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General Linear Models
General Linear Models
.
Explore 26 formal proofs and analytical renders within the discipline of General Linear Models .
Advanced
The Matrix Formulation of the General Linear Model: Y = Xβ + ϵ and its Fundamental Assumptions
Master the matrix formulation of the General Linear Model,
Y
=
X
β
+
ϵ
Y = X\beta + \epsilon
Y
=
X
β
+
ϵ
, and its fundamental assumptions. Rigorous yet intuitive content for BSc Math/Stats students.
Study Proof →
Foundational
Derivation of the Ordinary Least Squares (OLS) Estimator: β̂ = (X'X)⁻¹X'Y
Master the OLS estimator derivation:
β
^
=
(
X
′
X
)
−
1
X
′
Y
\hat{\beta} = (X'X)^{-1}X'Y
β
^
=
(
X
′
X
)
−
1
X
′
Y
. Explore the geometric orthogonality, matrix calculus, and Gauss-Markov foundations.
Study Proof →
Foundational
Proof of Unbiasedness of the OLS Estimator: E(β̂) = β
Master the rigorous proof of OLS estimator unbiasedness,
E
(
β
^
)
=
β
E(\hat{\boldsymbol{\beta}}) = \boldsymbol{\beta}
E
(
β
^
)
=
β
. Understand critical assumptions, geometric intuition, and common pitfalls for robust linear modeling.
Study Proof →
Foundational
Derivation of the Variance-Covariance Matrix of the OLS Estimator: Var(β̂) = σ²(X'X)⁻¹
A rigorous derivation of the Variance-Covariance matrix for the OLS estimator, exploring the geometric impact of data configuration on statistical precision.
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Foundational
The Gauss-Markov Theorem: Proof that OLS is the Best Linear Unbiased Estimator (BLUE)
Master the Gauss-Markov Theorem: Understand why OLS is the Best Linear Unbiased Estimator (BLUE) under key assumptions for robust statistical inference.
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Foundational
Properties and Derivation of the Hat Matrix (H): Symmetry, Idempotence, and its Role in Leverage
Explore the Hat Matrix (H) in GLMs: derivation, symmetry, and idempotence. Understand its role in leverage and impact on OLS fitted values for BSc students.
Study Proof →
Foundational
Derivation and Properties of OLS Residuals: e = (I-H)Y, including E(e) = 0 and Var(e) = σ²(I-H)
Explore the derivation and properties of OLS residuals:
e
=
(
I
−
H
)
Y
e = (I-H)Y
e
=
(
I
−
H
)
Y
,
E
(
e
)
=
0
E(e) = 0
E
(
e
)
=
0
, and
V
a
r
(
e
)
=
s
i
g
m
a
2
(
I
−
H
)
Var(e) = \\sigma^2(I-H)
V
a
r
(
e
)
=
s
i
g
m
a
2
(
I
−
H
)
. Understand their geometric meaning and statistical implications for model diagnostics.
Study Proof →
Foundational
Geometric Interpretation of OLS: Projection onto the Column Space of X
Master the geometric interpretation of OLS as an orthogonal projection of the response vector onto the column space of the design matrix.
Study Proof →
Foundational
Decomposition of Total Sum of Squares: SST = SSR + SSE and its Implications for R²
Unpack the core decomposition of total sum of squares (SST=SSR+SSE) in linear regression, its geometric intuition, and the implications for R² for BSc Math/Stats students.
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Foundational
Derivation of the F-statistic for Overall Model Significance and its Distribution
Derive the F-statistic for overall model significance in General Linear Models, understanding its distribution, geometric intuition, and practical implications for BSc students.
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Foundational
Proof of the Distribution of Sums of Squares under Normality Assumptions (Chi-squared and F distributions)
Master the rigorous proof of Chi-squared and F distributions for sums of squares under normality assumptions, crucial for General Linear Models and statistical inference.
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Foundational
The t-statistic for Individual Regression Coefficients: Derivation and its Distribution
Master the derivation and distribution of the t-statistic in GLMs. Explore the geometry, the role of variance estimation, and its t-distribution convergence.
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Advanced
Theoretical Basis of Influence Diagnostics: Cook's Distance, DFFITS, and DFBETAS
Master influence diagnostics: Cook's Distance, DFFITS, and DFBETAS. Learn the geometric and theoretical basis for detecting influential data in linear models.
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Foundational
Model Selection Criteria: Derivation and Comparison of AIC and BIC
Master the rigorous derivation and institutional intuition behind AIC and BIC. Learn how to navigate the bias-variance trade-off in General Linear Models.
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Foundational
Lagrange Multiplier Approach for Testing Linear Restrictions on Regression Coefficients
Explore the Lagrange Multiplier test for linear restrictions in regression. Master the geometry of constraints and the formal theory of the score statistic.
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Foundational
Heteroscedasticity-Consistent (White) Standard Errors: Derivation and Rationale for Robust Inference
Master the derivation and logic of White's Sandwich Estimator to ensure robust statistical inference in the presence of heteroscedasticity in linear models.
Study Proof →
Foundational
The Principle of Maximum Likelihood Estimation (MLE) in GLM for Normally Distributed Errors
Master Maximum Likelihood Estimation for GLMs with normally distributed errors. Explore the intersection of Gaussian geometry and statistical inference.
Study Proof →
Foundational
Introduction to Time Series: Definitions of Stationarity and Autocorrelation Functions (ACF/PACF)
Master time series fundamentals: stationarity definitions, ACF/PACF intuition, and their crucial role in statistical modeling.
Study Proof →
Foundational
The Autoregressive (AR) Model: Definition, Properties, and Conditions for Stationarity
Master the AR(p) model with our rigorous guide on stationarity conditions, characteristic polynomials, and the underlying stochastic mechanics for math students.
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Foundational
The Box-Jenkins Methodology (ARIMA): Theoretical Steps of Identification, Estimation, and Diagnostic Checking
Master the Box-Jenkins ARIMA methodology with rigorous theoretical foundations in identification, MLE estimation, and diagnostic residual analysis.
Study Proof →
Foundational
Derivation of the Coefficient of Determination (R²): Interpretation and Relationship to Correlation
Master the derivation and interpretation of the Coefficient of Determination (R²) in GLM. Explore variance decomposition, geometric orthogonality, and pitfalls.
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Foundational
Construction of Confidence Intervals for Regression Coefficients and Predictions
Master the rigorous construction of confidence and prediction intervals in GLMs. Understand the geometric mechanics of uncertainty and variance propagation.
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Foundational
Mathematical Consequences of Perfect Multicollinearity on OLS Estimation
Explore the mathematical mechanics of perfect multicollinearity in OLS estimation. Understand why rank deficiency leads to non-invertibility and model failure.
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Foundational
The Theoretical Basis and Derivation of the Variance Inflation Factor (VIF)
Master the derivation and theoretical underpinnings of the Variance Inflation Factor (VIF). Understand multicollinearity's impact on coefficient stability.
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Foundational
The Box-Cox Transformation: Theoretical Background for Power Transformations to Achieve Model Assumptions
Master the Box-Cox transformation for General Linear Models. Learn the theoretical power transformation framework to stabilize variance and ensure normality.
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Advanced
Formulation of One-Way ANOVA as a General Linear Model using Dummy Variables
Master the formulation of One-Way ANOVA as a General Linear Model using dummy variables, covering design matrices, geometric projections, and rank constraints.
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