Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Probability and Statistics ↓


Statistical Inference


Statistical inference is a method of making judgements or predictions about a population based on a sample of data taken from that population. It is a fundamental aspect of statistics and deals with drawing conclusions about the characteristics or parameters of a larger group by examining a smaller subgroup. The process involves hypothesis testing, estimation, and calculation of confidence intervals.

Key concepts of statistical inference

To understand statistical inference, it is important to first understand some basic concepts:

Population and sample

The population includes all the data points or items we are interested in studying, while the sample is a subset of the population that we actually observe and analyze. For example, if a car manufacturer wants to test the average fuel efficiency of a new model, the population would include all the units produced, and the sample could be 100 cars tested for fuel efficiency.

Parameters and statistics

A parameter is a measure that describes a characteristic of a population, such as the mean or standard deviation. In contrast, a statistic is a measure that describes a characteristic of a sample. For example, if the average height of a sample of 100 random people is 5'7", that average is a statistic.

Sample distribution

The sampling distribution is the distribution of a given statistic based on a random sample. It is an important concept because it allows us to understand how a statistic may vary from sample to sample, helping us make inferences about a population parameter.

0 population distribution

This graph shows the population distribution with random sample data points indicated in red circles.

Procedures in statistical inference

Statistical inference typically involves several procedures:

Point estimation

Point estimation involves the use of sample data to calculate a single value (known as a point estimate) that serves as a "best guess" or estimate of an unknown population parameter. Common point estimators are the sample mean, sample variance, and sample proportion.

For example, if we want to estimate the average height of all adult men in a city, we can use the average height of a sample of 100 adult men in that city. If the average height of the sample is 70 inches, our point estimate for the population average is also 70 inches.

Interval estimation

Unlike point estimation, interval estimation provides a range of values (an interval) and an associated confidence level that the parameter lies within this interval. This is known as the confidence interval.

[ text{confidence interval} = left( bar{x} - Z cdot frac{sigma}{sqrt{n}}, bar{x} + Z cdot frac{sigma}{sqrt{n}} right) ]

Here, ( bar{x} ) is the sample mean, ( Z ) is the Z-score from the standard normal distribution based on the desired confidence level, ( sigma ) is the population standard deviation, and ( n ) is the sample size.

Hypothesis testing

Hypothesis testing is a method of making decisions using data, whether from a controlled experiment or an observational study. A hypothesis is an assumption or statement about a population parameter. Hypothesis testing defines the framework for deciding whether to reject or accept these assumptions.

H_0: mu = mu_0 \
H_a: mu neq mu_0

Here, ( H_0 ) represents the null hypothesis, which states no effect or no difference, and ( H_a ) represents the alternative hypothesis, which states some effect or difference.

This process involves determining the p-value, which is the probability of obtaining test results at least as extreme as the observed results under the assumption that the null hypothesis is true.

Common methods used in statistical inference

Several methods are used in statistical inference to draw conclusions from data:

Bayesian inference

Bayesian inference involves updating the probability of a hypothesis as more evidence or information becomes available. It relies heavily on Bayes' theorem:

[ P(H|E) = frac{P(E|H) cdot P(H)}{P(E)} ]

where ( P(H|E) ) is the posterior probability, ( P(E|H) ) is the likelihood, ( P(H) ) is the prior probability, and ( P(E) ) is the marginal probability.

Frequentist estimation

Frequentist inference draws conclusions from sample data by emphasizing the frequency or proportion of the data. Frequentists design hypothesis tests and calculate confidence intervals without the use of prior probabilities.

Maximum likelihood estimation

Maximum likelihood estimation (MLE) is used to estimate the parameters of a statistical model. The method of MLE involves finding the values of the parameters that maximize the probability of the occurrence of the observed data.

If we have a sample data set and a statistical model, the likelihood function measures how well the model explains the observed data. It is expressed as:

L(theta | x) = prod_{i=1}^{n} f(x_i | theta)

where ( theta ) is a parameter, ( X ) is the data, and ( f(x_i | theta) ) is the probability of observing a data point ( x_i ) given ( theta ) .

Examples of statistical inference

Let us look at some examples to understand these concepts better:

Example 1: Estimating the average height

Suppose we want to determine the average height of all students in a university. Instead of measuring each student, we decide to take a sample of 100 students.

Sample data: [68, 70, 65, 72, 69, 71, 66, 73, 67, 70, ...] // continues for 100 entries

The average (mean) of this sample provides a point estimate for the average height of the population. Calculating the sample mean will allow us to draw a conclusion:

Sample mean = (68 + 70 + 65 + 72 + 69 + 71 + 66 + 73 + 67 + 70 + ...) / 100 = 69.5 inches

Thus, we estimate that the average height of all university students will be around 69.5 inches.

Example 2: Hypothesis testing for drug effectiveness

A pharmaceutical company believes that their new drug lowers blood pressure. To test this, they conducted a trial on 200 patients, half of whom were given the drug and the other half a placebo. The company hypothesized that:

H_0: Delta = 0 ,(text{The drug has no effect}) \
H_a: Delta neq 0 ,(text{the drug has an effect})

Based on the test data, the company calculates a p-value to determine the probability of observing results as extreme as the recorded results, assuming the null hypothesis is true. A common threshold p-value is 0.05:

If the p-value is < 0.05, reject ( H_0 ); otherwise, do not reject ( H_0 ).

When the p-value is less than 0.05, the company can conclude that the drug is effective in lowering blood pressure.

Conclusion

Statistical inference is instrumental in research and data analysis, bridging the gap between descriptive statistics and the real world. It provides tools and methods that allow us to make informed conclusions and predictions about populations using sample data. Mastering statistical inference techniques is crucial for data scientists, researchers, economists, and many other professionals who rely on data-driven decision making.


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Statistical Inference

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