GATE ST key terms

A rule or function that maps observed sample data to a numerical guess about an unknown population parameter. It is a random variable before data are collected and takes a specific value once the sample is drawn.

A statistic that captures all information in the sample about a parameter, so the conditional distribution of the data given the statistic does not depend on that parameter. Verified through the factorization criterion.

A property of a family of distributions where the only unbiased estimator of zero based on the statistic is the zero function almost surely. Combined with sufficiency, it yields minimum-variance unbiased estimators via Lehmann-Scheffe.

The uniformly minimum variance unbiased estimator: among all unbiased estimators of a parameter, the one with the smallest variance for every possible parameter value. Often obtained as a function of a complete sufficient statistic.

A result stating that conditioning any unbiased estimator on a sufficient statistic produces a new unbiased estimator whose variance is no larger, and typically strictly smaller unless the original was already a function of the statistic.

A lower bound on the variance of any unbiased estimator, equal to the reciprocal of the Fisher information for a scalar parameter. Estimators attaining it are called efficient.

A measure of how much information a sample carries about an unknown parameter, defined as the variance of the score function or minus the expected second derivative of the log-likelihood. It grows with sample size.

The parameter value that maximizes the likelihood or log-likelihood of the observed sample. Under regularity conditions it is consistent, asymptotically normal, and asymptotically efficient.

An estimation technique that equates sample moments to theoretical moments expressed as functions of parameters, then solves the resulting equations. It is simple but often less efficient than maximum likelihood.

A property of an estimator whose expected value equals the true parameter for every possible parameter value. It concerns the average over repeated sampling, not the accuracy of any single estimate.

The expected squared difference between an estimator and the true parameter, decomposing into variance plus squared bias. It is a common criterion for comparing estimators, allowing biased estimators to outperform unbiased ones.

A property whereby an estimator converges in probability to the true parameter as sample size grows without bound. Sufficient conditions include vanishing bias and variance, or application of the continuous mapping theorem.

The property that a suitably standardized estimator converges in distribution to a normal law as sample size increases. It underpins construction of large-sample confidence intervals and Wald tests.

A technique for deriving the asymptotic distribution of a smooth function of an asymptotically normal estimator, using a first-order Taylor expansion and the gradient. It yields the limiting variance via the derivative.

A result allowing convergence in distribution to be combined with convergence in probability, so that sums and products of such sequences retain their limiting distributions. It justifies replacing unknown variances with consistent estimates.

An interval computed from data that covers the true parameter with a stated probability under repeated sampling. The coverage probability refers to the procedure, not to any single realized interval.

A function of the data and parameter whose distribution does not depend on the parameter. It enables exact interval construction by inverting probability statements about the pivot.

The probability that a confidence procedure produces an interval containing the true parameter, evaluated over repeated samples. Nominal coverage is the target; actual coverage may differ in small samples or under model misspecification.

A decision framework comparing a null hypothesis with an alternative using a test statistic and a rejection region. Errors of the first and second kind are controlled through significance level and power.

The probability, computed under the null hypothesis, of obtaining a test statistic at least as extreme as that observed. It is not the probability that the null hypothesis is true.

A function giving the probability of rejecting the null hypothesis for each possible value of the parameter. Its value at the null is the significance level, and one minus power at an alternative is the type II error rate.

A result characterizing the most powerful test of a simple null against a simple alternative as a likelihood ratio test, with the threshold set to achieve a given significance level.

A test whose power is at least that of every other test of the same size for every parameter value in the alternative. Existence is guaranteed in monotone likelihood ratio families for one-sided alternatives.

A test comparing the maximized likelihood under the null with that under the full model. Under regularity conditions, minus twice the log ratio converges to a chi-square distribution with degrees of freedom equal to the parameter restriction.

A property of a family of distributions where the ratio of densities for two parameter values is monotone in a sufficient statistic. It guarantees stochastic ordering and existence of uniformly most powerful one-sided tests.

A class of distributions whose density factors as an exponential of a linear combination of sufficient statistics. Members possess convenient sufficiency, completeness, and moment-generating properties used throughout estimation theory.

The derivative of the log-likelihood with respect to the parameter. Its expectation is zero at the true parameter, and its variance equals the Fisher information.

A relative measure of estimator quality, often the ratio of the Cramer-Rao lower bound to an estimator's variance. An efficient estimator attains the bound asymptotically or exactly.

The principle that reducing an estimator's bias often increases its variance and vice versa, so total mean squared error may be minimized by accepting some bias. Central to regularization and shrinkage methods.

An estimator minimizing posterior expected loss, typically the posterior mean under squared error loss. It depends on a prior distribution and is admissible under mild conditions.

A prior distribution that, when combined with a given likelihood, yields a posterior in the same family. It simplifies Bayesian updating to algebraic updating of hyperparameters.

A Bayesian interval containing the parameter with a stated posterior probability. Unlike a confidence interval, its interpretation is direct given the prior and model.

A decision-theoretic property of an estimator or test whose worst-case risk over the parameter space is as small as possible. Minimax rules are often derived as limits of Bayes rules.

A property of a decision rule that no other rule dominates it uniformly in risk across all parameter values. Inadmissible estimators can always be improved upon everywhere.