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In statistics, statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation.^{[1]} More substantially, the terms statistical inference, statistical induction and inferential statistics are used to describe systems of procedures that can be used to draw conclusions from datasets arising from systems affected by random variation,^{[2]} such as observational errors, random sampling, or random experimentation.^{[1]} Initial requirements of such a system of procedures for inference and induction are that the system should produce reasonable answers when applied to welldefined situations and that it should be general enough to be applied across a range of situations. Inferential statistics are used to test hypotheses and make estimations using sample data.
The outcome of statistical inference may be an answer to the question "what should be done next?", where this might be a decision about making further experiments or surveys, or about drawing a conclusion before implementing some organizational or governmental policy.
For the most part, statistical inference makes propositions about populations, using data drawn from the population of interest via some form of random sampling. More generally, data about a random process is obtained from its observed behavior during a finite period of time. Given a parameter or hypothesis about which one wishes to make inference, statistical inference most often uses:
The conclusion of a statistical inference is a statistical proposition.^{[citation needed]} Some common forms of statistical proposition are:
Statistical inference is generally distinguished from descriptive statistics. In simple terms, descriptive statistics can be thought of as being just a straightforward presentation of facts, in which modeling decisions made by a data analyst have had minimal influence.
Any statistical inference requires some assumptions. A statistical model is a set of assumptions concerning the generation of the observed data and similar data. Descriptions of statistical models usually emphasize the role of population quantities of interest, about which we wish to draw inference.^{[4]} Descriptive statistics are typically used as a preliminary step before more formal inferences are drawn.^{[5]}
Statisticians distinguish between three levels of modeling assumptions;
Whatever level of assumption is made, correctly calibrated inference in general requires these assumptions to be correct; i.e., that the datagenerating mechanisms really have been correctly specified.
Incorrect assumptions of 'simple' random sampling can invalidate statistical inference.^{[7]} More complex semi and fully parametric assumptions are also cause for concern. For example, incorrectly assuming the Cox model can in some cases lead to faulty conclusions.^{[8]} Incorrect assumptions of Normality in the population also invalidates some forms of regressionbased inference.^{[9]} The use of any parametric model is viewed skeptically by most experts in sampling human populations: "most sampling statisticians, when they deal with confidence intervals at all, limit themselves to statements about [estimators] based on very large samples, where the central limit theorem ensures that these [estimators] will have distributions that are nearly normal."^{[10]} In particular, a normal distribution "would be a totally unrealistic and catastrophically unwise assumption to make if we were dealing with any kind of economic population."^{[10]} Here, the central limit theorem states that the distribution of the sample mean "for very large samples" is approximately normally distributed, if the distribution is not heavy tailed.
Given the difficulty in specifying exact distributions of sample statistics, many methods have been developed for approximating these.
With finite samples, approximation results measure how close a limiting distribution approaches the statistic's sample distribution: For example, with 10,000 independent samples the normal distribution approximates (to two digits of accuracy) the distribution of the sample mean for many population distributions, by the Berry–Esseen theorem.^{[11]} Yet for many practical purposes, the normal approximation provides a good approximation to the samplemean's distribution when there are 10 (or more) independent samples, according to simulation studies and statisticians' experience.^{[11]} Following Kolmogorov's work in the 1950s, advanced statistics uses approximation theory and functional analysis to quantify the error of approximation. In this approach, the metric geometry of probability distributions is studied; this approach quantifies approximation error with, for example, the Kullback–Leibler distance, Bregman divergence, and the Hellinger distance.^{[12]}^{[13]}^{[14]}
With indefinitely large samples, limiting results like the central limit theorem describe the sample statistic's limiting distribution, if one exists. Limiting results are not statements about finite samples, and indeed are irrelevant to finite samples.^{[15]}^{[16]}^{[17]} However, the asymptotic theory of limiting distributions is often invoked for work with finite samples. For example, limiting results are often invoked to justify the generalized method of moments and the use of generalized estimating equations, which are popular in econometrics and biostatistics. The magnitude of the difference between the limiting distribution and the true distribution (formally, the 'error' of the approximation) can be assessed using simulation.^{[18]} The heuristic application of limiting results to finite samples is common practice in many applications, especially with lowdimensional models with logconcave likelihoods (such as with oneparameter exponential families).
For a given dataset that was produced by a randomization design, the randomization distribution of a statistic (under the nullhypothesis) is defined by evaluating the test statistic for all of the plans that could have been generated by the randomization design. In frequentist inference, randomization allows inferences to be based on the randomization distribution rather than a subjective model, and this is important especially in survey sampling and design of experiments.^{[19]}^{[20]} Statistical inference from randomized studies is also more straightforward than many other situations.^{[21]}^{[22]}^{[23]} In Bayesian inference, randomization is also of importance: in survey sampling, use of sampling without replacement ensures the exchangeability of the sample with the population; in randomized experiments, randomization warrants a missing at random assumption for covariate information.^{[24]}
Objective randomization allows properly inductive procedures.^{[25]}^{[26]}^{[27]}^{[28]} Many statisticians prefer randomizationbased analysis of data that was generated by welldefined randomization procedures.^{[29]} (However, it is true that in fields of science with developed theoretical knowledge and experimental control, randomized experiments may increase the costs of experimentation without improving the quality of inferences.^{[30]}^{[31]}) Similarly, results from randomized experiments are recommended by leading statistical authorities as allowing inferences with greater reliability than do observational studies of the same phenomena.^{[32]} However, a good observational study may be better than a bad randomized experiment.
The statistical analysis of a randomized experiment may be based on the randomization scheme stated in the experimental protocol and does not need a subjective model.^{[33]}^{[34]}
However, at any time, some hypotheses cannot be tested using objective statistical models, which accurately describe randomized experiments or random samples. In some cases, such randomized studies are uneconomical or unethical.
It is standard practice to refer to a statistical model, often a linear model, when analyzing data from randomized experiments. However, the randomization scheme guides the choice of a statistical model. It is not possible to choose an appropriate model without knowing the randomization scheme.^{[20]} Seriously misleading results can be obtained analyzing data from randomized experiments while ignoring the experimental protocol; common mistakes include forgetting the blocking used in an experiment and confusing repeated measurements on the same experimental unit with independent replicates of the treatment applied to different experimental units.^{[35]}
Different schools of statistical inference have become established. These schools (or 'paradigms') are not mutually exclusive, and methods which work well under one paradigm often have attractive interpretations under other paradigms. The two main paradigms in use are frequentist and Bayesian inference, which are both summarized below.
This paradigm calibrates the production of propositions^{[clarification needed (complicated jargon)]} by considering (notional) repeated sampling of datasets similar to the one at hand. By considering its characteristics under repeated sample, the frequentist properties of any statistical inference procedure can be described — although in practice this quantification may be challenging.
One interpretation of frequentist inference (or classical inference) is that it is applicable only in terms of frequency probability; that is, in terms of repeated sampling from a population. However, the approach of Neyman^{[36]} develops these procedures in terms of preexperiment probabilities. That is, before undertaking an experiment, one decides on a rule for coming to a conclusion such that the probability of being correct is controlled in a suitable way: such a probability need not have a frequentist or repeated sampling interpretation. In contrast, Bayesian inference works in terms of conditional probabilities (i.e., probabilities conditional on the observed data), compared to the marginal (but conditioned on unknown parameters) probabilities used in the frequentist approach.
The frequentist procedures of significance testing and confidence intervals can be constructed without regard to utility functions. However, some elements of frequentist statistics, such as statistical decision theory, do incorporate utility functions.^{[citation needed]} In particular, frequentist developments of optimal inference (such as minimumvariance unbiased estimators, or uniformly most powerful testing) make use of loss functions, which play the role of (negative) utility functions. Loss functions need not be explicitly stated for statistical theorists to prove that a statistical procedure has an optimality property.^{[37]} However, lossfunctions are often useful for stating optimality properties: for example, medianunbiased estimators are optimal under absolute value loss functions, in that they minimize expected loss, and least squares estimators are optimal under squared error loss functions, in that they minimize expected loss.
While statisticians using frequentist inference must choose for themselves the parameters of interest, and the estimators/test statistic to be used, the absence of obviously explicit utilities and prior distributions has helped frequentist procedures to become widely viewed as 'objective'.^{[citation needed]}
The Bayesian calculus describes degrees of belief using the 'language' of probability; beliefs are positive, integrate to one, and obey probability axioms. Bayesian inference uses the available posterior beliefs as the basis for making statistical propositions. There are several different justifications for using the Bayesian approach.
Many informal Bayesian inferences are based on "intuitively reasonable" summaries of the posterior. For example, the posterior mean, median and mode, highest posterior density intervals, and Bayes Factors can all be motivated in this way. While a user's utility function need not be stated for this sort of inference, these summaries do all depend (to some extent) on stated prior beliefs, and are generally viewed as subjective conclusions. (Methods of prior construction which do not require external input have been proposed but not yet fully developed.)
Formally, Bayesian inference is calibrated with reference to an explicitly stated utility, or loss function; the 'Bayes rule' is the one which maximizes expected utility, averaged over the posterior uncertainty. Formal Bayesian inference therefore automatically provides optimal decisions in a decision theoretic sense. Given assumptions, data and utility, Bayesian inference can be made for essentially any problem, although not every statistical inference need have a Bayesian interpretation. Analyses which are not formally Bayesian can be (logically) incoherent; a feature of Bayesian procedures which use proper priors (i.e., those integrable to one) is that they are guaranteed to be coherent. Some advocates of Bayesian inference assert that inference must take place in this decisiontheoretic framework, and that Bayesian inference should not conclude with the evaluation and summarization of posterior beliefs.
Other forms of statistical inference have been developed from ideas in information theory^{[38]} and the theory of Kolmogorov complexity.^{[39]} For example, the minimum description length (MDL) principle selects statistical models that maximally compress the data; inference proceeds without assuming counterfactual or nonfalsifiable 'datagenerating mechanisms' or probability models for the data, as might be done in frequentist or Bayesian approaches.
However, if a 'data generating mechanism' does exist in reality, then according to Shannon's source coding theorem it provides the MDL description of the data, on average and asymptotically.^{[40]} In minimizing description length (or descriptive complexity), MDL estimation is similar to maximum likelihood estimation and maximum a posteriori estimation (using maximumentropy Bayesian priors). However, MDL avoids assuming that the underlying probability model is known; the MDL principle can also be applied without assumptions that e.g. the data arose from independent sampling.^{[40]}^{[41]} The MDL principle has been applied in communicationcoding theory in information theory, in linear regression, and in timeseries analysis (particularly for choosing the degrees of the polynomials in Autoregressive moving average (ARMA) models).^{[41]}
Informationtheoretic statistical inference has been popular in data mining, which has become a common approach for very large observational and heterogeneous datasets made possible by the computer revolution and internet.^{[39]}
The evaluation of statistical inferential procedures often uses techniques or criteria from computational complexity theory or numerical analysis.^{[42]}^{[43]}
Fiducial inference was an approach to statistical inference based on fiducial probability, also known as a "fiducial distribution". In subsequent work, this approach has been called illdefined, extremely limited in applicability, and even fallacious.^{[44]}^{[45]} However this argument is the same as that which shows^{[46]} that a socalled confidence distribution is not a valid probability distribution and, since this has not invalidated the application of confidence intervals, it does not necessarily invalidate conclusions drawn from fiducial arguments.
Developing ideas of Fisher and of Pitman from 1938 to 1939,^{[47]} George A. Barnard developed "structural inference" or "pivotal inference",^{[48]} an approach using invariant probabilities on group families. Barnard reformulated the arguments behind fiducial inference on a restricted class of models on which "fiducial" procedures would be welldefined and useful.
The topics below are usually included in the area of statistical inference.
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