The Central Idea

A population parameter is usually unknown.

We observe a sample and use a statistic to estimate that parameter.

A point estimator is the rule used to calculate an estimate.

A point estimate is the numerical value produced after the sample has been observed.

The distinction is important:

\[ \text{estimator} \longrightarrow \text{rule}, \] \[ \text{estimate} \longrightarrow \text{realized number}. \]

Parameters

A parameter is a numerical characteristic of a population or probability model.

Common examples include:

The parameter is fixed for the population being studied, even though its value may be unknown to us.

Statistics

A statistic is a numerical quantity calculated from sample data.

Examples include:

\[ \bar{X}, \qquad \hat{p}, \qquad S^2, \qquad S. \]

Before the sample is observed, these statistics are random variables because their values depend on which observations enter the sample.

Estimator versus Estimate

Suppose we estimate a population mean using the sample mean:

\[ \hat{\mu}=\bar{X}. \]

The expression \(\bar{X}\) is the estimator.

If the observed sample produces

\[ \bar{x}=42.7, \]

then \(42.7\) is the point estimate.

Point Estimation

Point estimation uses a single numerical value to estimate a population parameter.

Examples include:

\[ \hat{\mu}=\bar{x}, \] \[ \hat{p}=\frac{x}{n}, \] \[ \widehat{\sigma^2}=s^2. \]

Point estimates are compact and easy to report, but by themselves they do not express uncertainty.

Estimating a Population Mean

The standard point estimator of the population mean \(\mu\) is the sample mean:

\[ \boxed{ \bar{X} = \frac{1}{n} \sum_{i=1}^{n}X_i }. \]

After observing the sample,

\[ \boxed{ \bar{x} = \frac{1}{n} \sum_{i=1}^{n}x_i }. \]

Example: Estimating a Mean

Suppose a sample contains

\[ 12,\quad 15,\quad 11,\quad 18,\quad 14. \]

Then

\[ \bar{x} = \frac{12+15+11+18+14}{5} = \frac{70}{5} = 14. \]

The point estimate of the population mean is

\[ \boxed{14}. \]

Estimating a Population Proportion

Suppose \(X\) units in a sample of size \(n\) have a characteristic of interest.

The sample proportion is

\[ \boxed{ \hat{p} = \frac{X}{n} }. \]

It is the standard point estimator of the population proportion \(p\).

Example: Estimating a Proportion

Suppose \(168\) of \(240\) sampled customers say they would purchase a product again.

Then

\[ \hat{p} = \frac{168}{240} = 0.70. \]

The point estimate of the population proportion is

\[ \boxed{0.70}. \]

Estimating a Population Variance

For independent observations from a population with finite variance, the usual sample variance is

\[ \boxed{ S^2 = \frac{1}{n-1} \sum_{i=1}^{n}(X_i-\bar{X})^2 }. \]

Under standard conditions,

\[ E(S^2)=\sigma^2. \]

Thus \(S^2\) is an unbiased estimator of the population variance.

Why the Sample Variance Uses \(n-1\)

The deviations from the sample mean satisfy

\[ \sum_{i=1}^{n}(X_i-\bar{X})=0. \]

Once \(n-1\) deviations are known, the last is determined.

Dividing by \(n-1\) rather than \(n\) corrects the downward bias that would otherwise occur when estimating \(\sigma^2\) from deviations around the sample mean.

Estimating a Difference in Means

To estimate

\[ \mu_1-\mu_2, \]

a natural estimator is

\[ \boxed{ \bar{X}_1-\bar{X}_2 }. \]

For independent samples,

\[ E(\bar{X}_1-\bar{X}_2) = \mu_1-\mu_2. \]

Estimating a Difference in Proportions

To estimate

\[ p_1-p_2, \]

use

\[ \boxed{ \hat{p}_1-\hat{p}_2 }. \]

Under independent sampling,

\[ E(\hat{p}_1-\hat{p}_2) = p_1-p_2. \]

Desirable Properties of Estimators

Point estimators are often compared using several properties:

No single property is sufficient in every problem.

Unbiased Estimators

An estimator \(\hat{\theta}\) is unbiased for \(\theta\) when

\[ \boxed{ E(\hat{\theta})=\theta }. \]

Its sampling distribution is centered at the parameter.

Unbiasedness does not imply low variability.

Bias

The bias of an estimator is

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

Bias measures systematic displacement of the estimator's sampling distribution from the parameter.

Example: A Biased Estimator

Suppose an estimator \(T\) satisfies

\[ E(T)=\theta+3. \]

Then

\[ \operatorname{Bias}(T) = (\theta+3)-\theta = \boxed{3}. \]

The estimator tends to overestimate the parameter by \(3\) units.

Variance of an Estimator

The variance of an estimator describes how much it changes across repeated samples:

\[ \operatorname{Var}(\hat{\theta}). \]

Its standard deviation is the standard error:

\[ \boxed{ \operatorname{SE}(\hat{\theta}) = \sqrt{\operatorname{Var}(\hat{\theta})} }. \]

Precision

An estimator with smaller standard error is generally described as more precise.

Precision concerns the spread of the sampling distribution.

It does not by itself guarantee that the estimator is centered correctly.

Mean Squared Error

Mean squared error evaluates total squared estimation error:

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

It combines variability and bias.

Bias-Variance Decomposition

The mean squared error can be written

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

A good estimator may therefore be chosen by balancing bias against variance.

Example: Comparing Estimators by MSE

Suppose Estimator A has

\[ \operatorname{Bias}(A)=0, \qquad \operatorname{Var}(A)=16. \]

Then

\[ \operatorname{MSE}(A)=16. \]

Suppose Estimator B has

\[ \operatorname{Bias}(B)=2, \qquad \operatorname{Var}(B)=9. \]

Then

\[ \operatorname{MSE}(B)=9+4=13. \]

Estimator B has lower MSE even though it is biased.

Consistency

An estimator \(\hat{\theta}_n\) is consistent for \(\theta\) if it approaches \(\theta\) as sample size increases.

Informally,

\[ \boxed{ \hat{\theta}_n\to\theta } \]

as

\[ n\to\infty. \]

More precisely, consistency is commonly defined through convergence in probability.

Consistency in Probability

An estimator is consistent if, for every \(\varepsilon>0\),

\[ \boxed{ P\left( |\hat{\theta}_n-\theta|>\varepsilon \right) \to 0 } \]

as \(n\to\infty\).

The probability of a substantial estimation error becomes arbitrarily small for sufficiently large samples.

The Sample Mean Is Consistent

Under standard conditions,

\[ E(\bar{X})=\mu \]

and

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

As \(n\to\infty\),

\[ \frac{\sigma^2}{n}\to 0. \]

The sample mean therefore becomes increasingly concentrated around \(\mu\).

Efficiency

If two unbiased estimators estimate the same parameter, the estimator with the smaller variance is often called more efficient.

If

\[ \operatorname{Var}(T_1) < \operatorname{Var}(T_2), \]

then \(T_1\) is more precise and, in that comparison, more efficient.

Relative Efficiency

A simple comparison between two estimators is the ratio of their variances.

One convention is

\[ \boxed{ \operatorname{RE}(T_1,T_2) = \frac{\operatorname{Var}(T_2)} {\operatorname{Var}(T_1)} }. \]

If this ratio exceeds \(1\), then \(T_1\) has the smaller variance under that convention.

Sufficient Statistics

A statistic is called sufficient for a parameter when it captures all the information in the sample about that parameter within a specified probability model.

Sufficiency is a model-based concept.

It helps identify summaries that preserve the relevant parameter information without retaining every raw observation.

Example of Sufficiency

For independent Bernoulli observations with success probability \(p\), the total number of successes

\[ X=\sum_{i=1}^{n}X_i \]

is sufficient for \(p\).

Once the total number of successes is known, the ordering of those successes and failures contains no additional information about \(p\) under the Bernoulli model.

Robustness

An estimator is often called robust if its performance does not deteriorate severely when assumptions are mildly violated or when a few unusual observations are present.

Robustness can involve resistance to:

Mean versus Median as Estimators

The sample mean is often highly efficient for estimating a population mean under well-behaved symmetric models.

The sample median is typically more resistant to extreme observations.

Which estimator is preferable depends on:

Plug-In Estimation

A common principle is to estimate an unknown population quantity by substituting sample-based estimates for unknown parameters.

For example, if

\[ g(\mu)=\mu^2, \]

a simple plug-in estimator is

\[ g(\bar{X}) = \bar{X}^2. \]

Plug-in estimators are convenient, but they are not automatically unbiased.

A Function of an Unbiased Estimator Need Not Be Unbiased

Even if

\[ E(\bar{X})=\mu, \]

it does not generally follow that

\[ E(\bar{X}^2)=\mu^2. \]

In fact,

\[ E(\bar{X}^2) = \operatorname{Var}(\bar{X}) + [E(\bar{X})]^2. \]

Therefore,

\[ E(\bar{X}^2) = \frac{\sigma^2}{n}+\mu^2. \]

Method of Moments

The method of moments estimates parameters by matching sample moments to corresponding population moments.

The first population moment about zero is

\[ E(X), \]

while the corresponding sample moment is

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

Equating these can produce parameter estimates.

Example: Method of Moments for an Exponential Model

Suppose

\[ X\sim \operatorname{Exponential}(\lambda) \]

with

\[ E(X)=\frac{1}{\lambda}. \]

Set the population mean equal to the sample mean:

\[ \frac{1}{\lambda} = \bar{X}. \]

Solving gives the method-of-moments estimator

\[ \boxed{ \hat{\lambda}_{\text{MM}} = \frac{1}{\bar{X}} }. \]

Likelihood

Suppose observed data are

\[ x_1,\ldots,x_n \]

from a model depending on parameter \(\theta\).

The likelihood function treats the observed data as fixed and views the model probability or density as a function of \(\theta\).

In a continuous independent model,

\[ \boxed{ L(\theta) = \prod_{i=1}^{n}f(x_i;\theta) }. \]

Maximum Likelihood Estimation

The maximum likelihood estimator, or MLE, chooses the parameter value that maximizes the likelihood:

\[ \boxed{ \hat{\theta}_{\text{MLE}} = \arg\max_{\theta}L(\theta) }. \]

Informally, it selects the parameter value under which the observed data are most compatible with the specified model, in the likelihood sense.

Log-Likelihood

Products in likelihood functions can be difficult to differentiate.

Because the logarithm is strictly increasing, maximizing

\[ L(\theta) \]

is equivalent to maximizing

\[ \ell(\theta) = \log L(\theta). \]

For independent observations,

\[ \boxed{ \ell(\theta) = \sum_{i=1}^{n}\log f(x_i;\theta) }. \]

Example: MLE for a Bernoulli Proportion

Suppose

\[ X_1,\ldots,X_n \]

are independent Bernoulli observations with success probability \(p\).

Let

\[ x=\sum_{i=1}^{n}x_i. \]

The likelihood is

\[ L(p) = p^x(1-p)^{n-x}. \]

Deriving the Bernoulli MLE

The log-likelihood is

\[ \ell(p) = x\log p + (n-x)\log(1-p). \]

Differentiate:

\[ \ell'(p) = \frac{x}{p} - \frac{n-x}{1-p}. \]

Setting the derivative equal to zero gives

\[ x(1-p)=p(n-x). \]

Therefore,

\[ x=np \]

and

\[ \boxed{ \hat{p}_{\text{MLE}} = \frac{x}{n} }. \]

Example: MLE for a Normal Mean

Suppose

\[ X_1,\ldots,X_n \]

are independent normal observations with unknown mean \(\mu\) and known variance \(\sigma^2\).

Maximizing the likelihood leads to

\[ \boxed{ \hat{\mu}_{\text{MLE}} = \bar{X} }. \]

MLEs Are Not Automatically Unbiased

Maximum likelihood estimators often have excellent large-sample properties, but an MLE need not be unbiased in finite samples.

For example, the maximum likelihood estimator of a normal population variance when the mean is unknown uses denominator \(n\), not \(n-1\):

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

This estimator is biased downward for finite \(n\).

Large-Sample Properties of MLEs

Under suitable regularity conditions, maximum likelihood estimators often have useful asymptotic properties.

These can include:

The exact conditions matter, so these statements are not universal for every model.

Asymptotic Normality

In many regular parametric models,

\[ \hat{\theta} \]

becomes approximately normal for large \(n\).

Schematically,

\[ \hat{\theta} \approx N\left( \theta, \operatorname{Var}(\hat{\theta}) \right). \]

This approximation supports many large-sample confidence intervals and hypothesis tests.

Fisher Information

In regular likelihood models, Fisher information measures how much information the data distribution contains about a parameter.

For one observation, one common form is

\[ \boxed{ I(\theta) = E\left[ \left( \frac{\partial}{\partial\theta} \log f(X;\theta) \right)^2 \right] }. \]

Larger information is associated with the possibility of more precise estimation under the model.

Information from Independent Observations

For \(n\) independent identically distributed observations,

\[ I_n(\theta)=nI_1(\theta). \]

Information grows linearly with sample size.

Correspondingly, standard errors often decrease on the order of

\[ \frac{1}{\sqrt{n}}. \]

Cramér-Rao Lower Bound

Under suitable regularity conditions, an unbiased estimator \(T\) of a scalar parameter \(\theta\) satisfies

\[ \boxed{ \operatorname{Var}(T) \ge \frac{1}{I_n(\theta)} }. \]

This is the Cramér-Rao lower bound.

It provides a theoretical benchmark for how small the variance of an unbiased estimator can be in a regular parametric model.

Efficient Estimators and Information Bounds

If an unbiased estimator attains the Cramér-Rao lower bound under the required conditions, it is called efficient in that sense.

Efficiency is always relative to a model, parameter, and class of estimators.

Standard Error of a Point Estimator

A point estimate should usually be accompanied by information about its uncertainty.

The standard error of an estimator is

\[ \operatorname{SE}(\hat{\theta}) = \sqrt{\operatorname{Var}(\hat{\theta})}. \]

For the sample mean,

\[ \operatorname{SE}(\bar{X}) = \frac{\sigma}{\sqrt{n}} \]

under independent sampling from a population with standard deviation \(\sigma\).

Estimated Standard Error

Because \(\sigma\) is usually unknown, we often estimate the standard error of the sample mean by

\[ \boxed{ \widehat{\operatorname{SE}}(\bar{X}) = \frac{s}{\sqrt{n}} }. \]

The standard error itself may therefore be estimated from the data.

Estimated Standard Error for a Proportion

The true standard error of a sample proportion under independent Bernoulli sampling is

\[ \sqrt{\frac{p(1-p)}{n}}. \]

Because \(p\) is unknown, a common estimated standard error is

\[ \boxed{ \widehat{\operatorname{SE}}(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} }. \]

Point Estimates Do Not Show Their Own Uncertainty

Suppose two studies both estimate a population mean as

\[ 50. \]

One estimate may have standard error \(1\), while the other has standard error \(10\).

The point estimates are identical, but the inferential precision is very different.

Finite Population Estimation

When a simple random sample is drawn without replacement from a finite population, standard errors can include the finite population correction:

\[ \sqrt{\frac{N-n}{N-1}}. \]

For the sample mean,

\[ \operatorname{SE}(\bar{X}) = \frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}}. \]

Weighted Point Estimation

If observations have unequal sampling weights \(w_i\), a weighted mean can be written

\[ \boxed{ \bar{x}_w = \frac{\sum_{i=1}^{n}w_ix_i} {\sum_{i=1}^{n}w_i} }. \]

Weighted estimators can account for unequal selection probabilities or adjustment procedures.

Their standard errors must also reflect the weighting and sampling design.

Ratio Estimation

Sometimes an auxiliary variable is available and strongly related to the quantity of interest.

A ratio estimator may use

\[ \hat{R} = \frac{\bar{Y}}{\bar{X}}. \]

If a known population total or mean for \(X\) is available, that relationship can improve estimation of a corresponding \(Y\) quantity under suitable conditions.

Ratio Estimators Can Be Biased

Even when \(\bar{X}\) and \(\bar{Y}\) are unbiased estimators of their respective means,

\[ \frac{\bar{Y}}{\bar{X}} \]

is not generally an unbiased estimator of

\[ \frac{\mu_Y}{\mu_X}. \]

Nonlinear transformations can introduce bias.

Estimation under Model Assumptions

Some point estimators rely mainly on the sampling design.

Others rely heavily on a probability model.

For example, maximum likelihood estimation requires specifying a family of distributions.

Estimator quality can deteriorate if the model is seriously misspecified.

Design-Based and Model-Based Estimation

In design-based inference, randomness comes primarily from the sampling design.

In model-based inference, randomness is represented through a probability model for the observed variables.

Many practical analyses combine elements of both approaches.

Bayesian Point Estimation

Bayesian inference treats an unknown parameter as having a posterior distribution after combining prior information with observed data.

Common Bayesian point estimates include:

These can differ because they optimize different loss functions or summarize asymmetric posterior distributions differently.

Posterior Mean

Under squared-error loss, the Bayesian estimator that minimizes posterior expected loss is the posterior mean:

\[ \boxed{ \hat{\theta}_{\text{Bayes}} = E(\theta\mid \text{data}) }. \]

This illustrates how the choice of estimator can depend on the inferential framework and the loss function.

Loss Functions

A loss function assigns a numerical penalty to estimation error.

Squared-error loss is

\[ L(\hat{\theta},\theta) = (\hat{\theta}-\theta)^2. \]

Absolute-error loss is

\[ L(\hat{\theta},\theta) = |\hat{\theta}-\theta|. \]

Different losses can favor different estimators.

Shrinkage Estimation

Some estimators deliberately pull noisy estimates toward a central value.

This is called shrinkage.

Shrinkage may introduce bias while reducing variance enough to improve mean squared error.

The same bias-variance tradeoff encountered earlier appears here in a more deliberate form.

Robust Point Estimators

Robust estimators are designed to reduce sensitivity to unusual observations or mild departures from model assumptions.

Examples can include:

Trimmed Means

A trimmed mean removes a specified proportion of the smallest and largest observations before averaging the remainder.

This can reduce sensitivity to extreme values.

The resulting estimator may trade some efficiency under a perfectly normal model for greater robustness under contamination or heavy tails.

Estimator Choice Depends on the Target

Before selecting an estimator, identify the parameter being estimated.

The sample median is not an alternative estimator of the population mean merely because it is robust.

It naturally estimates a population median or another location feature, depending on the inferential setup.

The target parameter must remain explicit.

Estimator Choice Depends on the Data Structure

Independent observations, clustered data, repeated measurements, time series, censored data, and weighted survey data require different estimation strategies.

A formula derived for independent simple random sampling should not be applied automatically to dependent or complex data.

Estimator Choice Depends on Assumptions

A highly efficient estimator under one model may perform poorly under another.

Evaluating an estimator therefore requires understanding:

Common Error: Confusing an Estimator with an Estimate

The expression

\[ \bar{X} \]

is an estimator.

The observed value

\[ \bar{x}=27.4 \]

is an estimate.

The first is a random rule before sampling; the second is the realized numerical result.

Common Error: Believing an Unbiased Estimator Is Always Accurate

An unbiased estimator can have a very large variance.

Individual estimates may therefore frequently lie far from the true parameter even though the estimator is correctly centered in repeated sampling.

Common Error: Choosing the Estimator with the Smallest Bias Only

Bias is one component of estimation quality.

An estimator with slightly more bias may have much smaller variance and therefore lower mean squared error.

Estimator evaluation should consider the full inferential objective.

Common Error: Ignoring the Standard Error

Reporting

\[ \hat{\theta}=52 \]

without any measure of uncertainty can make the estimate appear more exact than the data justify.

Standard errors and interval estimates provide essential context.

Common Error: Assuming the MLE Is Always Unbiased

Maximum likelihood is a general estimation principle.

It does not guarantee finite-sample unbiasedness.

MLEs are often favored because of likelihood-based interpretation and useful large-sample properties, not because they are always unbiased.

Common Error: Ignoring Model Misspecification

A mathematically optimal estimator under the wrong probability model can produce misleading results.

Estimation formulas should therefore be evaluated together with the assumptions used to derive them.

Common Error: Treating More Decimal Places as More Information

A software package may report an estimate such as

\[ 3.84726191. \]

The number of displayed digits does not determine the precision of the estimator.

Inferential precision is governed by standard error and data quality, not by formatting.

A Reliable Procedure for Point Estimation

  1. State the target population or probability model.
  2. Identify the parameter to be estimated.
  3. Identify the sampling design and dependence structure.
  4. Select an estimator appropriate to the parameter and design.
  5. Compute the observed point estimate.
  6. Determine whether the estimator is biased or approximately unbiased.
  7. Estimate or derive its standard error.
  8. Consider consistency and efficiency when relevant.
  9. Check robustness and model assumptions.
  10. Report the estimate with appropriate uncertainty and limitations.

Example: Full Mean Estimation Workflow

A simple random sample of \(100\) observations produces

\[ \bar{x}=72, \qquad s=15. \]

The point estimate of the population mean is

\[ \boxed{\hat{\mu}=72}. \]

The estimated standard error is

\[ \widehat{\operatorname{SE}}(\bar{X}) = \frac{15}{\sqrt{100}} = 1.5. \]

The estimate is therefore \(72\), with sampling uncertainty on a standard error scale of approximately \(1.5\).

Example: Full Proportion Estimation Workflow

Suppose \(360\) of \(600\) sampled units have the characteristic of interest.

Then

\[ \hat{p} = \frac{360}{600} = 0.60. \]

The estimated standard error is

\[ \widehat{\operatorname{SE}}(\hat{p}) = \sqrt{\frac{0.60(0.40)}{600}} = \sqrt{0.0004} = 0.02. \]

Thus the point estimate is

\[ \boxed{0.60} \]

with estimated standard error

\[ \boxed{0.02}. \]

Practice 1: Estimator or Estimate?

A sample mean is defined by

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

Is \(\bar{X}\) an estimator or an estimate before the data are observed?

Practice 2: Estimate a Mean

A sample contains

\[ 8,\quad 11,\quad 9,\quad 12,\quad 10. \]

Find the point estimate of the population mean.

Practice 3: Estimate a Proportion

In a sample of \(250\) units, \(90\) have the characteristic of interest.

Find \(\hat{p}\).

Practice 4: Bias

An estimator satisfies

\[ E(T)=\theta-2. \]

Find its bias.

Practice 5: Mean Squared Error

An estimator has variance \(25\) and bias \(3\).

Find its mean squared error.

Practice 6: Precision

Two unbiased estimators estimate the same parameter. Estimator A has variance \(16\), and Estimator B has variance \(9\).

Which is more precise?

Practice 7: Estimated Standard Error of a Mean

A sample has

\[ s=24, \qquad n=64. \]

Find the estimated standard error of the sample mean.

Practice 8: Estimated Standard Error of a Proportion

Suppose

\[ \hat{p}=0.40, \qquad n=600. \]

Find the estimated standard error.

Practice 9: Method of Moments

Suppose a model has

\[ E(X)=2\theta. \]

Use the first method-of-moments equation to estimate \(\theta\).

Practice 10: Identify the Error

A student says, “Estimator A is unbiased, so it must have smaller mean squared error than every biased estimator.”

Explain the error.

Practice Answers

Practice 1

Before the data are observed, \(\bar{X}\) is an estimator.

Practice 2

\[ \bar{x} = \frac{8+11+9+12+10}{5} = \frac{50}{5} = \boxed{10}. \]

Practice 3

\[ \hat{p} = \frac{90}{250} = \boxed{0.36}. \]

Practice 4

\[ \operatorname{Bias}(T) = E(T)-\theta = (\theta-2)-\theta = \boxed{-2}. \]

Practice 5

\[ \operatorname{MSE} = 25+3^2 = \boxed{34}. \]

Practice 6

Estimator B is more precise because

\[ 9<16. \]

Practice 7

\[ \widehat{\operatorname{SE}}(\bar{X}) = \frac{24}{\sqrt{64}} = \frac{24}{8} = \boxed{3}. \]

Practice 8

\[ \widehat{\operatorname{SE}}(\hat{p}) = \sqrt{\frac{0.40(0.60)}{600}} = \sqrt{0.0004} = \boxed{0.02}. \]

Practice 9

Match the population moment to the sample moment:

\[ 2\theta = \bar{X}. \]

Therefore,

\[ \boxed{ \hat{\theta}_{\text{MM}} = \frac{\bar{X}}{2} }. \]

Practice 10

Unbiasedness removes the squared-bias term, but mean squared error also depends on variance.

A biased estimator can have sufficiently smaller variance that

\[ \operatorname{MSE} = \operatorname{Var} + \operatorname{Bias}^2 \]

is lower than that of an unbiased estimator.

What to Remember

Point estimation uses sample statistics to produce single-number estimates of unknown population parameters.

The most important habit is: do not evaluate a point estimate only by its observed value; evaluate the estimator by how it behaves across repeated samples.

Where We Go Next

A point estimate gives one plausible value for an unknown parameter, but it does not by itself show the range of values that are reasonably compatible with the sampling uncertainty.

The next lesson, Confidence Intervals, uses point estimates, standard errors, and sampling distributions to construct intervals that quantify uncertainty around unknown population parameters.