MFIN6201 Chap.2 Random Sampling and Estimator Behaviour
Random Sampling and Estimator Behaviour
This chapter treats an estimator as a random variable across repeated samples. It distinguishes population parameters from sample statistics and separates unbiasedness, variance, standard error and consistency. The sample mean provides the main engine: under iid sampling it is unbiased, its variance falls with sample size, and its standard error shrinks at the square-root rate rather than directly with n.
The population parameter is fixed even though it is unknown; randomness enters through which observations appear in the sample. Applying the same estimator to every possible sample creates the sampling distribution. Its centre addresses bias and its spread addresses precision. An unbiased rule can miss in one realised sample, and a precise rule can still target the wrong quantity.
Iid sampling combines a common population law with independence across draws, allowing covariance terms in the sample-mean variance to vanish. If returns are serially dependent, the nominal row count exaggerates independent information. Consistency is a large-sample property rather than a guarantee of small-sample accuracy.
The sample variance uses a degrees-of-freedom correction because the estimated sample mean constrains the deviations. These distinctions let you diagnose whether a proposed remedy needs a better estimator, a larger sample or a variance calculation that respects dependence.
What this chapter covers
- 01
Population and sample distinctions
- 02
Sampling distributions
- 03
Unbiasedness and precision
- 04
Variance of the sample mean
- 05
Square-root sample-size scaling
- 06
Consistency under large samples
Worked example · free
Standard error after a larger sample
- +1Recognise that 256 is four times 64.
- +1Standard error scales with one over the square root of n, so divide by two.
- +1Compute 0.024/2=0.012.
Key terms
- Population parameter
- A fixed numerical feature of the population that the analysis targets.
- Sampling distribution
- The repeated-sample distribution of an estimator.
- Consistency
- Convergence of an estimator toward its target as sample size grows.
Random Sampling and Estimator Behaviour FAQ
Can an unbiased estimate be wrong in one sample?
Yes. Unbiasedness concerns the average estimate across repeated samples, not exact equality in every realised dataset. A single estimate can sit above or below the target even when the estimation rule is unbiased.
Why does standard error shrink slowly?
The standard error of an iid sample mean is proportional to one over the square root of sample size. Doubling observations therefore improves precision by a factor of the square root of two, while halving standard error requires four times as many observations.
What happens when observations are dependent?
Positive dependence can add covariance terms to the variance of the sample mean, making the iid formula too small. The row count then overstates the amount of independent information, so the variance method must reflect the actual sampling structure.
How are bias and precision different?
Bias concerns the centre of an estimator's sampling distribution relative to the target. Precision concerns its spread. An unbiased estimator can be noisy, while a slightly biased rule can be tightly concentrated around a nearby value.
Does consistency guarantee a good small sample?
No. Consistency describes convergence as sample size grows under maintained conditions. A consistent estimator can still have substantial finite-sample bias or variance, so current uncertainty must be assessed rather than replaced with an asymptotic slogan.
What is random in a sampling distribution?
The population parameter remains fixed. The sample composition changes under repeated draws, which changes the statistic computed by the estimation rule. That repeated-sample variation is the uncertainty summarised by the sampling distribution and standard error.
Why does sample variance use n minus one?
Estimating the sample mean imposes one restriction on the deviations because they must sum to zero. Dividing by n minus one corrects for that lost degree of freedom and makes the usual variance estimator unbiased under iid sampling. This correction matters most when the sample is small.
Exam move
Keep four properties in separate columns: target, bias, variance and large-sample behaviour. Practise sample-size changes without recalculating from scratch, then explain why dependence changes the variance formula. When reading an estimator claim, ask what was repeatedly sampled and whether the design justifies iid reasoning. Draw repeated samples conceptually rather than treating the observed dataset as the population.
For every standard-error calculation, write the population spread in the numerator and the square root of sample size in the denominator. Rehearse doubling, quadrupling and multiplying n by nine so the scaling rule becomes automatic. Then break iid deliberately: add positive serial covariance and explain why the simple variance formula is too small.
Compare unbiasedness with consistency using plain-language definitions and an example where a single unbiased estimate misses. Re-derive the sample-variance degrees-of-freedom correction at least once, but prioritise its interpretation in revision. When a question proposes more observations, state which features of the population and design are being held constant.
Finish each answer with the repeated-sample claim it supports, because a formula without that probability statement is incomplete.
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