Understanding Parameter Estimation: Point vs Interval Methods

Statistical inference involves deducing the distribution and numerical characteristics of a population from a sample. A fundamental issue in statistical inference is parameter estimation.

Parameter estimation has two types: point estimation, which uses a sample statistic as the estimate of an unknown parameter, and interval estimation, which uses an interval formed by two statistics to estimate the unknown parameter.

When estimating the population mean, using the sample mean as the estimate is point estimation. Before performing confidence interval estimation, a confidence level must be specified, e.g., 1‑alpha, where alpha is the significance level used in hypothesis testing. The confidence level corresponds to the significance level. The general formula for a confidence interval is: point estimate ± critical value × standard error of the sample mean.

The critical value is the value corresponding to the chosen significance level for a two‑tailed test. It may be denoted as z_{α/2} or t_{α/2}, depending on the distribution used.

Example: An investment analyst selects a random sample from equity funds and computes the average Sharpe ratio. The sample size is 100 and the sample mean Sharpe ratio is (value omitted). The sample standard deviation is (value omitted). Using a critical value based on the standard normal distribution, the analyst calculates and interprets the confidence interval for the population mean Sharpe ratio. The critical value is (value), yielding a confidence interval of (lower, upper), i.e., (interval). The analyst can state with (confidence level) confidence that this interval contains the true population mean.

Reference

Zhu Shunquan, Economic and Financial Data Analysis and Its Python Application