• Point Estimation involves using sample data to estimate population parameters.
• Common estimators include:
  • Mean (\(\bar{x}\)) for estimating population mean (\(\mu\))
  • Variance (\(s^2\)) for estimating population variance (\(\sigma^2\))
• Point estimates provide a single value as an estimate of an unknown parameter.

1. Unbiasedness: An estimator \(\hat{\theta}\) is unbiased if its expected value equals the true parameter: \[ E[\hat{\theta}] = \theta \]

2. Consistency: An estimator is consistent if it converges in probability to the true parameter as the sample size increases: \[ \lim_{n \to \infty} P(|\hat{\theta}_n - \theta| < \epsilon) = 1 \]

3. Efficiency: Among unbiased estimators, the one with the smallest variance is most efficient.

• We’ll estimate the population mean (\(\mu\)) using the sample mean (\(\bar{x}\)).

• Formula for sample mean: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i \]

• Similarly, the sample variance is given by: \[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]

Here’s R code to simulate data from a normal distribution: # Simulating Data in R

Here’s R code to simulate data from a normal distribution:

set.seed(123)
n <- 100
mu <- 50
sigma <- 10
sample_data <- rnorm(n, mean = mu, sd = sigma)
mean(sample_data)

## [1] 50.90406

We can simulate the sampling distribution of the sample mean by taking multiple samples and calculating their means.

• Point Estimation provides a single value as an estimate of an unknown population parameter.

• Important properties of estimators include unbiasedness, consistency, and efficiency.

• Simulations and visualizations help us understand the behavior of estimators.

• “An Introduction to Statistical Learning” by Gareth James et al.

• R Documentation and ggplot2, plotly libraries