Earth Stats Week #3 Readings

A set of 35 vocabulary flashcards based on the statistical inference lecture transcript covering parametric/nonparametric statistics, hypothesis testing, t-tests, and F-tests.

Parametric Model

A family of distributions fully parameterized by a finite set of unknown parameters $\theta = (\theta_1, \dots, \theta_d)$.

Maximum Likelihood Estimation (MLE)

A frequentist method that selects the parameter value $\theta$ maximizing the probability density or mass function of the observed i.i.d. sample data.

Method of Moments (MoM)

An estimation approach where parameter estimators are created by substituting sample moments into functions $g_1, \dots, g_d$ derived from population moments.

Least Squares Estimation (LSE)

A regression estimation method that selects parameters to minimize the sum of squared errors $\sum_{i=1}^n (Y_i - f_\theta(X_i))^2$.

Maximum a Posteriori (MAP) Estimation

A Bayesian estimation method where the parameter estimate is chosen as the mode (peak) of the posterior distribution $p_\theta \propto L_\theta \cdot \pi$.

UMVUE

Uniformly Minimum-Variance Unbiased Estimator; an unbiased estimator of a parameter function $q(\theta)$ that achieves the lowest variance among all unbiased estimators.

Rao–Blackwell Theorem

A theorem establishing that if a UMVU estimator with finite variance exists, it is unique.

Lehmann–Scheffé Theorem

A theorem stating that if a complete sufficient statistic $T$ exists, any unbiased estimator of the form $f(T)$ is the unique UMVUE.

BLUE

Best Linear Unbiased Estimator; an estimator restricted to linear combinations of data $\hat{\theta} = \sum_{i=1}^n \alpha_i X_i$ that achieves minimum variance.

Gauss–Markov Theorem

A theorem stating that in a regression setting, the least-squares estimator is the unique Best Linear Unbiased Estimator (BLUE).

Cramér–Rao Bound

The lower bound on the variance of any unbiased estimator $T$ of $q(\theta)$, given by $\nabla q(\theta) I(\theta)^{-1} \nabla q(\theta)^T$.

Fisher Information Matrix

A matrix $I(\theta)$ used in regular statistical models to calculate the Cramér–Rao lower bound for estimator variance.

Null Hypothesis ($H_0$)

The default statistical hypothesis that typically represents a statement of 'no effect' or 'no difference' in population parameters.

Alternative Hypothesis ($H_1$)

The hypothesis that is mutually exclusive with the null hypothesis $H_0$ and represents its negation.

p-value

The conditional probability of observing a test statistic at least as extreme as the calculated value, assuming the null hypothesis $H_0$ is true.

Type I Error

The error committed when a true null hypothesis $H_0$ is incorrectly rejected, occurring with probability $\alpha$ (significance level).

Type II Error

The error committed when a false null hypothesis $H_0$ fails to be rejected, occurring with probability $\beta$.

Statistical Power

The probability of correctly rejecting a false null hypothesis $H_0$, mathematically calculated as $1 - \beta$.

Simple Hypothesis

A hypothesis that fully specifies the population distribution, causing the sampling distribution of any statistic to depend solely on sample size.

Composite Hypothesis

A hypothesis that does not fully specify the population distribution, such as specifying a normal distribution with a known mean but unknown variance.

Uniformly Most Powerful (UMP) Test

A test that achieves the greatest power across all possible alternative parameter values contained in $H_1$ for a given significance level.

Critical Region

The set of test-statistic values for which the null hypothesis $H_0$ is rejected (also known as the region of rejection).

Specificity

In biostatistics, the complement of the false-positive rate (size) of a hypothesis test.

Two-Tailed Test

A hypothesis test appropriate when an estimated statistic is considered significant in either direction (both greater than or less than a reference range).

One-Tailed Test

A hypothesis test appropriate when only a departure from $H_0$ in a single, specified direction is considered significant.

Student's t-Test

A hypothesis test whose statistic follows a Student's t-distribution under $H_0$, used when population standard deviation is unknown and estimated from data.

Paired Samples t-Test

A t-test conducted on matched pairs or repeated measures from the same subjects, possessing $n/2 - 1$ degrees of freedom for $n$ total observations.

William Sealy Gosset

The Guinness Brewery chemist who derived the t-statistic for small samples and published under the pseudonym 'Student' in 1908.

F-Test

A hypothesis test that compares ratios of variability components or sample variances using the F-distribution.

ANOVA

Analysis of Variance; a framework using an F-test to compare whether population means across several normal groups with equal variance are equal.

Homoscedasticity

The assumption of homogeneity of variance across groups, which is required for standard ANOVA and F-tests.

Distribution-Free Techniques

Nonparametric methods that do not rely on the assumption that data are drawn from a specific parametric family of distributions.

Flexible-Structure Techniques

Nonparametric techniques where model structure is not fixed in advance and can expand in size and complexity as needed by the data.

Order Statistics

Statistics defined purely as functions of the ordinal ranking of sample observations, serving as an example of a distribution-free technique.

Dirichlet Process

A stochastic process that governs the rate at which latent variables grow in nonparametric hierarchical Bayesian models.