Glossary: M07 — Estimation and Inference

Module: M07 Formulas: Formula Sheet Concept page: Sampling and Estimation

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Population

The complete set of all items of interest to an analyst. Characterized by parameters such as the population mean $\mu$ and variance ${\sigma }^2$.

LOS: 7.a | Contrast: Sample

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Sample

A subset of the population selected for analysis. Used to estimate population parameters when examining the entire population is impractical or impossible.

LOS: 7.a | Related: Sampling Error, Sampling Distribution

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Parameter

A descriptive measure of an entire population (e.g., $\mu$, ${\sigma }^2$). Parameters are typically unknown and must be estimated from sample data.

LOS: 7.a | Contrast: Statistic

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Statistic

A descriptive measure computed from a sample (e.g., sample mean $\bar{X}$, sample variance $s^2$). Used as an estimate of the corresponding population Parameter.

LOS: 7.a | Related: Point Estimate, Unbiased Estimator

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Sampling Error

The difference between the value of a statistic computed from a sample and the true value of the population parameter.

$\text{Sampling error}=\bar{X}-\mu$

LOS: 7.b | Note: Sampling error does not imply a mistake; it is the natural variation due to working with a subset of the population.

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Simple Random Sampling

A sampling method in which every member of the population has an equal probability of being selected, and each sample of size $n$ is equally likely to be chosen.

LOS: 7.c | Related: Probability Sampling

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Systematic Sampling

A sampling method in which every $k$-th member of the population is selected after a random starting point. Efficient for large populations with a natural ordering.

LOS: 7.c | Risk: Can introduce bias if the population has a periodic pattern matching the sampling interval.

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Stratified Random Sampling

A sampling method in which the population is divided into subgroups (strata) based on a characteristic, and random samples are drawn from each stratum in proportion to its size.

LOS: 7.c | Advantage: Ensures representation of all subgroups; reduces variance. | Application: Index construction, where bond/stock subgroups must be represented.

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Cluster Sampling

A sampling method in which the population is divided into clusters, a random selection of clusters is chosen, and all members of selected clusters are included.

LOS: 7.c | Contrast: Stratified Random Sampling samples from each stratum; cluster sampling selects entire clusters.

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Convenience Sampling

A non-probability sampling method in which the sample consists of members of the population that are easy to obtain. Subject to selection bias.

LOS: 7.c | Risk: May not be representative of the population. | Related: Non-Probability Sampling

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Judgmental Sampling

A non-probability sampling method in which the analyst uses personal judgment to select which population members to include in the sample.

LOS: 7.c | Limitation: Highly subjective and potentially biased. | Related: Non-Probability Sampling

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Probability Sampling

Any sampling method in which each member of the population has a known, non-zero probability of being selected. Includes Simple Random Sampling, Systematic Sampling, Stratified Random Sampling, and Cluster Sampling.

LOS: 7.c | Contrast: Non-Probability Sampling

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Non-Probability Sampling

Sampling methods where some members of the population have no chance of being selected, or the probability of selection is unknown. Includes Convenience Sampling and Judgmental Sampling.

LOS: 7.c | Risk: Results may not be generalizable to the population.

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Sampling Distribution

The probability distribution of a statistic (e.g., sample mean $\bar{X}$) computed from all possible samples of a given size $n$ drawn from a population.

LOS: 7.d | Key concept: The Central Limit Theorem (CLT) describes the sampling distribution of the mean.

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Central Limit Theorem (CLT)

States that for a population with mean $\mu$ and variance ${\sigma }^2$, the sampling distribution of the sample mean $\bar{X}$ approaches a normal distribution as $n$ increases, regardless of the population’s distribution.

$\bar{X}\sim N\left(\mu ,\frac{{\sigma }^2}{n}\right)\text{approximately, for large }n\text{ (typically }n\ge 30\text{)}$

LOS: 7.d | Importance: Justifies the use of normal-based inference even when the population is non-normal. | Related: Standard Error, Confidence Interval

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Standard Error

The standard deviation of the sampling distribution of a statistic. For the sample mean:

${\sigma }_{\bar{X}}=\frac{\sigma }{\sqrt{n}}{s}_{\bar{X}}=\frac{s}{\sqrt{n}}\text{ (when }\sigma \text{ unknown)}$

LOS: 7.d | Key: Standard error decreases as sample size $n$ increases — larger samples produce more precise estimates.

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Point Estimate

A single value used to estimate an unknown population parameter. For example, $\bar{X}$ as an estimate of $\mu$.

LOS: 7.e | Limitation: Does not communicate the uncertainty of the estimate. | Contrast: Confidence Interval

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Confidence Interval

A range of values constructed such that the probability is $(1-\alpha )$ that the interval contains the true population parameter. Formula for the population mean:

$\bar{X}\pm z_{\alpha /2}\frac{\sigma }{\sqrt{n}}\text{(when }\sigma \text{ known)}$

$\bar{X}\pm t_{\alpha /2,n-1}\frac{s}{\sqrt{n}}\text{(when }\sigma \text{ unknown)}$

LOS: 7.e | Key: A 95% CI does NOT mean there is a 95% probability the true mean falls in THIS interval; rather, 95% of such intervals constructed over many samples will contain the true mean.

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Level of Significance

Denoted $\alpha$. The probability of making a Type I error — rejecting a true null hypothesis. The complement of the degree of confidence.

$\alpha =1-\text{Degree of Confidence}$

LOS: 7.e | Common values: $\alpha =0.10$, $0.05$, $0.01$. | Related: Hypothesis Testing

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Degree of Confidence

The probability $(1-\alpha )$ that the confidence interval contains the true parameter value. Also called the confidence level.

LOS: 7.e | Common values: 90%, 95%, 99%.

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Reliability Factor

The critical value ($z^{\ast }$ or $t^{\ast }$) used to construct a confidence interval. Determines the interval’s width.

LOS: 7.e | Common reliability factors: $z_{0.025}=1.96$ for 95% CI; $z_{0.005}=2.576$ for 99% CI.

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Unbiased Estimator

An estimator whose expected value equals the population parameter it estimates. No systematic over- or under-estimation.

$E(\hat{\theta })=\theta$

LOS: 7.f | Example: Sample mean $\bar{X}$ is unbiased for $\mu$; sample variance $s^2$ (with $n-1$ denominator) is unbiased for ${\sigma }^2$.

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Efficient Estimator

Among unbiased estimators, the one with the smallest variance. The most precise unbiased estimator.

LOS: 7.f | Related: Unbiased Estimator, Consistent Estimator

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Consistent Estimator

An estimator that converges to the true population parameter as sample size increases — i.e., becomes more accurate with larger samples.

LOS: 7.f | Key: A consistent estimator may be biased in small samples but converges to the true value as $n\to \infty$.

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Data Snooping Bias

Bias that arises from repeatedly testing the same data set until a pattern or significant result is found. The discovered pattern may be spurious and not generalize to new data.

LOS: 7.g | In practice: Mining historical data to find investment strategies may identify patterns that worked in the past purely by chance.

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Sample Selection Bias

Bias that arises when some members of the population are systematically excluded from the sample, making the sample unrepresentative.

LOS: 7.g | Related: Survivorship Bias

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Survivorship Bias

A form of Sample Selection Bias in which only surviving (successful) entities are included in a sample — failed funds or firms that no longer exist are excluded.

LOS: 7.g | Effect: Overstates historical performance; understates risk. | Example: Mutual fund databases that exclude defunct funds show artificially high average returns.

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Look-Ahead Bias

Bias introduced when a study uses information that was not available at the time the investment decision would have been made. Overstates historical performance.

LOS: 7.g | Example: Using full-year earnings (reported in February) to make investment decisions retroactively dated to January of that year.

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Time Period Bias

Bias arising from selecting a sample period that is too short, too long, or not representative of all market conditions. Results may not generalize to other periods.

LOS: 7.g | Example: A strategy back-tested only during a bull market may appear superior but perform poorly in bear markets.

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Backfill Bias

Bias that arises when a database is filled with the historical returns of funds only after those funds are added to the database. Funds with poor early histories are less likely to be added — inflating reported performance.

LOS: 7.g | Also called: Instant history bias. | Related: Survivorship Bias, Sample Selection Bias