The Theory of Unbiased Estimators | IB数学：无偏估计量理论

📚 The Theory of Unbiased Estimators | IB数学：无偏估计量理论

In statistics, an estimator is a rule for calculating an estimate of a parameter from sample data. When studying inference in IB Mathematics, you are often asked to judge whether a proposed estimator is biased or unbiased. Unbiasedness is a guarantee that the estimator does not systematically overestimate or underestimate the true parameter.

在统计学中，估计量是根据样本数据计算参数估计值的规则。在 IB 数学学习推断时，你常被要求判断某个提出的估计量是有偏还是无偏。无偏性意味着估计量不会系统性地高估或低估真实参数。

1. Introduction to Estimation | 估计简介

Statistical inference uses sample data to draw conclusions about a population. The numerical feature we wish to know, such as the population mean μ or population variance σ², is called a parameter. A formula that uses the sample observations to approximate this parameter is called an estimator.

统计推断利用样本数据对总体作出结论。我们希望知道的总体数值特征，如总体均值 μ 或总体方差 σ²，称为参数。一种利用样本观测值来近似该参数的公式称为估计量。

For example, to estimate the population mean, the natural estimator is the sample mean X̄ = (X₁ + X₂ + ⋯ + Xₙ)/n. Likewise, the population variance σ² is usually estimated by a sample-based quantity such as s². In IB exams, you may be given a statistic and asked to show whether it is unbiased for a parameter.

例如，要估计总体均值，自然的估计量是样本均值 X̄ = (X₁ + X₂ + ⋯ + Xₙ)/n。同理，总体方差 σ² 通常用基于样本的量（如 s²）来估计。在 IB 考试中，你可能会被给出一个统计量，并要求证明它是否为某个参数的无偏估计量。

2. Unbiased Estimator: Definition | 无偏估计量的定义

An estimator θ̂ of a parameter θ is called unbiased if the expected value of the estimator equals the true parameter value for every possible value of θ.

若参数 θ 的估计量 θ̂ 满足：对 θ 的所有可能取值，都有估计量的期望值等于真实参数值，则称该估计量为无偏估计量。

E(θ̂) = θ

If E(θ̂) ≠ θ, the estimator is said to be biased. The size of this difference is called the bias, denoted B(θ̂) = E(θ̂) − θ. An estimator with zero bias is called unbiased; if the bias tends to zero as the sample size grows, the estimator is called asymptotically unbiased.

如果 E(θ̂) ≠ θ，则称该估计量是有偏的。这一差值的大小称为偏差，记作 B(θ̂) = E(θ̂) − θ。偏差为零的估计量称为无偏估计量；若随着样本容量的增大偏差趋于零，则称该估计量为渐近无偏。

3. Bias and Mean Square Error | 偏差与均方误差

Bias alone does not tell the full story of an estimator’s quality. A good estimator also has low variance. The mean square error (MSE) combines bias and variance:

仅有偏差并不能完全说明估计量的优劣。好的估计量还应当有较小的方差。均方误差（MSE）综合了偏差与方差：

MSE(θ̂) = Var(θ̂) + [B(θ̂)]²

This identity shows that an estimator may be biased but still have a smaller MSE than any unbiased alternative, because its variance is much smaller. Nevertheless, the IB syllabus focuses on unbiasedness as a fundamental criterion, especially when comparing estimators.

这一恒等式表明：一个估计量可以有偏，但因方差更小而比任何无偏替代估计量具有更小的均方误差。不过，IB 大纲将无偏性作为基本准则，尤其是在比较估计量时。

In exam contexts, you are usually not required to calculate MSE, but you should understand that unbiasedness is assessed through E(θ̂) alone. The variance of θ̂ is studied separately under the heading of efficiency.

在考试中，通常不要求计算 MSE，但你应当理解无偏性仅通过 E(θ̂) 来判断。θ̂ 的方差则放在“有效性”这一主题下单独研究。

4. Why Unbiasedness Matters in IB | 为什么无偏性在 IB 中重要

Unbiasedness is one of the most frequently tested concepts in IB Statistics. A common question gives a statistic such as T = aX₁ + bX₂ or a modified sample variance, then asks you to find a constant so that T is unbiased.

无偏性是 IB 统计中最常考的概念之一。常见题目会给出一个统计量，例如 T = aX₁ + bX₂ 或修正后的样本方差，然后要求你求出某个常数，使 T 成为无偏估计量。

The main reason unbiasedness is important is that, on average, an unbiased estimator hits the target parameter. Although the estimator from a single sample may be far from the true value, if you repeated sampling many times, the average of the estimates would converge to the true parameter.

无偏性之所以重要，是因为平均而言，无偏估计量会命中目标参数。虽然单次样本得到的估计值可能远离真实值，但如果重复抽样很多次，所有估计值的平均值将收敛于真实参数。

Direction for revision: Learn to compute E(estimator) by treating the sample observations Xᵢ as random variables.
中文复习方向：学会通过把样本观测值 Xᵢ 视为随机变量来计算 E(估计量)。
Use linearity of expectation: E(aX + bY) = aE(X) + bE(Y).
利用期望的线性性质：E(aX + bY) = aE(X) + bE(Y)。

5. Standard Unbiased Estimators | 标准无偏估计量

The following table lists common parameters and their usual unbiased estimators. For a random sample X₁, X₂, …, Xₙ from a population with mean μ and variance σ², the sample mean is always unbiased for μ. The sample variance with denominator n − 1 is unbiased for σ².

下表列出了常用参数及其通常的无偏估计量。对于来自均值为 μ、方差为 σ² 的总体的随机样本 X₁, X₂, …, Xₙ，样本均值总是 μ 的无偏估计量。分母为 n − 1 的样本方差是 σ² 的无偏估计量。

Parameter 参数	Unbiased Estimator 无偏估计量	Expectation 期望
Mean μ	X̄ = (Σxᵢ)/n	E(X̄) = μ
Variance σ²	s² = Σ(xᵢ − X̄)²/(n − 1)	E(s²) = σ²
Proportion p	p̂ = X/n (number of successes)	E(p̂) = p

Notice that the sample proportion p̂ is unbiased because each observation is a Bernoulli variable with mean p. This result is used in confidence interval questions, where the centre of the interval is p̂.

注意，样本比例 p̂ 是无偏的，因为每个观测值都是均值为 p 的伯努利变量。这一结果用于置信区间问题中，区间的中心就是 p̂。

6. Proof that the Sample Variance (n − 1) is Unbiased | 证明样本方差（n − 1）是无偏的

The most important proof in this topic is showing that s² with denominator n − 1 satisfies E(s²) = σ². The proof uses a clever algebraic decomposition of the sum of squared deviations around the sample mean.

本主题最重要的证明是：分母为 n − 1 的 s² 满足 E(s²) = σ²。该证明用了一个巧妙代数分解：围绕样本均值的离差平方和。

Start with the identity:

首先从恒等式开始：

Σ(Xᵢ − X̄)² = Σ(Xᵢ − μ)² − n(X̄ − μ)²

Taking expectations of both sides:

两边同时取期望：

E[Σ(Xᵢ − X̄)²] = E[Σ(Xᵢ − μ)²] − nE[(X̄ − μ)²]

Since each Xᵢ has variance σ², we have E[Σ(Xᵢ − μ)²] = nσ². Also, Var(X̄) = σ²/n, so E[(X̄ − μ)²] = σ²/n. Substituting gives:

因为每个 Xᵢ 的方差为 σ²，所以 E[Σ(Xᵢ − μ)²] = nσ²。又因 Var(X̄) = σ²/n，故 E[(X̄ − μ)²] = σ²/n。代入得：

E[Σ(Xᵢ − X̄)²] = nσ² − n(σ²/n) = (n − 1)σ²

Dividing by n − 1 gives E(s²) = σ². Hence s² is an unbiased estimator of σ². This is why the IB formula booklet uses (n − 1) rather than n.

除以 n − 1 得 E(s²) = σ²。因此 s² 是 σ² 的无偏估计量。这就是为什么 IB 公式手册使用 n − 1 而不是 n 的原因。

7. The Biased Estimator with Denominator n | 分母为 n 的有偏估计量

If you use the sample variance with denominator n instead of n − 1, the resulting estimator is biased. Let σ̂² = Σ(Xᵢ − X̄)²/n. From the previous proof,

若使用分母为 n 的样本方差代替 n − 1，所得估计量是有偏的。设 σ̂² = Σ(Xᵢ − X̄)²/n。由前面的证明，

E(σ̂²) = (n − 1)σ²/n = σ² − σ²/n

Thus the bias is B(σ̂²) = −σ²/n. The estimator underestimates the true variance because the sample mean is closer to the data than the population mean, reducing the average squared deviation.

因此偏差为 B(σ̂²) = −σ²/n。该估计量会低估真实方差，因为样本均值比总体均值更贴近数据，从而平均离差平方更小。

In IB papers, a common question is: “Explain why the sample variance with denominator n is biased.” The correct explanation is that the sample mean minimises the sum of squared deviations, so deviations from X̄ are, on average, smaller than deviations from μ.

IB 试卷中常见问题：“解释为什么分母为 n 的样本方差是有偏的。”正确解释是：样本均值使离差平方和最小，因此相对于 X̄ 的离差平均小于相对于 μ 的离差。

8. Unbiasedness in Exam Problems | 考试题中的无偏性

Typical IB exam questions revolve around a statistic built from a sample. You are asked to find a constant, show unbiasedness, or compare two estimators. A standard example:

IB 典型考题围绕由样本构成的统计量展开。题目要求你求常数、证明无偏性或比较两个估计量。一个标准例题：

Example (typical style): Let X₁ and X₂ be an independent sample from a population with mean μ and variance σ². Show that T = (2X₁ + X₂)/3 is an unbiased estimator of μ.

例题（典型风格）：设 X₁ 与 X₂ 是来自均值为 μ、方差为 σ² 的总体的独立样本。证明 T = (2X₁ + X₂)/3 是 μ 的无偏估计量。

Solution: By linearity of expectation, E(T) = (2μ + μ)/3 = μ. So T is unbiased. Note that we never use the variance in this calculation.

解答：由期望的线性性质，E(T) = (2μ + μ)/3 = μ。因此 T 是无偏的。注意在计算中完全不需要使用方差。

Another typical question: find the value of k such that k(X₁ + X₂) is unbiased for μ. This is a pure algebra problem: solve k(2μ) = μ, hence k = 1/2.

另一种典型题：求 k 的值，使 k(X₁ + X₂) 是 μ 的无偏估计量。这是纯粹的代数问题：解 k(2μ) = μ，得 k = 1/2。

9. Linear Combinations of Estimators | 估计量的线性组合

If T₁ and T₂ are unbiased estimators of θ, then any weighted average T = wT₁ + (1 − w)T₂ is also unbiased, provided the weights sum to 1. This is because

若 T₁ 和 T₂ 都是 θ 的无偏估计量，则任何加权平均 T = wT₁ + (1 − w)T₂ 也是无偏的，前提是权重之和为 1。这是因为

E(T) = wE(T₁) + (1 − w)E(T₂) = wθ + (1 − w)θ = θ

This idea appears in IB when combining two independent estimates of the same parameter, for example using two smaller samples. Choosing the value of w that minimises Var(T) leads to the concept of the best linear unbiased estimator (BLUE).

这个思想在 IB 中出现在将同一参数的两个独立估计值合并时，例如使用两个较小样本。选择使 Var(T) 最小的 w 值，会引出最佳线性无偏估计量（BLUE）的概念。

To minimise Var(T) = w²Var(T₁) + (1 − w)²Var(T₂), you set the derivative equal to zero. The optimal weight is inversely proportional to the variances, giving greater weight to the more precise estimator.

为使 Var(T) = w²Var(T₁) + (1 − w)²Var(T₂) 最小，可令导数为零。最优权重与方差成反比，即给精度更高的估计量更大的权重。

10. Efficiency and Best Unbiased Estimators | 有效性与最佳无偏估计量

Unbiasedness alone does not tell us which estimator is best. Among all unbiased estimators, we prefer the one with the smallest variance. This property is called efficiency. An unbiased estimator with minimum variance is said to be efficient.

仅有无偏性并不能说明哪个估计量最好。在所有无偏估计量中，我们偏好方差最小者。这一性质称为有效性。方差最小的无偏估计量称为有效估计量。

For example, suppose X₁ and X₂ are independent observations with common mean μ and variance σ². Compare T₁ = (X₁ + X₂)/2 and T₂ = (X₁ + 2X₂)/3. Both are unbiased, but Var(T₁) = σ²/2 whereas Var(T₂) = (1 + 4)σ²/9 = 5σ²/9. Since σ²/2 < 5σ²/9, T₁ is more efficient.

例如，设 X₁ 和 X₂ 是独立观测值，共同均值为 μ，方差为 σ²。比较 T₁ = (X₁ + X₂)/2 与 T₂ = (X₁ + 2X₂)/3。两者都无偏，但 Var(T₁) = σ²/2，而 Var(T₂) = (1 + 4)σ²/9 = 5σ²/9。因为 σ²/2 < 5σ²/9，所以 T₁ 更有效。

In IB, you may be asked to compare the variances of two unbiased estimators and state which one is better. Always remember: unbiasedness plus low variance gives a reliable estimator.

在 IB 中，你可能会被要求比较两个无偏估计量的方差，并指出哪一个更好。永远记住：无偏性加上低方差才能得到可靠的估计量。

11. Summary and Common Mistakes | 总结与常见错误

Unbiased estimators are a cornerstone of statistical inference in IB Mathematics. You should be able to compute E(estimator), prove unbiasedness, identify a biased form, and compare estimators by variance.

无偏估计量是 IB 数学统计推断的基石。你应当能够计算 E(估计量)、证明无偏性、识别有偏形式，并通过方差比较估计量。

Mistake 1: Using E(ΣXᵢ²) = nμ². Correct: E(Xᵢ²) = μ² + σ².
中文常见错误 1：误写 E(ΣXᵢ²) = nμ²。正确：E(Xᵢ²) = μ² + σ²。
Mistake 2: Forgetting that X̄ is a random variable in its own right; Var(X̄) = σ²/n.
中文常见错误 2：忘记 X̄ 本身也是随机变量，其方差为 Var(X̄) = σ²/n。
Mistake 3: Claiming a denominator of n is unbiased because “both formulas converge to the same value as n → ∞”. Convergence does not imply unbiasedness for finite n.
中文常见错误 3：认为分母为 n 时无偏，因为“两个公式在 n → ∞ 时收敛于同一值”。收敛性并不等于有限 n 时无偏。

Finally, always write the definition of unbiasedness before performing any calculation. In structured IB questions, showing E(θ̂) = θ explicitly earns full marks.

最后，在任意计算之前先写出无偏性的定义。在 IB 结构化题目中，显式写出 E(θ̂) = θ 可以获得满分。

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The Theory of Unbiased Estimators | IB数学：无偏估计量理论