second derivative is not a function of the data, the expectation is straightforward:
$$ I(\theta) = -E\left[\frac{\partial^2 \ell}{\partial \theta^2}\right] = -E\left[-\frac{n}{\theta^2}\right] = \frac{n}{\theta^2} $$

All three statistics asymptotically follow a chi-squared distribution with 1 degree of freedom, $\chi_1^2$, under the null hypothesis $H_0$. The Wald test measures the distance between the MLE ($\hat{\theta}$) and the hypothesized value ($\theta_0$), scaled by the curvature of the likelihood function at the MLE. The formula is $W = (\hat{\theta} - \theta_0)^2 I(\hat{\theta})$.
\begin{align*}
    I(\hat{\theta}) &= \frac{n}{\hat{\theta}^2} = \frac{n}{(1/\bar{x})^2} = n\bar{x}^2 \\
    W &= \left(\frac{1}{\bar{x}} - \theta_0\right)^2 (n\bar{x}^2) \\
      &= \left(\frac{1 - \theta_0\bar{x}}{\bar{x}}\right)^2 (n\bar{x}^2) \\
      &= n(1 - \theta_0\bar{x})^2
\end{align*}
The Score test evaluates how close the slope of the log-likelihood function is to zero at the hypothesized value $\theta_0$. The formula is $S = [U(\theta_0)]^2 [I(\theta_0)]^{-1}$.
\begin{align*}
    U(\theta_0) &= \frac{n}{\theta_0} - \sum x_i = \frac{n}{\theta_0} - n\bar{x} = n\left(\frac{1 - \theta_0\bar{x}}{\theta_0}\right) \\
    I(\theta_0) &= \frac{n}{\theta_0^2} \\
    S &= \left[n\left(\frac{1 - \theta_0\bar{x}}{\theta_0}\right)\right]^2 \left(\frac{n}{\theta_0^2}\right)^{-1} \\
      &= \frac{n^2(1 - \theta_0\bar{x})^2}{\theta_0^2} \cdot \frac{\theta_0^2}{n} \\
      &= n(1 - \theta_0\bar{x})^2
\end{align*}
The LR test compares the maximized value of the likelihood function under the null hypothesis, $L(\theta_0)$, with its globally maximized value, $L(\hat{\theta})$. The formula is $LR = 2[\ell(\hat{\theta}) - \ell(\theta_0)]$.
\begin{align*}
    \ell(\hat{\theta}) &= n \log(\hat{\theta}) - \hat{\theta}\sum x_i = n \log\left(\frac{1}{\bar{x}}\right) - \left(\frac{1}{\bar{x}}\right)(n\bar{x}) = -n\log(\bar{x}) - n \\
    \ell(\theta_0) &= n \log(\theta_0) - \theta_0\sum x_i = n\log(\theta_0) - n\theta_0\bar{x} \\
    LR &= 2\left[(-n\log(\bar{x}) - n) - (n\log(\theta_0) - n\theta_0\bar{x})\right] \\
       &= 2n\left[-\log(\bar{x}) - 1 - \log(\theta_0) + \theta_0\bar{x}\right] \\
       &= 2n\left[\theta_0\bar{x} - 1 - (\log(\bar{x}) + \log(\theta_0))\right] \\
       &= 2n[\theta_0\bar{x} - 1 - \log(\theta_0\bar{x})]
\end{align*}

\textbf{Bonus:} For the exponential distribution model we analyzed, the Wald and Score test statistics simplify to the identical algebraic formula, while the Likelihood Ratio (LR) statistic has a distinct form.


2. $X_{i} \stackrel{\text { i.i.d. }}{\sim} f(x \mid \theta)$ 
$$
f(x \mid \theta)=\theta c^{\theta} x^{-(\theta+1)} \mathbb{I}\{x>c\} \quad \text { (Pareto distribution) }
$$
where $c$ is a known constant and $\theta$ is unknown.

## Solution

The likelihood function, $L(\theta | \mathbf{x})$, is the product of the individual PDFs:
$$ L(\theta | \mathbf{x}) = \prod_{i=1}^{n} \theta c^{\theta} x_i^{-(\theta+1)} = \theta^n c^{n\theta} \left(\prod_{i=1}^{n} x_i\right)^{-(\theta+1)} $$
The log-likelihood, $\ell(\theta | \mathbf{x})$, is:
\begin{align*}
    \ell(\theta | \mathbf{x}) &= \log\left( \theta^n c^{n\theta} \left(\prod_{i=1}^{n} x_i\right)^{-(\theta+1)} \right) \\
    &= n\log(\theta) + n\theta\log(c) - (\theta+1)\sum_{i=1}^{n}\log(x_i) \\
    &= n\log(\theta) + \theta\left(n\log(c) - \sum_{i=1}^{n}\log(x_i)\right) - \sum_{i=1}^{n}\log(x_i)
\end{align*}

The first derivative of the log-likelihood with respect to $\theta$.
$$ U(\theta) = \frac{\partial \ell}{\partial \theta} = \frac{n}{\theta} + n\log(c) - \sum_{i=1}^{n} \log(x_i) $$


Set the score function to zero to find the MLE.
$$ \frac{n}{\hat{\theta}} + n\log(c) - \sum_{i=1}^{n} \log(x_i) = 0 $$
$$ \frac{n}{\hat{\theta}} = \sum_{i=1}^{n} \log(x_i) - n\log(c) = \sum_{i=1}^{n} (\log(x_i) - \log(c)) = \sum_{i=1}^{n} \log\left(\frac{x_i}{c}\right) $$
$$ \hat{\theta} = \frac{n}{\sum_{i=1}^{n} \log(x_i/c)} $$

The negative expected value of the second derivative of the log-likelihood.
$$ \frac{\partial^2 \ell}{\partial \theta^2} = -\frac{n}{\theta^2} $$
Since the second derivative is a constant with respect to the data, the Fisher Information is:
$$ I(\theta) = -E\left[-\frac{n}{\theta^2}\right] = \frac{n}{\theta^2} $$


All three statistics asymptotically follow a $\chi_1^2$ distribution under $H_0$. The Wald test is based on the distance between $\hat{\theta}$ and $\theta_0$.
\begin{align*}
    W &= (\hat{\theta} - \theta_0)^2 I(\hat{\theta}) \\
      &= (\hat{\theta} - \theta_0)^2 \left(\frac{n}{\hat{\theta}^2}\right) \\
      &= n\left(\frac{\hat{\theta} - \theta_0}{\hat{\theta}}\right)^2 = n\left(1 - \frac{\theta_0}{\hat{\theta}}\right)^2
\end{align*}

The Score test is based on the slope of the log-likelihood at $\theta_0$.
\begin{align*}
    U(\theta_0) &= \frac{n}{\theta_0} + n\log(c) - \sum \log(x_i) = \frac{n}{\theta_0} - \sum \log(x_i/c) = \frac{n}{\theta_0} - \frac{n}{\hat{\theta}} \\
    S &= [U(\theta_0)]^2 [I(\theta_0)]^{-1} \\
      &= \left[n\left(\frac{1}{\theta_0} - \frac{1}{\hat{\theta}}\right)\right]^2 \left(\frac{n}{\theta_0^2}\right)^{-1} \\
      &= n^2\frac{(\hat{\theta}-\theta_0)^2}{\theta_0^2 \hat{\theta}^2} \frac{\theta_0^2}{n} = n\frac{(\hat{\theta}-\theta_0)^2}{\hat{\theta}^2} = n\left(1 - \frac{\theta_0}{\hat{\theta}}\right)^2
\end{align*}

The LR test compares the log-likelihood values at $\hat{\theta}$ and $\theta_0$.
\begin{align*}
    \ell(\theta) &= n\log\theta - \theta \frac{n}{\hat{\theta}} - \sum\log(x_i) \\
    \ell(\hat{\theta}) &= n\log(\hat{\theta}) - \hat{\theta} \frac{n}{\hat{\theta}} - \sum\log(x_i) = n\log(\hat{\theta}) - n - \sum\log(x_i) \\
    \ell(\theta_0) &= n\log(\theta_0) - \theta_0 \frac{n}{\hat{\theta}} - \sum\log(x_i) \\
    \ell(\hat{\theta}) - \ell(\theta_0) &= (n\log\hat{\theta} - n) - (n\log\theta_0 - n\frac{\theta_0}{\hat{\theta}}) \\
    &= n\left[\log\left(\frac{\hat{\theta}}{\theta_0}\right) - 1 + \frac{\theta_0}{\hat{\theta}}\right] \\
    LR &= 2[\ell(\hat{\theta}) - \ell(\theta_0)] = 2n\left[\log\left(\frac{\hat{\theta}}{\theta_0}\right) + \frac{\theta_0}{\hat{\theta}} - 1\right]
\end{align*}


# Part 2: Test equivalence

Let $\theta$ be a scalar parameter and suppose we test
$$
H_{0}: \theta=\theta_{0} \quad \text { versus } \quad H_{1}: \theta \neq \theta_{0} .
$$
Let $W$ be the Wald test statistic and let $\lambda$ be the likelihood ratio test statistic. Show that these tests are equivalent in the sense that
$$
\frac{W}{\lambda} \stackrel{\mathrm{P}}{\longrightarrow} 1
$$
as $n \rightarrow \infty$. Hint: Use a Taylor expansion of the log-likelihood $\ell(\theta)$ to show that
$$
\lambda \approx\left(\sqrt{n}\left(\widehat{\theta}-\theta_{0}\right)\right)^{2}\left(-\frac{1}{n} \ell^{\prime \prime}(\widehat{\theta})\right)
$$

## Solution

The \textbf{Likelihood Ratio (LR) statistic}, $\lambda$, is:
$$ \lambda = 2(\ell(\hat{\theta}) - \ell(\theta_{0})) $$
The \textbf{Wald statistic}, $W$, using the observed Fisher information, is:
$$ W = (\hat{\theta} - \theta_0)^2 \left(-\ell''(\hat{\theta})\right) $$
where $\ell(\theta)$ is the log-likelihood, $\ell''(\theta)$ is its second derivative, and $\hat{\theta}$ is the MLE.


We perform a second-order Taylor expansion of $\ell(\theta_0)$ around the MLE $\hat{\theta}$, including the remainder term in \textbf{little-o notation}:
$$ \ell(\theta_0) = \ell(\hat{\theta}) + \ell'(\hat{\theta})(\theta_0 - \hat{\theta}) + \frac{1}{2}\ell''(\hat{\theta})(\theta_0 - \hat{\theta})^2 + o_p\left((\hat{\theta} - \theta_0)^2\right) $$
The term $o_p\left((\hat{\theta} - \theta_0)^2\right)$ is a random quantity that converges to zero in probability faster than $(\hat{\theta} - \theta_0)^2$. By definition of the MLE, $\ell'(\hat{\theta}) = 0$, which simplifies the expansion to:
$$ \ell(\theta_0) = \ell(\hat{\theta}) + \frac{1}{2}\ell''(\hat{\theta})(\hat{\theta} - \theta_0)^2 + o_p\left((\hat{\theta} - \theta_0)^2\right) $$

Rearranging the equation above allows us to express $\lambda$ explicitly.
$$ \ell(\hat{\theta}) - \ell(\theta_0) = -\frac{1}{2}\ell''(\hat{\theta})(\hat{\theta} - \theta_0)^2 - o_p\left((\hat{\theta} - \theta_0)^2\right) $$
Multiplying by 2 gives the expression for $\lambda$:
\begin{align*}
    \lambda = 2(\ell(\hat{\theta}) - \ell(\theta_0)) &= (\hat{\theta} - \theta_0)^2\left(-\ell''(\hat{\theta})\right) - 2 \cdot o_p\left((\hat{\theta} - \theta_0)^2\right) \\
    \lambda &= W + o_p\left((\hat{\theta} - \theta_0)^2\right)
\end{align*}


Finally, we analyze the ratio $\frac{W}{\lambda}$ as $n \to \infty$.
$$ \frac{W}{\lambda} = \frac{W}{W + o_p\left((\hat{\theta} - \theta_0)^2\right)} $$
Divide the numerator and denominator by $W$:
$$ \frac{W}{\lambda} = \frac{1}{1 + \frac{o_p\left((\hat{\theta} - \theta_0)^2\right)}{W}} $$
We examine the remainder term in the denominator:
$$ \frac{o_p\left((\hat{\theta} - \theta_0)^2\right)}{W} = \frac{o_p\left((\hat{\theta} - \theta_0)^2\right)}{(\hat{\theta} - \theta_0)^2\left(-\ell''(\hat{\theta})\right)} $$

By standard MLE theory, $-\frac{1}{n}\ell''(\hat{\theta}) \stackrel{P}{\to} I(\theta_0)$, where $I(\theta_0)$ is a positive constant. This implies $-\ell''(\hat{\theta}) = O_p(n)$. Therefore, the term simplifies to:
$$ \frac{o_p(1)}{-\ell''(\hat{\theta})} = \frac{o_p(1)}{O_p(n)} \stackrel{P}{\longrightarrow} 0 $$
The remainder term vanishes as $n \to \infty$. The denominator of our ratio converges to 1:
$$ 1 + \frac{o_p\left((\hat{\theta} - \theta_0)^2\right)}{W} \stackrel{P}{\longrightarrow} 1 + 0 = 1 $$
This explicitly shows that:
$$ \frac{W}{\lambda} \stackrel{\mathrm{P}}{\longrightarrow} 1 $$

### Justification: Convergence to Fisher Information
The key result that $-\frac{1}{n}\ell''(\hat{\theta}) \stackrel{P}{\to} I(\theta_0)$ is established in two main steps:
\begin{enumerate}
\item \textbf{Law of Large Numbers:} The term $-\frac{1}{n}\ell''(\theta_0)$, evaluated at the true parameter $\theta_0$, is an average of i.i.d. random variables. By the Law of Large Numbers, this average converges in probability to its expectation, which is the definition of the Fisher Information, $I(\theta_0)$.
$$ -\frac{1}{n}\ell''(\theta_0) = -\frac{1}{n}\sum_{i=1}^{n} \frac{\partial^2}{\partial \theta^2}\log f(X_i; \theta_0) \stackrel{P}{\longrightarrow} -E\left[ \frac{\partial^2}{\partial \theta^2}\log f(X; \theta_0) \right] = I(\theta_0) $$
\item \textbf{Consistency of the MLE:} The MLE is a consistent estimator, meaning $\hat{\theta} \stackrel{P}{\to} \theta_0$. Because the function is continuous, the \textbf{Continuous Mapping Theorem} allows us to substitute the consistent estimator $\hat{\theta}$ for the true parameter $\theta_0$ in the limit, and the convergence result still holds.
\end{enumerate}
This confirms that $-\frac{1}{n}\ell''(\hat{\theta}) \stackrel{P}{\to} I(\theta_0)$, which is a positive constant.

