Question

Let \(X\) be Cauchy-distributed random variable with PDF $$ f(x ; \theta)=\frac{1}{\pi} \frac{1}{\left(1+(x-\theta)^{2}\right)}, \quad-\infty

Short answer

The Cramer-Rao Lower Bound for the estimation of the location parameter \( \theta \) is \( \pi/2 \).

Step-by-step solution

Calculation of derivative of Log-likelihood

The first step involves calculating the derivative of the log-likelihood function. The likelihood function is obtained from the Probability Density Function (PDF) and its natural logarithm will provide us with the log-likelihood function which is \( L(\theta) = \log(f(x ; \theta)) = \log \left( \frac{1}{\pi} \right) - \log \left[1+(x-\theta)^{2}\right] \). Differentiating \( L( \theta ) \) with respect to \( \theta \), we get: \( L'(\theta) = 2(x-\theta) / [1+(x-\theta)^{2}] \)

Computing the Fisher Information

After obtaining the derivative of the log-likelihood function, the next step is to calculate the Fisher Information. The Fisher Information for the parameter \( \theta \) is calculated using the formula \( I(\theta) = E[(L'(\theta))^2] \) which is the expectation of the square of the derivative of the log-likelihood function computed in Step 1. To compute the expectation, we need to integrate the function across all values of x ((-\infty, \infty) in this case), multiplied by the function itself.Carrying out this calculation, we get \( I(\theta) = 2/\pi \)

Calculation of Cramer-Rao Lower Bound

The final step is to calculate the Cramer-Rao Lower Bound (CRLB) using the formula \( CRLB = 1/nI(\theta) \), where n is the sample size. Since in this case we are considering a single observation, n=1. Hence, the CRLB for this sample is \( CRLB = 1/I(\theta) = \pi/2 \)

Key concepts

Fisher Information

Fisher Information is a critical concept in the field of statistical estimation. It provides a measure of how much information a random variable carries about an unknown parameter, in this case, the parameter \( \theta \).

Informally, it tells us how "pointy" or concentrated the likelihood function is near the true parameter value. The sharper the peak, the more Fisher Information there is about the parameter, indicating a more precise estimate.

• Mathematically, for a parameter \( \theta \), Fisher Information is given by the expected value of the squared derivative of the log-likelihood function: \( I(\theta) = E[(L'(\theta))^2] \).
• This expectation is taken over the distribution of the data, which means it integrates the squared log-likelihood derivative over all possible values of the data, weighted by the likelihood function itself.

The larger the value of Fisher Information, the more accurately we can estimate \( \theta \). It plays a crucial role in deriving the Cramer-Rao Lower Bound, which indicates the minimum variance of an unbiased estimator.

Cauchy Distribution

The Cauchy distribution is known for its unique properties which make it intriguing yet tricky for statistical analysis. It's a continuous probability distribution like the more familiar normal distribution but with significant differences.

Here are some interesting facets of the Cauchy distribution:

• The probability density function (PDF) of the Cauchy distribution is given by \( f(x; \theta) = \frac{1}{\pi} \frac{1}{1 + (x-\theta)^2} \), where \( \theta \) is the location parameter.
• Unlike the normal distribution, the Cauchy distribution does not have a defined mean or variance. Its tails are heavier, meaning it allows for a higher probability of extreme values.
• This distribution is often used in scenarios where outliers are expected, making standard deviation an unreliable measure of variability.

Cauchy's properties make parameter estimation typical challenging, necessitating techniques like the Cramer-Rao Lower Bound to assess estimator performance.

Likelihood Function

The likelihood function is a foundational concept in statistics used in parameter estimation. It is fundamentally the probability of observing your data given a particular parameter value.

Here's a breakdown of how it works:

• Derived from the PDF, the likelihood function for the Cauchy distribution is \( L(\theta) = f(x; \theta) \). This function inserts observed data into the PDF.
• The log-likelihood is often used to simplify mathematical operations. For the Cauchy distribution, the log-likelihood \( L(\theta) \) is expressed as \( \log \left( \frac{1}{\pi} \right) - \log[1 + (x-\theta)^2] \).

Maximizing the likelihood or log-likelihood function helps find the parameter that makes the observed data most probable, a method known as Maximum Likelihood Estimation (MLE). Understanding the shape and peak of this function is crucial for efficient parameter estimation.

Parameter Estimation

Parameter Estimation is the process through which we aim to determine the parameters of the underlying distribution that best explain the observed data. It's a core objective of statistical analysis and involves several mean techniques implemented in the exercise.

Consider the following aspects:

• We use analytical approaches like Maximum Likelihood Estimation (MLE) to obtain parameter estimates. By adjusting the parameter \( \theta \), MLE finds the value that maximizes the likelihood function, thereby considering it the most probable given the data.
• For distributions like the Cauchy distribution, conventional measures like the mean are not useful. Here, estimating the location parameter \( \theta \) is key to deriving useful information.
• The Cramer-Rao Lower Bound (CRLB) is crucial as it provides the theoretical lower bound for the variance of any unbiased estimator. In this context, it helps assess the minimum variance achievable when estimating the location parameter.

A clear understanding of these processes ensures that statistics can provide reliable and insightful results, reflecting the true nature of the data generation process.