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Mathematical statistics is the application ofprobability theory and other mathematical concepts tostatistics, as opposed to techniques for collecting statistical data.[1] Specific mathematical techniques that are commonly used in statistics includemathematical analysis,linear algebra,stochastic analysis,differential equations, andmeasure theory.[2][3]
Statistical data collection is concerned with the planning of studies, especially with thedesign of randomized experiments and with the planning ofsurveys usingrandom sampling. The initial analysis of the data often follows the study protocol specified prior to the study being conducted. The data from a study can also be analyzed to consider secondary hypotheses inspired by the initial results, or to suggest new studies. A secondary analysis of the data from a planned study uses tools fromdata analysis, and the process of doing this is mathematical statistics.
Data analysis is divided into:
While the tools of data analysis work best on data from randomized studies, they are also applied to other kinds of data. For example, fromnatural experiments andobservational studies, in which case the inference is dependent on the model chosen by the statistician, and so subjective.[4][5]
The following are some of the important topics in mathematical statistics:[6][7]
Aprobability distribution is afunction that assigns aprobability to eachmeasurable subset of the possible outcomes of a randomexperiment,survey, or procedure ofstatistical inference. Examples are found in experiments whosesample space is non-numerical, where the distribution would be acategorical distribution; experiments whose sample space is encoded by discreterandom variables, where the distribution can be specified by aprobability mass function; and experiments with sample spaces encoded by continuous random variables, where the distribution can be specified by aprobability density function. More complex experiments, such as those involvingstochastic processes defined incontinuous time, may demand the use of more generalprobability measures.
A probability distribution can either beunivariate ormultivariate. A univariate distribution gives the probabilities of a singlerandom variable taking on various alternative values; a multivariate distribution (ajoint probability distribution) gives the probabilities of arandom vector—a set of two or more random variables—taking on various combinations of values. Important and commonly encountered univariate probability distributions include thebinomial distribution, thehypergeometric distribution, and thenormal distribution. Themultivariate normal distribution is a commonly encountered multivariate distribution.
Statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation.[8] Initial requirements of such a system of procedures forinference andinduction are that the system should produce reasonable answers when applied to well-defined 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. Whereasdescriptive statistics describe a sample, inferential statistics infer predictions about a larger population that the sample represents.
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:
Instatistics,regression analysis is a statistical process for estimating the relationships among variables. It includes many ways for modeling and analyzing several variables, when the focus is on the relationship between adependent variable and one or moreindependent variables. More specifically, regression analysis helps one understand how the typical value of the dependent variable (or 'criterion variable') changes when any one of the independent variables is varied, while the other independent variables are held fixed. Most commonly, regression analysis estimates theconditional expectation of the dependent variable given the independent variables – that is, theaverage value of the dependent variable when the independent variables are fixed. Less commonly, the focus is on aquantile, or otherlocation parameter of the conditional distribution of the dependent variable given the independent variables. In all cases, the estimation target is afunction of the independent variables called theregression function. In regression analysis, it is also of interest to characterize the variation of the dependent variable around the regression function which can be described by aprobability distribution.
Many techniques for carrying out regression analysis have been developed. Familiar methods, such aslinear regression, areparametric, in that the regression function is defined in terms of a finite number of unknownparameters that are estimated from thedata (e.g. usingordinary least squares).Nonparametric regression refers to techniques that allow the regression function to lie in a specified set offunctions, which may beinfinite-dimensional.
Nonparametric statistics are values calculated from data in a way that is not based onparameterized families ofprobability distributions. They include bothdescriptive andinferential statistics. The typical parameters are the expectations, variance, etc. Unlikeparametric statistics, nonparametric statistics make no assumptions about theprobability distributions of the variables being assessed.[9]
Non-parametric methods are widely used for studying populations that take on a ranked order (such as movie reviews receiving one to four stars). The use of non-parametric methods may be necessary when data have aranking but no clear numerical interpretation, such as when assessingpreferences. In terms oflevels of measurement, non-parametric methods result in "ordinal" data.
As non-parametric methods make fewer assumptions, their applicability is much wider than the corresponding parametric methods. In particular, they may be applied in situations where less is known about the application in question. Also, due to the reliance on fewer assumptions, non-parametric methods are morerobust.
One drawback of non-parametric methods is that since they do not rely on assumptions, they are generally lesspowerful than their parametric counterparts.[10] Low power non-parametric tests are problematic because a common use of these methods is for when a sample has a low sample size.[10] Many parametric methods are proven to be the most powerful tests through methods such as theNeyman–Pearson lemma and theLikelihood-ratio test.
Another justification for the use of non-parametric methods is simplicity. In certain cases, even when the use of parametric methods is justified, non-parametric methods may be easier to use. Due both to this simplicity and to their greater robustness, non-parametric methods are seen by some statisticians as leaving less room for improper use and misunderstanding.
Mathematical statistics is a key subset of the discipline ofstatistics.Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions.
Mathematicians and statisticians likeGauss,Laplace, andC. S. Peirce useddecision theory withprobability distributions andloss functions (orutility functions). The decision-theoretic approach to statistical inference was reinvigorated byAbraham Wald and his successors[11][12][13][14][15][16][17] and makes extensive use ofscientific computing,analysis, andoptimization; for thedesign of experiments, statisticians usealgebra andcombinatorics. But while statistical practice often relies onprobability anddecision theory, their application can be controversial[5]
See Dover reprint, 2004:ISBN 0-486-43912-7
