Sampling, Estimation and Statistical Inference

Introduction and Context

Financial decisions are made with incomplete information. An analyst may wish to estimate the average return of a security, the default rate of a loan portfolio, the volatility of an index, or the proportion of customers likely to refinance. In most cases, observing the entire population is impossible or too costly. Statistical inference provides a disciplined method for using a sample to learn about an underlying population while explicitly recognizing uncertainty.

This chapter develops the logic of sampling distributions, the central limit theorem, point and interval estimation, maximum likelihood estimation, confidence intervals, and sample-size planning. These ideas support empirical asset pricing, risk management, credit analysis, portfolio evaluation, auditing, financial forecasting, and policy research.

Learning Objectives

After studying this chapter, you should be able to:

• distinguish a population, sample, parameter, and statistic;
• explain sampling distributions and the central limit theorem;
• calculate and interpret point estimates and standard errors;
• construct confidence intervals for means, proportions, and variances;
• explain maximum likelihood estimation and apply it to common financial models;
• determine sample sizes subject to precision and confidence requirements;
• identify assumptions, sources of bias, and ethical issues in financial inference.

1. Populations, Samples and Statistical Inference

A population is the complete set of units relevant to a question. For example, the population may be all daily returns of a stock during a specified period, all borrowers in a bank’s portfolio, or all listed firms in an exchange. A sample is a subset selected from that population.

A numerical characteristic of a population is a parameter. Common parameters include the population mean μ, variance σ², standard deviation σ, and proportion p. A numerical characteristic calculated from a sample is a statistic, such as the sample mean x̄, sample variance s², or sample proportion p̂.

The inferential problem has three stages:

1. Sampling: obtain observations using a defensible design.
2. Estimation: use sample statistics to estimate unknown parameters.
3. Inference: quantify uncertainty and evaluate claims about parameters.

A representative sample is not necessarily a large sample. A very large sample selected through a biased process can produce worse inference than a smaller probability sample. For example, surveying only highly active online traders may seriously overstate the risk tolerance of all investors.

Sampling designs

• Simple random sampling: every possible sample of a given size has equal probability of selection.
• Stratified sampling: the population is divided into relevant groups, or strata, and observations are sampled within each group. This is useful when banks need representation across borrower grades.
• Cluster sampling: naturally occurring groups are sampled, and units within selected groups are observed. It can reduce cost but may increase sampling variance.
• Systematic sampling: every kth unit is selected after a random starting point.
• Convenience sampling: units are selected because they are easy to access. It is inexpensive but often unsuitable for generalization.

Random sampling reduces selection bias but does not eliminate measurement error, nonresponse bias, survivorship bias, or model misspecification.

2. Sampling Distributions

A sampling distribution is the probability distribution of a statistic over all possible samples of a fixed size drawn under a specified sampling process. It is not the distribution of individual observations. This distinction is central: an asset’s daily returns may be skewed, while the distribution of sample means may be approximately normal.

For a random sample X₁, X₂, …, Xₙ from a population with mean μ and variance σ², the sample mean is

where n is sample size. Under random sampling,

and, when observations are independent,

The standard error, SE(x̄), measures the typical sampling fluctuation of the sample mean. It is not the standard deviation of individual observations. If σ is unknown, it is commonly estimated by

where s is the sample standard deviation.

Finite population correction

When sampling without replacement from a finite population of size N, and the sampling fraction is not negligible, the standard error is adjusted to

The factor is the finite population correction. It approaches one when n is small relative to N and approaches zero when nearly the entire population is observed.

3. The Central Limit Theorem

The central limit theorem (CLT) states that, under appropriate conditions, the standardized sample mean approaches a standard normal distribution as sample size increases, even when the population distribution is not normal:

The notation means convergence in distribution. The CLT is most reliable when observations are independent and identically distributed, have finite variance, and the sample is sufficiently large. “Sufficiently large” depends on skewness, tail thickness, dependence, and the statistic being used. In financial returns, volatility clustering and extreme observations can make simple CLT approximations less reliable.

If the population itself is normal and σ is known, the sample mean is exactly normal for any n. If σ is estimated and the population is normal, the standardized statistic follows Student’s t distribution:

with n − 1 degrees of freedom. The t distribution has heavier tails than the standard normal distribution, reflecting additional uncertainty from estimating σ.

4. Point Estimation

A point estimator is a rule for using sample data to estimate a parameter. Its realized value is a point estimate. For example, x̄ estimates μ and p̂ estimates p.

Important properties include:

• Unbiasedness: an estimator θ̂ is unbiased for θ if E(θ̂) = θ.
• Consistency: θ̂ becomes arbitrarily close to θ as n increases.
• Efficiency: among estimators with relevant properties, an estimator has smaller variance.
• Sufficiency: an estimator captures all sample information relevant to a parameter under a specified model.

Unbiasedness is not the only criterion. A slightly biased estimator can have much smaller variance and therefore lower mean squared error (MSE):

The decomposition shows that precision and bias must be evaluated together.

Estimating a population mean

The sample mean is generally unbiased for μ under random sampling. If observations are independent, its variance decreases in proportion to 1/n. Therefore, quadrupling the sample size halves the standard error, not the standard deviation of the observations.

Estimating a population proportion

For a binary outcome, let p̂ = x/n, where x is the number of successes and n is sample size. Then

The large-sample standard error is

The normal approximation is usually appropriate when expected successes and failures are both sufficiently large. For small samples or proportions near zero or one, Wilson or exact binomial intervals are generally preferable to the simple Wald interval.

Estimating variance

For an independent normal sample, the unbiased sample variance is

The denominator n − 1, rather than n, corrects the downward bias created by estimating μ from the same data. Under normality,

This chi-square relationship supports confidence intervals for σ², but it is sensitive to non-normality.

5. Interval Estimation and Confidence Intervals

A confidence interval combines a point estimate with a margin of error. A general form is

A 95% confidence interval is constructed by a procedure that captures the fixed population parameter in 95% of repeated samples, assuming the model and sampling procedure are valid. It is not technically correct to say that there is a 95% probability that a fixed parameter lies inside a particular frequentist interval. The parameter is fixed; the interval-producing procedure is random.

Mean with known population standard deviation

When σ is known and the population is normal, or a CLT approximation is justified, a 100(1 − α)% interval for μ is

where z₁₋α/₂ is the standard-normal critical value. For 95% confidence, z₀.₉₇₅ ≈ 1.96.

Mean with unknown standard deviation

When σ is unknown and the population is normal, use

where t₁₋α/₂,ₙ₋₁ is the t critical value with n − 1 degrees of freedom. For large n, the t critical value approaches the normal critical value.

Worked example: mean return

Suppose 36 monthly returns have sample mean 1.2% and sample standard deviation 4.8%. Assuming independent observations and a sufficiently stable distribution, a 95% t interval is approximately

The standard error is 0.8%, and the margin of error is approximately 1.624%. Thus the interval is approximately −0.424% to 2.824% per month. The result indicates substantial uncertainty: the data are compatible with a slightly negative as well as a positive average monthly return.

Confidence interval for a proportion

A commonly used large-sample interval is

It can perform poorly for small n or extreme p̂. In professional reporting, the Wilson interval is often preferred because it has better coverage properties. The selected method should be disclosed.

Confidence interval for variance

For a normal population, an interval for σ² is

The order of the chi-square quantiles produces the lower and upper limits because the chi-square distribution is asymmetric. This interval should not be used mechanically when returns are strongly non-normal.

Difference between two means

For independent groups, an estimate of μ₁ − μ₂ is x̄₁ − x̄₂. If equal population variances are assumed, a pooled variance may be used; otherwise, Welch’s method is usually safer:

The corresponding degrees of freedom are approximated by the Welch–Satterthwaite formula. Independence, appropriate sampling, and correct treatment of paired observations are essential.

6. Maximum Likelihood Estimation

Maximum likelihood estimation (MLE) selects parameter values that make the observed data most plausible under a specified probability model. If observations x₁, …, xₙ have density or probability function f(xᵢ; θ), the likelihood is

Because products can be numerically inconvenient, analysts maximize the log-likelihood:

The MLE is θ̂ = arg maxθ ℓ(θ). The method requires a correctly specified likelihood and appropriate assumptions about independence or dependence.

Normal-model example

If Xᵢ are independent N(μ, σ²), the MLEs are

and

The MLE of σ² uses n in the denominator and is biased downward in finite samples, unlike the unbiased sample variance using n − 1. MLE is not automatically unbiased, but under regularity conditions it is consistent and asymptotically efficient.

In finance, MLE is used in volatility models, credit-default models, duration models, regime-switching models, and generalized linear models. Numerical optimization may be required, and local maxima, boundary estimates, weak identification, or convergence failure must be investigated rather than ignored.

7. Sample-Size Planning

Sample size should be chosen before data collection when possible. It depends on confidence level, desired precision, population variability, study design, cost, and the consequences of error.

For a mean with known or planning value σ and desired margin of error E,

The result is rounded upward. If σ is unknown, use a pilot estimate, prior evidence, or a conservative planning value.

For a proportion with desired margin E,

When no prior value of p is available, p = 0.5 gives the largest variance and therefore the most conservative sample size.

Worked sample-size example

A bank wants to estimate a default proportion within ±3 percentage points at 95% confidence, with no prior estimate of p. Using p = 0.5,

The bank should plan for at least 1,068 completed observations. If it expects a 10% nonresponse or unusable-record rate, the number invited should be approximately

Sample-size calculations address random sampling error, not systematic bias. A larger biased sample remains biased.

For a finite population, a common correction for a proportion is

where n₀ is the initial infinite-population requirement and N is population size. Complex designs may require a design effect, often expressed as n_effective = n/design effect. Clustering generally increases the required nominal sample.

8. Applied Case: Estimating a Credit Portfolio’s Default Rate

Background

A bank is reviewing a portfolio of 50,000 small-business loans. Management wants an estimate of the one-year default rate to support capital planning and risk-based pricing.

Problem

Reviewing every loan is expensive. The bank proposes selecting 1,200 accounts from its loan-management system and calculating the proportion that defaulted within one year.

Analysis and approach

The risk team first defines default consistently, including the observation window and treatment of restructurings. It then stratifies the population by internal risk grade and loan size. Stratified random sampling ensures that high-risk and low-risk loans are represented. Sampling weights are recorded because the sample fractions may differ across strata.

Suppose 96 of 1,200 sampled loans defaulted. The unweighted estimate is

or 8%. A simple approximate standard error is

A rough 95% interval is 8% ± 1.96 × 0.784 percentage points, or approximately 6.46% to 9.54%, before considering stratification and finite-population adjustments. The final bank report should use survey-weighted estimation and a variance method appropriate to the design.

Outcome

Management uses the estimate as an input to expected-loss and capital analysis, not as a guaranteed future default rate. Sensitivity analysis is performed for economic deterioration, changes in underwriting, and possible reporting delays.

Learning points

• Clear definitions are as important as calculations.
• Stratification can improve representation and precision.
• A confidence interval quantifies sampling uncertainty, not recession risk or model risk.
• Operational decisions should incorporate parameter uncertainty and scenario uncertainty separately.

9. Assumptions, Limitations and Professional Governance

Inference can fail through:

• Selection bias: sampled units differ systematically from the target population.
• Nonresponse and missing data: unavailable observations are related to the outcome.
• Survivorship bias: failed or delisted firms are excluded from performance analysis.
• Look-ahead bias: information unavailable at the decision date is used in estimation.
• Dependence: time-series observations may exhibit autocorrelation or volatility clustering.
• Non-normality and heavy tails: normal-based intervals may understate uncertainty.
• Multiple testing: searching many hypotheses increases false discoveries.
• P-hacking: choices are made after seeing results to obtain statistical significance.
• Model risk: a precisely estimated parameter may belong to an inappropriate model.

Financial institutions should document the target population, sampling frame, selection method, exclusions, missing-data treatment, estimator, confidence level, software, and validation procedures. Material analyses should be reproducible and independently reviewed. Privacy rules require lawful collection, minimization of personal data, access controls, and careful handling of sensitive borrower information.

Technology can improve sampling frames, automate data-quality checks, and support bootstrap or simulation-based inference. AI systems, however, can amplify historical bias, obscure sampling decisions, and generate unjustified precision. Human oversight, model validation, audit trails, and monitoring for drift remain necessary. Resampling methods such as the bootstrap are valuable when analytic formulas are unreliable, but bootstrap validity also depends on an appropriate resampling scheme, especially for dependent time series.

Chapter Summary

Statistical inference uses sample information to learn about population parameters. Sampling distributions describe how statistics vary across repeated samples, while the central limit theorem explains why sample means often become approximately normal. The sample mean, sample proportion, and sample variance are central point estimators, each requiring appropriate assumptions.

Confidence intervals combine estimates, critical values, and standard errors. t intervals are appropriate for means with unknown standard deviation under normality, while proportion and variance intervals require their own conditions. Maximum likelihood chooses parameter values that maximize the likelihood of observed data and supports many modern financial models, although it depends on model specification and numerical reliability.

Sample-size planning links confidence, variability, and desired precision. It cannot correct systematic bias. Professional inference therefore requires sound sampling design, transparent assumptions, sensitivity analysis, ethical data governance, and careful interpretation of uncertainty.

Key Terms

Population: Complete set of units relevant to an investigation.

Sample: Subset of a population used for analysis.

Parameter: Numerical characteristic of a population.

Statistic: Numerical characteristic calculated from a sample.

Sampling distribution: Distribution of a statistic across repeated samples.

Standard error: Standard deviation of an estimator’s sampling distribution.

Central limit theorem: Result describing approximate normality of standardized sample means under suitable conditions.

Estimator: Rule for calculating an estimate from data.

Unbiasedness: Property that an estimator’s expected value equals the parameter.

Consistency: Convergence of an estimator to the parameter as sample size increases.

Efficiency: Relative precision of an estimator, often measured by variance.

Mean squared error: Variance plus squared bias.

Confidence interval: Interval generated by a procedure with specified long-run coverage.

Critical value: Quantile used to determine an interval’s margin of error.

Maximum likelihood estimation: Parameter estimation by maximizing the likelihood of observed data.

Finite population correction: Adjustment for sampling without replacement from a finite population.

Selection bias: Systematic difference caused by the sampling process.

Design effect: Inflation of variance caused by a complex sampling design.

Model risk: Risk arising from an unsuitable or incorrectly implemented model.

Review Questions

Short Answer Questions

1. Distinguish a parameter from a statistic and give one financial example of each.
2. What does the standard error of a sample mean measure?
3. State the main idea of the central limit theorem.
4. Why does the unbiased sample variance use n − 1 rather than n?
5. What is the difference between a point estimate and a confidence interval?

Long Answer Questions

1. Explain the assumptions underlying a confidence interval for a population mean and discuss how financial-return characteristics may violate them.
2. Compare simple random, stratified, cluster, and convenience sampling in a banking application.
3. Explain maximum likelihood estimation, including its likelihood and log-likelihood functions, and discuss its strengths and limitations.
4. Derive and interpret the relationship between sample size and the margin of error for estimating a population proportion.
5. Discuss how selection bias, survivorship bias, and look-ahead bias can distort empirical investment research.

Analytical / Application Questions

1. A sample of 64 monthly returns has mean 0.9% and standard deviation 4.0%. Construct an approximate 95% t confidence interval for the population mean monthly return and interpret it.
2. A lender observes 45 defaults in a random sample of 750 accounts. Estimate the default rate and calculate a large-sample 95% confidence interval. State whether the approximation is defensible.
3. A portfolio study uses daily returns over ten years but excludes securities that were delisted. Identify the principal inferential problem, explain its effect, and propose a revised sampling or data strategy.

Multiple Choice Questions with Answers

Question 1

Which statement best defines a sampling distribution?

A. The distribution of individual observations in the population

B. The distribution of a statistic across repeated samples

C. The distribution of unknown population parameters

D. The distribution of measurement errors only

Correct Answer: B. The distribution of a statistic across repeated samples

Question 2

If the population standard deviation is σ and observations are independent, the standard error of the sample mean is:

A. σ × n

B. σ/n

C. σ/√n

D. √n/σ

Correct Answer: C. σ/√n

Question 3

A key implication of the central limit theorem is that:

A. All financial returns are normally distributed

B. Sample means can become approximately normal under suitable conditions

C. The population variance becomes zero as n increases

D. Sampling bias disappears for any large sample

Correct Answer: B. Sample means can become approximately normal under suitable conditions

Question 4

For a normal population with unknown σ, a confidence interval for μ generally uses:

A. A chi-square distribution

B. A Student’s t distribution

C. A binomial distribution

D. A uniform distribution

Correct Answer: B. A Student’s t distribution

Question 5

A sample has n = 100 and s = 20. The estimated standard error of the sample mean is:

A. 0.2

B. 2

C. 20

D. 200

Correct Answer: B. 2

Question 6

In maximum likelihood estimation, parameters are selected to:

A. Minimize the sample size

B. Maximize the likelihood of the observed data

C. Guarantee an unbiased estimator

D. Eliminate all sampling variation

Correct Answer: B. Maximize the likelihood of the observed data

Question 7

A 95% confidence interval means that:

A. There is a 95% probability that the fixed parameter lies in this particular interval

B. 95% of observations lie inside the interval

C. The procedure captures the true parameter in 95% of repeated samples

D. The estimate has a 95% probability of being correct

Correct Answer: C. The procedure captures the true parameter in 95% of repeated samples

Question 8

If the desired margin of error for a mean is cut in half, with all else unchanged, the required sample size becomes:

A. Half as large

B. Twice as large

C. Four times as large

D. Unchanged

Correct Answer: C. Four times as large

Question 9

For planning a proportion sample size with no prior estimate of p, the conservative choice is usually:

A. p = 0.01

B. p = 0.10

C. p = 0.50

D. p = 1.00

Correct Answer: C. p = 0.50

Question 10

Excluding firms that failed or were delisted from an investment-performance sample creates:

A. Finite population correction

B. Survivorship bias

C. Stratified sampling

D. Measurement invariance

Correct Answer: B. Survivorship bias