IB DP Maths: AI HL复习笔记4.6.2 Unbiased Estimates

What is an unbiased estimator of a population parameter?

• An estimator is a random variable that is used to estimate a population parameter
  • An estimate is the value produced by the estimator when a sample is used
• An estimator is called unbiased if its expected value is equal to the population parameter
  • An estimate from an unbiased estimator is called an unbiased estimate
  • This means that the mean of the unbiased estimates will get closer to the population parameter as more samples are taken

• The sample mean is an unbiased estimate for the population mean
• The sample variance is not an unbiased estimate for the population variance
  • On average the sample variance will underestimate the population variance
  • As the sample size increases the sample variance gets closer to the unbiased estimate

What are the formulae for unbiased estimates of the mean and variance of a population?

• A sample of n data values (x₁, x₂, ... etc) can be used to find unbiased estimates for the mean and variance of the population

Is s_n-1 an unbiased estimate for the standard deviation?

• Unfortunately s_n-1 is not an unbiased estimate for the standard deviation of the population
• It is better to work with the unbiased variance rather than standard deviation
• There is not a formula for an unbiased estimate for the standard deviation that works for all populations
  • Therefore you will not be asked to find one in your exam

How do I show the sample mean is an unbiased estimate for the population mean?

• You do not need to learn this proof
  • It is simply here to help with your understanding
• Suppose the population of X has mean μ and variance σ²
• Take a sample of n observations
  • X_1, X_2, ..., X_n
  • E(X_i) = μ
• Using the formula for a linear combination of n independent variables:Why is there a divisor of n-1 in the unbiased estimate for the variance?

• You do not need to learn this proof
  • It is simply here to help with your understanding
• Suppose the population of X has mean μ and variance σ²
• Take a sample of n observations
  • X_1, X_2, ..., X_n
  • E(X_i) = μ
  • Var(X_i) = σ²
• Using the formula for a linear combination of n independent variables: