Understanding Parameter Estimation: Point vs Interval and Confidence Intervals

Statistical inference involves inferring 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, the sample mean serves as a point estimate. Before constructing a confidence interval, a confidence level must be specified, typically denoted as 1‑α, where α is the significance level used in hypothesis testing. The confidence level is thus complementary 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 corresponds 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.

For example, suppose an investment analyst selects a random sample of 100 equity funds and computes an average Sharpe ratio of (value). The sample standard deviation is (value). Using a critical value based on the standard normal distribution, the analyst calculates the confidence interval for the population mean Sharpe ratio. The critical value is (value), yielding a confidence interval of (lower bound) to (upper bound), i.e., (interval). The analyst can state with (1‑α) confidence that this interval contains the true population mean.

Reference:

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