Hypothesis testing allows researchers to evaluate claims about a population using sample data. We begin with a null hypothesis, $H_0$, which represents the claim being tested, and an alternative hypothesis, $H_a$, which represents a competing claim. Because decisions are based on sample data, errors...
The chi-square ($\chi^2$) test is a family of statistical tests for categorical count data. These tests compare observed frequencies with the frequencies expected under a null hypothesis...
A confidence interval (CI) is a range of plausible values for a population parameter, such as a mean or proportion, calculated from sample data. It supplements a point estimate by showing the uncertainty around that estimate...
Hypothesis testing is a statistical tool used to draw conclusions about populations based on sample data. It is widely applied in scientific research, from evaluating new treatments in clinical trials to studying customer behavior in business analytics...
Statistical hypothesis testing is a method for using sample data to make inferences about a population. Understanding null and alternative hypotheses, along with how p-values are calculated and interpreted, is essential for applying hypothesis tests correctly...
A parameter is an unknown feature of a population or statistical model, such as a mean $\mu$, variance $\sigma^2$, probability $p$, or regression coefficient $\beta$. A point estimator is a rule or statistic computed from sample data and used to estimate that parameter...
Stationarity describes which probabilistic features of a time series remain stable when the time origin is shifted. Weak stationarity focuses on a constant mean and variance together with an autocovariance that depends only on lag, while strict stationarity requires the full joint distribution to be...
Autoregressive models describe a time series whose current value depends linearly on its own recent history plus a new innovation. The lag coefficients determine how strongly past observations persist into the present and whether the process tends to decay smoothly, alternate, or oscillate...
Frequency-domain analysis describes a time series by the cycle lengths that contribute to its variation rather than by relationships at individual lags. The periodogram and spectral density provide a complementary view to the ACF, translating repeated temporal structure into peaks at corresponding f...
A stochastic process is a collection of random variables indexed by time, and an observed time series is one realized path from that process. This viewpoint separates the data we actually see from the probabilistic mechanism used to describe possible paths and future uncertainty...
Dynamic regression combines external predictors with time-series structure so that the conditional mean can respond to explanatory variables without assuming the remaining errors are independent. Lagged predictors can represent delayed effects, while ARMA or ARIMA errors capture serial dependence le...
Multivariate time-series models describe several evolving variables jointly so that cross-lag feedback, shared shocks, and predictive relationships can be represented within one system. They are useful when separate univariate models would ignore information carried by the histories of related serie...
A random walk models a level that changes by accumulating new shocks over time. Each innovation is temporary as an increment but permanent in the level, so the process can drift far from its starting point even when the expected one-step change is zero...
A difference equation describes how a discrete-time quantity evolves from its earlier values. Rather than specifying every term directly, it gives a recursive rule, so the behavior of the sequence is determined by the recursion together with its initial conditions...
Financial time series often separate naturally into a price level, a return process, and a time-varying scale of uncertainty. Returns may show little predictable movement in their conditional mean while their magnitude clusters over time, producing periods of calm and periods of elevated volatility...
State-space models separate a time series into an unobserved state that evolves through time and an observation process that measures that state with noise. This framework is flexible enough to represent latent levels, trends, seasonal components, regression effects, missing observations, and many f...
Autocovariance measures how a time series varies jointly with lagged versions of itself. Lag 0 is the variance, while nonzero lags show the direction and scale of linear dependence between observations separated in time, expressed in the squared units of the original series...
The autocorrelation function measures the unit-free linear association between a time series and lagged versions of itself. The partial autocorrelation function asks a narrower question: how much direct linear relationship remains at a given lag after the intervening lags have been accounted for...
ARMA models describe stationary linear dependence through a combination of autoregressive terms, which use past observations, and moving-average terms, which use current and past innovations. ARIMA extends that structure by differencing a series with a stochastic trend, while SARIMA adds seasonal di...
Trend and seasonality are systematic forms of time structure that operate on different scales. A trend is a persistent long-run movement in level or slope, while seasonality is a pattern that repeats at a fixed and known period such as day of week, month of year, or quarter...
Regression with ARMA errors separates two kinds of structure that often appear together in time series. The regression explains how observed predictors shift the conditional mean, while the ARMA component models serial dependence that remains in the unexplained part...
A time-series forecast is a conditional statement about a future observation given the information available at a specific forecast origin. The horizon, target functional, and information set are part of the forecast itself, so a point prediction without its timing and uncertainty is an incomplete d...
A moving-average model represents the current observation as a finite weighted combination of the current innovation and a fixed number of past innovations. Because each shock enters the model for only a limited number of periods, an MA($q$) process has a finite direct shock response and a theoretic...
A time series is an ordered set of observations together with an information structure: timestamps, sampling rules, and sequence position determine what can depend on what and what information is available for prediction. Treating the same values as an unordered sample would discard much of the stru...
The Yule-Walker equations connect an autoregressive model's coefficients to its autocovariances and autocorrelations. They translate a recursion written in terms of lagged observations into a set of moment relationships that can be studied theoretically or estimated from data...
The backward shift operator is a compact notation for referring to earlier observations in a time series. Instead of writing each lag separately, it lets lagged values, differences, and model equations be expressed as polynomials in a single operator...
Sequences and series provide the convergence language behind many time-series representations. A sequence describes an ordered set of terms, while a series asks whether the cumulative effect of those terms approaches a finite limit...
Time-series modeling is an iterative process of specifying structure, estimating parameters, diagnosing what remains unexplained, and testing forecasts on future-like data. Choosing an equation is only one step; transformations, initial conditions, residual behavior, and the forecast information set...
Statistical moments summarize features such as the center, spread, and joint variation of a distribution. In time series, those same quantities acquire a time dimension: the mean and variance may change across the record, and covariance becomes a function of lag as well as scale...
Randomness tests look for specific kinds of structure that should not appear under a stated null model. In time-series work, they are most useful as diagnostics: one test may target linear autocorrelation, another unusual turning-point behavior, another monotone trend, and another dependence in squa...
Probability theory is built on a small set of principles, called axioms, that define how probability measures behave. These axioms, formalized by the Russian mathematician Andrey Kolmogorov, provide the foundation for the rules used throughout probability theory...
Geometric probability is a fascinating branch of probability theory where outcomes are associated with geometric figures and their measures—such as lengths, areas, and volumes—rather than discrete numerical outcomes. It often deals with continuous random variables and employs integral calculus to ca...
The law of total probability allows for the computation of the probability of an event A based on a set of mutually exclusive and exhaustive events. It's particularly useful when the overall sample space is divided into several distinct scenarios, or partitions, that cover all possible outcomes. The...
Conditional Probability is the likelihood of an event occurring given that another event has already occurred. It is denoted as $P(A|B)$, representing the probability of event $A$ happening, assuming event $B$ has already taken place. This concept is crucial in understanding dependent events in prob...
Probability theory offers a structured approach to assessing the probability of events, allowing for logical and systematic reasoning about their likelihood...