Outline for a Statistics 104 course, along with worked-out examples for each topic. This course typically builds upon the fundamentals covered in Statistics 100 and introduces more advanced topics in statistical analysis and inference.
Statistics 104 Course Outline
1. Advanced Probability Theory
- Probability Distributions and Densities: Understanding various distributions and their properties.
- Moment Generating Functions: Tools for finding moments of a distribution.
- Law of Large Numbers: Theoretical foundation for convergence of sample averages.
- Central Limit Theorem: Further applications and implications.
2. Estimation Theory
- Point Estimators: Properties of estimators (unbiasedness, efficiency, consistency).
- Interval Estimators: Constructing confidence intervals.
- Maximum Likelihood Estimation (MLE): Method for estimating parameters.
- Bayesian Estimation: Incorporating prior information into estimation.
3. Hypothesis Testing
- Neyman-Pearson Lemma: Framework for hypothesis testing.
- Likelihood Ratio Tests: Comparing the fit of models.
- Non-parametric Tests: Tests that do not assume a specific distribution.
4. Regression Analysis
- Multiple Linear Regression: Extending simple linear regression to multiple predictors.
- Logistic Regression: Modeling binary outcomes.
- Residual Analysis: Diagnosing and improving model fit.
- Model Selection: Criteria and methods for choosing the best model (AIC, BIC).
5. Analysis of Variance (ANOVA)
- One-way and Two-way ANOVA: Testing for differences across groups.
- MANOVA: Multivariate analysis of variance.
- Repeated Measures ANOVA: Analysis of data collected over time.
6. Time Series Analysis
- Components of Time Series: Trend, seasonality, cyclic patterns.
- ARIMA Models: AutoRegressive Integrated Moving Average models for forecasting.
- Exponential Smoothing: Techniques for smoothing time series data.
- Seasonal Decomposition: Decomposing time series data into its components.
7. Multivariate Analysis
- Principal Component Analysis (PCA): Reducing dimensionality of data.
- Factor Analysis: Identifying underlying relationships between variables.
- Cluster Analysis: Grouping similar observations together.
- Discriminant Analysis: Classifying observations into predefined classes.
8. Non-parametric Statistics
- Bootstrap Methods: Resampling techniques for estimating distributions.
- Jackknife Estimation: Technique for reducing bias and estimating variance.
- Kruskal-Wallis Test: Non-parametric version of ANOVA.
- Spearman’s Rank Correlation: Measure of rank correlation.
9. Advanced Statistical Software Usage
- Advanced R Programming: Writing efficient R code for complex analyses.
- Python for Statistical Analysis: Using Python libraries for statistical modeling.
- Machine Learning Integration: Applying machine learning algorithms to statistical problems.
- Data Visualization Techniques: Advanced methods for visualizing complex data.
Worked-out Examples
1. Advanced Probability Theory
Example: Calculating the moment generating function (MGF) for a normal distribution.
The MGF for a normal random variable ( X ) with mean ( ) and variance ( ^2 ) is given by: [ M_X(t) = (t + ^2 t^2 ) ]
2. Estimation Theory
Example: Maximum Likelihood Estimation for a normal distribution.
Given a sample ( x_1, x_2, …, x_n ) from a normal distribution with unknown mean ( ) and variance ( ^2 ), the likelihood function is: [ L(, ^2) = _{i=1}^n (-) ]
Taking the log of the likelihood function: [ (, ^2) = - (2) - (^2) - _{i=1}^n (x_i - )^2 ]
Maximizing the log-likelihood function with respect to ( ) and ( ^2 ): [ = {i=1}^n x_i ] [ = {i=1}^n (x_i - )^2 ]
3. Hypothesis Testing
Example: Likelihood Ratio Test for comparing two nested models.
Given a full model ( L_1 ) and a reduced model ( L_0 ), the test statistic is: [ = ]
For large samples, ( -2 ) follows a chi-square distribution with degrees of freedom equal to the difference in the number of parameters between the two models.
4. Regression Analysis
Example: Multiple Linear Regression with three predictors.
Model: ( Y = _0 + _1 X_1 + _2 X_2 + _3 X_3 + )
Fitting the model to data and finding estimates for ( _0, _1, _2, _3 ), we obtain the regression equation: [ = + X_1 + X_2 + X_3 ]
5. Analysis of Variance (ANOVA)
Example: One-way ANOVA to test for differences in means across three groups.
Suppose we have three groups with sample data:
- Group 1: ( X_1 = {5, 7, 8} )
- Group 2: ( X_2 = {6, 9, 10} )
- Group 3: ( X_3 = {4, 6, 7} )
Calculate the group means, overall mean, and sum of squares: [ = {i=1}^k n_i ({X}i - {X})^2 ] [ = {i=1}^k {j=1}^{n_i} (X_{ij} - {X}_i)^2 ] [ = + ]
F-statistic: [ F = ]
Where: [ = ] [ = ]
6. Time Series Analysis
Example: Fitting an ARIMA(1,1,1) model to time series data.
Model: ( (1 - _1 B)(1 - B)Y_t = (1 + _1 B) _t )
Estimate parameters ( _1 ) and ( _1 ) using maximum likelihood or other estimation techniques.
7. Multivariate Analysis
Example: Principal Component Analysis (PCA) on a dataset with three variables.
Standardize the data and calculate the covariance matrix. Find the eigenvalues and eigenvectors of the covariance matrix. The principal components are given by the eigenvectors.
8. Non-parametric Statistics
Example: Bootstrap method for estimating the standard error of the mean.
Resample the dataset with replacement many times, calculate the mean for each resample, and then estimate the standard error from the distribution of the resampled means.
9. Advanced Statistical Software Usage
Example: Using Python to fit a logistic regression model.
import pandas as pd
from sklearn.linear_model import LogisticRegression
# Load data
data = pd.read_csv('data.csv')
X = data[['predictor1', 'predictor2', 'predictor3']]
y = data['outcome']
# Fit model
model = LogisticRegression()
model.fit(X, y)
# Model coefficients
print(model.coef_)
print(model.intercept_)This outline and examples provide a comprehensive foundation in advanced statistical analysis and inference, preparing students for practical applications in various fields.
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