Estimators, Bias and Standard Error | 估计量、偏差与标准误

📚 Estimators, Bias and Standard Error | 估计量、偏差与标准误

In A-Level Statistics, we rarely have data for an entire population. We therefore take a random sample and use a statistic, such as the sample mean, to estimate an unknown population parameter. This article explains what makes an estimator good, what bias means, and how the standard error measures the precision of an estimate.

在 A-Level 统计中，我们很少能获得整个总体的数据。因此，我们抽取随机样本，并使用样本均值等统计量来估计未知的总体参数。本文解释什么是好的估计量、偏差的含义，以及标准误如何衡量估计值的精确度。

1. Population Parameters and Sample Statistics | 总体参数与样本统计量

A population parameter is a fixed number that describes a whole population. Common examples are the population mean μ, population variance σ² and population proportion p. These values are usually unknown because a census is too costly or impossible.

总体参数是描述整个总体的固定数值。常见的例子有总体均值 μ、总体方差 σ² 和总体比例 p。这些值通常是未知的，因为普查成本太高或不可能完成。

A sample statistic is a random quantity calculated from a sample. For example, the sample mean X̄ and sample variance S² are statistics. We use them as estimators of the corresponding population parameters.

样本统计量是从样本中计算出来的随机量。例如样本均值 X̄ 和样本方差 S² 都是统计量。我们把它们用作相应总体参数的估计量。

2. What Is an Estimator? | 什么是估计量

An estimator is a rule or formula that tells us how to use sample data to produce an estimate of a population parameter. If θ is a parameter, we write its estimator as θ̂ (theta-hat). For example, X̄ is an estimator for μ.

估计量是一种规则或公式，告诉我们应该如何使用样本数据来产生总体参数的估计值。如果 θ 是参数，我们把它的估计量记为 θ̂（theta 帽）。例如，X̄ 是 μ 的估计量。

An estimate is the numerical value obtained when the estimator is applied to a particular sample. Different samples give different estimates, so an estimator is a random variable.

估计值是将估计量应用于某个特定样本后得到的数值。不同样本会给出不同估计值，因此估计量是一个随机变量。

3. Unbiased Estimators | 无偏估计量

An estimator θ̂ is unbiased for θ if its expected value equals the true parameter value:

E(θ̂) = θ

如果估计量 θ̂ 的期望值等于真实参数值：

E(θ̂) = θ

This means that, on average over all possible samples, the estimator hits the target parameter exactly. It does not guarantee that one particular estimate is correct, but it avoids systematic error.

这意味着在所有可能样本的平均意义上，该估计量恰好命中目标参数。它不能保证某一次具体估计正确，但能避免系统性误差。

If E(θ̂) > θ, the estimator tends to overestimate; if E(θ̂) < θ, it tends to underestimate. Both cases are described as biased.

如果 E(θ̂) > θ，该估计量倾向于高估；如果 E(θ̂) < θ，则倾向于低估。这两种情况都称为有偏。

4. Bias of an Estimator | 估计量的偏差

The bias of an estimator is defined by:

Bias(θ̂) =