Point Estimation and Confidence Intervals

Introduction

Statistical inference uses information from a sample to learn about an unknown population or data-generating process. In artificial intelligence and machine learning, inference is used to estimate model parameters, quantify uncertainty in performance metrics, compare systems, and support decisions under incomplete information. This chapter develops two central tools of inference: point estimation, which produces a single numerical estimate, and interval estimation, which produces a range of plausible values.

Suppose observations \(X_1, X_2,\ldots,X_n\) are generated from a population distribution indexed by an unknown parameter \(\theta\). An estimator uses the observed data to estimate \(\theta\). A good estimator should be accurate on average, stable across repeated samples, and computationally practical. No estimator is best in every setting; its quality depends on the criterion, assumptions, and purpose of analysis.

Learning Objectives

By the end of this chapter, students should be able to:

1. Distinguish parameters, statistics, estimators, and estimates.

2. Assess estimators using bias, variance, mean squared error, consistency, efficiency, and sufficiency.

3. Construct estimators using the method of moments and maximum likelihood.

4. Derive and interpret standard errors for common estimators.

5. Construct and interpret confidence intervals for means, proportions, variances, and model parameters.

6. Explain when normal, \(t\), bootstrap, and asymptotic methods are appropriate.

7. Identify limitations, assumptions, and ethical risks in uncertainty quantification.

1. Statistical Foundations of Estimation

A population is the complete set of units or possible outcomes of interest. A sample is a collection of observations drawn from that population. A parameter is a fixed but unknown numerical feature of the population, such as a mean \(\mu\), variance \(\sigma^2\), or success probability \(p\). A statistic is a quantity calculated from sample data and therefore varies from sample to sample.

An estimator is a rule or function of the data used to estimate a parameter. It is usually denoted with a hat, such as \(\hat{\theta}\). Once data have been observed, the numerical value produced by the rule is called an estimate. For example, if \(X_1,\ldots,X_n\) are observations, the sample mean

\[

\bar X=\frac{1}{n}\sum_{i=1}^n X_i

\]

is an estimator of the population mean \(\mu\). The symbol \(\bar X\) represents the random estimator before data are observed; a value such as 72.4 is the resulting estimate.

The quality of an estimator is evaluated under a sampling model. This requires assumptions about how the data were generated. A common foundation is that \(X_1,\ldots,X_n\) are independent and identically distributed (i.i.d.) from a distribution with finite mean and variance. Independence means that one observation does not provide information about another after the model conditions are specified. Identical distribution means that all observations follow the same probability law.

2. Criteria for Assessing Estimators

2.1 Bias

The bias of an estimator \(\hat\theta\) for parameter \(\theta\) is

\[

\operatorname{Bias}(\hat\theta)=E(\hat\theta)-\theta.

\]

An estimator is unbiased if \(E(\hat\theta)=\theta\). Unbiasedness concerns the estimator’s average behavior over repeated samples; it does not guarantee that a particular estimate is close to the parameter.

For an i.i.d. sample with finite mean \(\mu\),

\[

E(\bar X)=E\left(\frac{1}{n}\sum_{i=1}^nX_i\right)=\frac{1}{n}\sum_{i=1}^nE(X_i)=\mu,

\]

so the sample mean is unbiased for \(\mu\). If the population variance is \(\sigma^2\), the estimator

\[

\frac{1}{n}\sum_{i=1}^n(X_i-\bar X)^2

\]

is biased downward for \(\sigma^2\). The usual sample variance corrects this bias:

\[

S^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\bar X)^2,

\]

provided the observations are i.i.d. with finite variance.

Bias is not always undesirable. A slightly biased estimator may have much lower variance and therefore lower overall error.

2.2 Variance and Mean Squared Error

The variance of an estimator measures its sampling variability:

\[

\operatorname{Var}(\hat\theta)=E[(\hat\theta-E\hat\theta)^2].

\]

The mean squared error (MSE) combines variability and bias:

\[

\operatorname{MSE}(\hat\theta)=E[(\hat\theta-\theta)^2]

=\operatorname{Var}(\hat\theta)+[\operatorname{Bias}(\hat\theta)]^2.

\]

Thus, an estimator with small bias can still be inferior if its variance is large. In machine learning, regularized estimators often intentionally introduce bias to reduce variance and improve predictive performance.

For i.i.d. observations with variance \(\sigma^2\),

\[

\operatorname{Var}(\bar X)=\frac{\sigma^2}{n}.

\]

The standard deviation of an estimator is called its standard error. If \(\sigma\) is known, the standard error of \(\bar X\) is \(\sigma/\sqrt n\). If \(\sigma\) is unknown, it is commonly estimated by \(S/\sqrt n\).

2.3 Consistency

An estimator \(\hat\theta_n\) is consistent for \(\theta\) if it converges in probability to \(\theta\) as sample size increases:

\[

\hat\theta_n\xrightarrow{p}\theta \quad\text{as }n\to\infty.

\]

Informally, a consistent estimator becomes increasingly reliable with more data. Under i.i.d. sampling and finite variance, the law of large numbers gives \(\bar X\xrightarrow{p}\mu\). Consistency does not describe finite-sample performance and does not imply unbiasedness. An estimator may be biased for every finite sample but still become consistent if its bias and variability vanish as \(n\) grows.

2.4 Efficiency

Among unbiased estimators of the same parameter, an estimator with smaller variance is generally called more efficient. Relative efficiency can be expressed as

\[

\operatorname{RE}(\hat\theta_1,\hat\theta_2)

=\frac{\operatorname{Var}(\hat\theta_2)}{\operatorname{Var}(\hat\theta_1)},

\]

when both estimators are unbiased and the denominator estimator is used as the reference. Efficiency comparisons must specify the class of estimators and the sampling model.

For a regular parametric model, the Cramér–Rao lower bound states that the variance of an unbiased estimator of a scalar parameter is at least

\[

\frac{1}{I_n(\theta)},

\]

where \(I_n(\theta)\) is the Fisher information in the sample. Under standard differentiability and support conditions,

\[

I_n(\theta)=nI_1(\theta),

\]

where \(I_1(\theta)\) is the information in one observation. An estimator attaining the bound is efficient within the relevant class. The regularity conditions matter: the result cannot be applied mechanically to models with parameter-dependent support or other irregularities.

2.5 Sufficiency

A statistic \(T(X)\) is sufficient for \(\theta\) if, after observing \(T\), the remaining sample contains no additional information about \(\theta\), under the specified model. The Fisher–Neyman factorization theorem provides a practical test: if the likelihood can be written as

\[

f(x_1,\ldots,x_n;\theta)=g(T(x),\theta)h(x),

\]

where \(h\) does not depend on \(\theta\), then \(T\) is sufficient for \(\theta\).

For a normal sample with known variance, the sample mean is sufficient for the population mean. For a Bernoulli sample, the total number of successes \(\sum X_i\) is sufficient for \(p\). Sufficiency is a property of a statistic relative to a model, not an absolute property of data.

3. Constructing Point Estimators

3.1 Method of Moments

The method of moments equates population moments to corresponding sample moments. If the model has parameter \(\theta\) and the first population moment is \(E_\theta(X)=m_1(\theta)\), the first moment estimator solves

\[

m_1(\hat\theta)=\bar X.

\]

For multiple parameters, additional equations use sample moments such as \(n^{-1}\sum X_i^2\).

Example: exponential rate. Suppose \(X_1,\ldots,X_n\) are i.i.d. exponential with rate \(\lambda>0\). Since \(E(X)=1/\lambda\), the method-of-moments estimator satisfies

\[

\bar X=\frac{1}{\hat\lambda},

\qquad \hat\lambda_{MM}=\frac{1}{\bar X}.

\]

The method is often simple and useful for obtaining starting values for numerical algorithms. However, it may produce estimators with undesirable bias or variance, and moment equations may have multiple or no meaningful solutions.

3.2 Maximum Likelihood Estimation

For observed data \(x=(x_1,\ldots,x_n)\), the likelihood is the joint density or probability of the observations viewed as a function of \(\theta\):

\[

L(\theta;x)=f(x_1,\ldots,x_n;\theta).

\]

The maximum likelihood estimator (MLE) is

\[

\hat\theta_{ML}=\arg\max_\theta L(\theta;x).

\]

Because logarithms preserve order, one usually maximizes the log-likelihood

\[

\ell(\theta)=\log L(\theta;x).

\]

For independent observations, the likelihood is a product and the log-likelihood is a sum. Differentiating and solving the score equation \(\ell'(\theta)=0\), followed by checking parameter boundaries and the second-order condition, often gives the MLE.

Example: Bernoulli probability. Let \(X_i\sim\operatorname{Bernoulli}(p)\), independently, where \(X_i=1\) denotes success. If \(s=\sum x_i\), then

\[

L(p)=p^s(1-p)^{n-s},\quad 0<p<1,

\]

and

\[

\ell(p)=s\log p+(n-s)\log(1-p).

\]

The derivative is \(s/p-(n-s)/(1-p)\). Setting it to zero gives

\[

\hat p_{ML}=\frac{s}{n}=\bar X.

\]

The result is a maximum because the log-likelihood is concave on \((0,1)\). The MLE is also sufficient through the success count and is unbiased for \(p\).

Example: normal mean and variance. If \(X_i\sim N(\mu,\sigma^2)\), with both parameters unknown, the MLEs are

\[

\hat\mu_{ML}=\bar X,

\qquad

\hat\sigma^2_{ML}=\frac{1}{n}\sum_{i=1}^n(X_i-\bar X)^2.

\]

The variance MLE uses divisor \(n\), not \(n-1\), and is biased downward for finite \(n\). The unbiased sample variance uses divisor \(n-1\). This illustrates that maximum likelihood and unbiasedness are different objectives.

Under regularity conditions, MLEs are often consistent and asymptotically normal:

\[

\sqrt n(\hat\theta_{ML}-\theta)

\xrightarrow{d}N\left(0, I_1(\theta)^{-1}\right),

\]

for a scalar parameter, where \(I_1(\theta)\) is per-observation Fisher information. In finite samples, however, MLEs can be biased, unstable, or nonexistent. Logistic regression with complete separation is a notable example in which ordinary maximum likelihood coefficients may diverge.

3.3 Invariance of Maximum Likelihood

If \(\hat\theta\) is the MLE of \(\theta\), and \(\tau=g(\theta)\) is a one-to-one transformation, then the MLE of \(\tau\) is \(g(\hat\theta)\). For example, if \(\hat\lambda=1/\bar X\) estimates an exponential rate, the MLE of the mean lifetime \(1/\lambda\) is \(1/\hat\lambda=\bar X\). Care is needed when transformations are not one-to-one or when the parameter space has boundaries.

4. Standard Errors and Sampling Distributions

A sampling distribution describes how an estimator varies over hypothetical repeated samples. Standard errors quantify this variation and are essential for confidence intervals and hypothesis tests.

For the sample mean under i.i.d. sampling with finite variance,

\[

SE(\bar X)=\frac{\sigma}{\sqrt n},

\]

or, when \(\sigma\) is unknown,

\[

\widehat{SE}(\bar X)=\frac{S}{\sqrt n}.

\]

If the population is normal and \(\sigma\) is unknown, then

\[

\frac{\bar X-\mu}{S/\sqrt n}\sim t_{n-1}.

\]

For large samples, the central limit theorem often gives approximate normality of \(\bar X\), provided observations are independent and no single observation dominates the sum. Heavy tails, dependence, clustering, and small samples can invalidate a naive normal approximation.

For a sample proportion \(\hat p=X/n\), where \(X\sim\operatorname{Binomial}(n,p)\),

\[

E(\hat p)=p,

\qquad

\operatorname{Var}(\hat p)=\frac{p(1-p)}{n}.

\]

The estimated standard error is commonly \(\sqrt{\hat p(1-\hat p)/n}\), although this plug-in estimate performs poorly near 0 or 1 in small samples.

For a differentiable function \(g(\hat\theta)\), the delta method approximates variance by

\[

\operatorname{Var}[g(\hat\theta)]\approx [g'(\theta)]^2\operatorname{Var}(\hat\theta).

\]

In practice, \(\theta\) is replaced by \(\hat\theta\). For a vector parameter, the scalar derivative is replaced by a gradient and the variance by a covariance matrix.

5. Confidence Intervals

A \(100(1-\alpha)\%\) confidence interval is a procedure that, under repeated sampling and its assumptions, covers the fixed parameter in approximately \(100(1-\alpha)\%\) of samples. The parameter is not properly described as having a probability of lying inside a particular frequentist interval; after the data are observed, the interval either covers the parameter or does not. The confidence level refers to the long-run performance of the procedure.

5.1 Mean with Known Variance

If \(X_1,\ldots,X_n\) are i.i.d. normal with known standard deviation \(\sigma\), a confidence interval for \(\mu\) is

\[

\bar x\pm z_{1-\alpha/2}\frac{\sigma}{\sqrt n},

\]

where \(z_{1-\alpha/2}\) is the corresponding standard normal quantile. For a 95% interval, \(z_{0.975}\approx1.96\).

5.2 Mean with Unknown Variance

If the population is normal and \(\sigma\) is unknown, use

\[

\bar x\pm t_{n-1,1-\alpha/2}\frac{s}{\sqrt n}.

\]

The \(t\)-critical value has heavier tails than the normal critical value, especially for small samples. The interval is exact under normal sampling, but only approximate under nonnormal sampling unless other large-sample justification is available.

Worked example. A quality-control sample of \(n=25\) components has mean lifetime \(\bar x=102\) hours and sample standard deviation \(s=10\) hours. Assuming normal lifetimes, a 95% interval for the population mean uses \(t_{24,0.975}\approx2.064\):

\[

102\pm2.064\left(\frac{10}{5}\right)

=102\pm4.128.

\]

Thus the interval is approximately \((97.87,106.13)\) hours. The margin of error is 4.13 hours. It is not correct to say that 95% of individual component lifetimes lie in this interval; the interval concerns the population mean.

5.3 Proportions

The simple Wald interval is

\[

\hat p\pm z_{1-\alpha/2}\sqrt{\frac{\hat p(1-\hat p)}{n}}.

\]

It is convenient but can have poor coverage for small samples or proportions near 0 and 1, and it may extend outside \([0,1]\). Wilson intervals or exact binomial intervals are generally preferable in such cases. A normal approximation should not be selected solely because \(n\) is large; the expected numbers of successes and failures should also be adequate.

5.4 Differences in Means and Proportions

For two independent groups, a difference in means can be estimated by \(\bar X_1-\bar X_2\). When variances are not assumed equal, Welch’s interval is preferred:

\[

(\bar x_1-\bar x_2)\pm t_{\nu,1-\alpha/2}

\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}},

\]

where \(\nu\) is the Welch–Satterthwaite approximate degrees of freedom. Pairing, clustering, or repeated measures require methods that account for dependence; treating correlated observations as independent typically makes intervals too narrow.

5.5 Confidence Intervals for Model Parameters

For a scalar estimator that is approximately normal,

\[

\hat\theta\pm z_{1-\alpha/2}\,SE(\hat\theta)

\]

is a Wald interval. In regression and classification, standard errors may be obtained from the inverse observed information matrix, robust sandwich estimators, clustered estimators, or resampling. The choice must reflect the data-generating structure. A narrow interval can be misleading if the model is misspecified, the sample is biased, or the effective sample size is overestimated.

6. Bootstrap Methods

The bootstrap estimates the sampling distribution of a statistic by repeatedly resampling from the observed data. It is useful when a theoretical standard error is difficult to derive or when the statistic is nonlinear, such as a median, correlation, quantile, or machine-learning performance measure.

For the ordinary nonparametric bootstrap:

1. From the observed sample of size \(n\), draw a bootstrap sample of size \(n\) with replacement.

2. Compute the statistic \(\hat\theta^*\) for that sample.

3. Repeat the process \(B\) times.

4. Use the bootstrap values to estimate standard error or construct an interval.

The bootstrap standard error is

\[

\widehat{SE}_{boot}

=\sqrt{\frac{1}{B-1}\sum_{b=1}^B(\hat\theta_b^-\bar\theta^)^2},

\]

where \(\bar\theta^=B^{-1}\sum_{b=1}^B\hat\theta_b^\).

A percentile interval takes the empirical \(\alpha/2\) and \(1-\alpha/2\) quantiles of the bootstrap distribution. Other intervals include the basic bootstrap and bias-corrected and accelerated (BCa) interval. Percentile intervals are easy to explain but may have poor coverage for strongly skewed or biased statistics. BCa intervals can improve accuracy but require additional calculations.

The bootstrap must reproduce the relevant sampling design. For time series, resampling individual observations destroys dependence; block bootstrap methods are more appropriate. For clustered data, resample clusters rather than individual records. In predictive modeling, preprocessing and model fitting must be repeated within each resample to avoid leakage. A bootstrap cannot repair a systematically biased sample or an invalid estimand.

7. Illustrative Case: Uncertainty in a Machine-Learning Model’s Recall

Background. A hospital develops a classifier intended to identify patients requiring additional clinical review. The model is evaluated on 240 independently held-out patients, of whom 80 truly require review. It correctly identifies 68 of these patients, giving an observed recall of

\[

\hat p=\frac{68}{80}=0.85.

\]

Problem. The deployment team wants to know whether the apparent recall is sufficiently reliable for governance approval. Reporting only 85% hides sampling uncertainty.

Analysis and approach. A simple approximate standard error is

\[

\sqrt{\frac{0.85(1-0.85)}{80}}

=\sqrt{0.00159375}\approx0.0399.

\]

The approximate 95% Wald interval is therefore

\[

0.85\pm1.96(0.0399)\approx(0.772,0.928).

\]

Because the number of positive cases is only 80 and recall is a proportion, the team should preferably report a Wilson or exact binomial interval rather than relying only on the Wald interval. If patients come from several hospitals, a cluster-aware analysis is needed. If the test set was repeatedly used to tune the model, the nominal interval is also optimistic because the evaluation data are no longer a clean held-out sample.

A bootstrap analysis can resample the 80 positive cases and recompute recall, but it must preserve the definition of the metric and the sampling design. If the model threshold was selected using the same data, the threshold-selection process should be repeated inside each bootstrap replicate for a valid assessment of the complete development procedure.

Outcome. The governance committee does not interpret 85% as a guaranteed future recall. Instead, it records the point estimate, an appropriate confidence interval, subgroup intervals, the sampling design, and the possibility of distribution shift. It requests prospective monitoring after deployment.

Learning points. A performance estimate is incomplete without uncertainty information. The relevant sample size for recall is the number of actual positive cases, not necessarily the total test-set size. Model evaluation must distinguish random sampling error from dataset shift, selection bias, and dependence. Uncertainty reporting supports safer and more accountable AI decisions.

8. Challenges, Limitations, and Best Practices

Assumptions and model misspecification

Confidence intervals are conditional on assumptions such as independence, correct distributional form, random sampling, and stable measurement. Violations can produce intervals with poor coverage. Diagnostics, subject-matter knowledge, sensitivity analysis, and robust methods should accompany formula-based inference.

Multiple comparisons and selective reporting

If many parameters, subgroups, or metrics are examined and only favorable intervals are reported, the apparent evidence can be misleading. Multiplicity adjustments, preregistered analysis plans, and transparent reporting reduce selective inference.

Dependence and effective sample size

Repeated records from the same person, spatial data, time series, and network data are not generally i.i.d. The nominal number of records may substantially exceed the effective amount of information. Cluster-robust, hierarchical, or time-series methods may be necessary.

Statistical versus practical importance

A narrow interval around a tiny effect may be statistically precise but practically irrelevant. Conversely, a wide interval may reflect insufficient data rather than absence of an important effect. Interpretation should consider domain-specific thresholds, costs, benefits, and uncertainty.

Ethical and governance considerations

Uncertainty intervals should not be used to create false confidence, conceal subgroup disparities, or justify high-risk decisions without validation. AI systems require uncertainty assessment across relevant demographic and operational groups. Data privacy, reproducibility, documentation of preprocessing, and secure handling of resampling outputs are also important. A confidence interval quantifies one component of uncertainty; it does not measure fairness, causal validity, or safety by itself.

Best practices

• Define the estimand before selecting an estimator.
• State the sampling design and assumptions.
• Report point estimates together with standard errors or intervals.
• Use \(t\)-methods rather than normal methods when appropriate for small-sample means.
• Prefer Wilson or exact methods to the Wald interval for small or extreme proportions.
• Preserve clustering, temporal dependence, and preprocessing steps in bootstrap procedures.
• Distinguish statistical uncertainty from measurement error, selection bias, and distribution shift.
• Report sample sizes, missing-data handling, and analysis decisions transparently.
• Use simulation or sensitivity analysis when theoretical assumptions are doubtful.

Contemporary Developments

Modern statistical learning increasingly combines classical estimators with computational inference. Cross-fitting and sample splitting help reduce overfitting when estimating treatment effects or evaluating complex prediction pipelines. Influence-function methods provide efficient uncertainty estimates for many estimands. Bayesian credible intervals offer a different interpretation based on a prior and posterior distribution; they should not be confused with frequentist confidence intervals. Conformal prediction constructs prediction sets with finite-sample marginal coverage under exchangeability, addressing predictive uncertainty rather than directly estimating a population parameter. These methods expand the toolkit but do not eliminate the need to define the target, assess data quality, and examine assumptions.

Chapter Summary

Point estimation uses sample data to estimate unknown population parameters. Estimators are assessed using bias, variance, mean squared error, consistency, efficiency, and sufficiency. The method of moments is based on matching population and sample moments, while maximum likelihood selects the parameter value that makes the observed data most plausible under a specified model. Maximum likelihood often has strong large-sample properties but may be biased or unstable in finite samples.

Standard errors describe estimator variability and support interval estimation. Confidence intervals are procedures with stated long-run coverage under specified assumptions; they do not assign a frequentist probability to a fixed parameter after data are observed. Normal, \(t\), proportion, Welch, and asymptotic intervals address different settings. Bootstrap methods approximate sampling distributions by resampling and are especially useful for complex statistics, provided the resampling scheme respects dependence and the complete analysis procedure.

In AI and ML, uncertainty intervals are essential for responsible evaluation of model performance. They must be interpreted alongside sampling design, distribution shift, subgroup performance, practical consequences, and governance requirements.

Key Terms

Parameter: A fixed but generally unknown numerical feature of a population or model.

Statistic: A quantity computed from sample data.

Estimator: A rule that maps sample data to an estimate of a parameter.

Estimate: The numerical value produced by an estimator for observed data.

Bias: The difference between an estimator’s expected value and the target parameter.

Variance: The expected squared deviation of an estimator from its own mean.

Mean squared error: Variance plus squared bias.

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

Efficiency: Comparative precision of an estimator, often assessed through variance.

Sufficiency: A property whereby a statistic contains all sample information about a parameter under a model.

Likelihood: The joint probability or density of observed data viewed as a function of parameters.

Maximum likelihood estimator: The parameter value maximizing the likelihood.

Method of moments: An estimation method equating sample moments with population moments.

Standard error: The standard deviation, or an estimate of the standard deviation, of an estimator.

Confidence interval: A repeated-sampling procedure designed to cover a parameter at a stated long-run rate.

Bootstrap: A resampling method used to approximate an estimator’s sampling distribution.

Fisher information: A measure of the information about a parameter contained in a probability model.

Cramér–Rao lower bound: A lower bound on the variance of regular unbiased estimators.

Delta method: A Taylor-approximation method for obtaining the variance of a transformed estimator.

Estimand: The precisely defined quantity that an analysis aims to estimate.

Review Questions

Short Answer Questions

1. Distinguish an estimator from an estimate and provide an example.

2. Define bias and explain why an unbiased estimator is not necessarily optimal.

3. What does consistency mean in statistical estimation?

4. Why is the divisor \(n-1\) used in the usual sample variance?

5. State the correct repeated-sampling interpretation of a 95% confidence interval.

Long Answer Questions

1. Explain bias, variance, and mean squared error, and discuss the bias–variance trade-off in machine learning.

2. Derive the maximum likelihood estimator of a Bernoulli success probability.

3. Compare the method of moments and maximum likelihood methods, including advantages and limitations.

4. Explain sufficiency and use the factorization theorem to demonstrate sufficiency for a simple model.

5. Describe the assumptions behind a one-sample \(t\)-interval and explain how violations affect its reliability.

Analytical / Application Questions

1. A sample of 16 normally distributed measurements has mean 48 and standard deviation 8. Construct a 95% confidence interval for the population mean and interpret it correctly.

2. A classifier identifies 45 of 60 truly positive cases. Explain which uncertainty interval you would report and why a simple Wald interval may be unsuitable.

3. Design a bootstrap procedure for estimating uncertainty in the median prediction error of a model evaluated on clustered customer data. Explain how the clustering should affect resampling.

Multiple Choice Questions with Answers

Question 1

Which quantity is the mean squared error of an estimator?

A. Its variance minus its bias

B. Its variance plus its squared bias

C. Its bias divided by sample size

D. Its standard error plus its parameter value

Correct Answer: B. Its variance plus its squared bias

Question 2

For i.i.d. observations with population variance σ², what is Var(X̄)?

A. σ²n

B. σ²/(n−1)

C. σ²/n

D. σ/√n

Correct Answer: C. σ²/n

Question 3

Which estimator is the maximum likelihood estimator for a Bernoulli probability p?

A. n/s, where s is the number of successes

B. s/n

C. s/(n−1)

D. 1−s/n

Correct Answer: B. s/n

Question 4

If the population standard deviation is unknown and the population is normal, which distribution is used for a one-sample mean interval?

A. Uniform distribution

B. Binomial distribution

C. Student’s t distribution

D. Poisson distribution

Correct Answer: C. Student’s t distribution

Question 5

What does consistency of an estimator mean?

A. It is unbiased for every sample size

B. It has zero variance

C. It converges in probability to the target as sample size increases

D. It always equals the population parameter

Correct Answer: C. It converges in probability to the target as sample size increases

Question 6

A 95% frequentist confidence interval is best interpreted as:

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

B. 95% of individual observations lie in the interval

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

D. The sample mean has a 95% probability of being correct

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

Question 7

Why can the simple Wald interval for a proportion be problematic?

A. It always requires a known population variance

B. It can have poor coverage and extend outside [0,1]

C. It cannot be calculated for any sample size

D. It is based on the exact binomial distribution

Correct Answer: B. It can have poor coverage and extend outside [0,1]

Question 8

In an ordinary nonparametric bootstrap, each resample is formed by:

A. Sampling without replacement and reducing the sample size

B. Drawing with replacement from the observed data

C. Generating data from a uniform distribution

D. Replacing the parameter with zero

Correct Answer: B. Drawing with replacement from the observed data

Question 9

What is a sufficient statistic?

A. A statistic that always has zero bias

B. A statistic containing all sample information about a parameter under the model

C. A statistic with the largest possible variance

D. A statistic calculated only from normally distributed data

Correct Answer: B. A statistic containing all sample information about a parameter under the model

Question 10

For a 95% confidence interval with known σ for a normal population mean, the critical standard-normal value is approximately:

A. 0.95

B. 1.00

C. 1.645

D. 1.96

Correct Answer: D. 1.96