Mathematical Statistics - Wiki

8/16/2019 Mathematical Statistics - Wiki

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Illustration of linear regression on a data set.

Regression analysis is an important part of

mathematical statistics.

Mathematical statisticsFrom Wikipedia, the free encyclo pedia

Mathematical statistics is the application of

mathematics to statistics, which was originally

conceived as the science of the state — the collection

and analysis of facts about a country: its economy, land,

military, population, and so forth. Mathematical

techniques which are used for this include mathematical

analysis, linear algebra, stochastic analysis, differential

equations, and measure-theoretic probability theory.[1][2]

Contents

1 Introduction

2 Topics2.1 Probability distributions

2.1.1 Special distributions2.2 Statistical inferences2.3 Regression2.4 Nonparametric statistics

3 Statistics, mathematics, and mathematicalstatistics

4 See also5 References

6 Additional reading

Introduction

Statistical science is concerned with the planning of studies, especially with the design of r andomized

experiments and with the planning of surveys using r andom sampling. The initial analysis of the data from

properly randomized studies often follows the study protocol.

Of course, the data from a randomized study can be analyzed to consider secondary hypotheses or to

suggest new ideas. A secondary analysis of the data from a planned study uses tools from data analysis.

Data analysis is divided into:

descriptive statistics - the part of statistics that describes data, i.e. summarises the data and their typical properties.inferential statistics - the part of statistics that draws conclusions from data (using some model for thdata): For example, inferential statistics involves selecting a model for the data, checking whether thdata fulfill the conditions of a particular model, and with quantifying the involved uncertainty (e.g.using confidence intervals).

https://en.wikipedia.org/wiki/Confidence_intervalhttps://en.wikipedia.org/wiki/Inferential_statisticshttps://en.wikipedia.org/wiki/Inferential_statisticshttps://en.wikipedia.org/wiki/Descriptive_statisticshttps://en.wikipedia.org/wiki/Differential_equationshttps://en.wikipedia.org/wiki/File:Linear_regression.svghttps://en.wikipedia.org/wiki/File:Linear_regression.svghttps://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Statisticshttps://en.wikipedia.org/wiki/File:Linear_regression.svghttps://en.wikipedia.org/wiki/Confidence_intervalhttps://en.wikipedia.org/wiki/Inferential_statisticshttps://en.wikipedia.org/wiki/Descriptive_statisticshttps://en.wikipedia.org/wiki/Random_samplinghttps://en.wikipedia.org/wiki/Statistical_surveyhttps://en.wikipedia.org/wiki/Design_of_experimentshttps://en.wikipedia.org/wiki/Measure-theoretic_probability_theoryhttps://en.wikipedia.org/wiki/Differential_equationshttps://en.wikipedia.org/wiki/Stochastic_analysishttps://en.wikipedia.org/wiki/Linear_algebrahttps://en.wikipedia.org/wiki/Mathematical_analysishttps://en.wikipedia.org/wiki/Statisticshttps://en.wikipedia.org/wiki/Mathematicshttps://en.wikipedia.org/wiki/Regression_analysishttps://en.wikipedia.org/wiki/File:Linear_regression.svg


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While the tools of data analysis work best on data from randomized studies, they are also applied to other

kinds of data --- for example, from natural experiments and observational studies, in which case the

inference is dependent on the model chosen by the statistician, and so subjective.[3]

Mathematical statistics has been inspired by and has extended many options in applied statistics.

Topics

The following are some of the important topics in mathematical statistics:[4][5]

Probability distributions

A probability distribution assigns a probability to each measurable subset of the possible outcomes of a

random experiment, survey, or procedure of statistical inference. Examples are found in experiments whos

sample space is non-numerical, where the distribution would be a categorical distribution; experiments

whose sample space is encoded by discrete random variables, where the distribution can be specified by a

probability mass function; and experiments with sample spaces encoded by continuous random variables,

where the distribution can be specified by a probability density function. More complex experiments, suchas those involving stochastic processes defined in continuous time, may demand the use of more general

probability measures.

A probability distribution can either be univariate or multivariate. A univariate distribution gives the

probabilities of a single random variable taking on various alternative values; a multivariate distribution (a

oint probability distribution) gives the probabilities of a random vector—a set of two or more random

variables—taking on various combinations of values. Important and commonly encountered univariate

probability distributions include the binomial distribution, the hypergeometric distribution, and the normal

distribution. The multivariate normal distribution is a commonly encountered multivariate distribution.

Special distributions

Normal distribution (Gaussian distribution), the most common continuous distributionBernoulli distribution, for the outcome of a single Bernoulli trial (e.g. success/failure, yes/no)Binomial distribution, for the number of "positive occurrences" (e.g. successes, yes votes, etc.) givena fixed total number of independent occurrences

Negative binomial distribution, for binomial-type observations but where the quantity of interest isthe number of failures before a given number of successes occursGeometric distribution, for binomial-type observations but where the quantity of interest is the

number of failures before the first success; a special c*Discrete uniform distribution, for a finite set ovalues (e.g. the outcome of a fair die)Continuous uniform distribution, for continuously distributed valuesPoisson distribution, for the number of occurrences of a Poisson-type event in a given period of timeExponential distribution, for the time before the next Poisson-type event occursGamma distribution, for the time before the next k Poisson-type events occur Chi-squared distribution, the distribution of a sum of squared standard normal variables; useful e.g.for inference regarding the sample variance of normally distributed samples (see chi-squared test)Student's t distribution, the distribution of the ratio of a standard normal variable and the square rootof a scaled chi squared variable; useful for inference regarding the mean of normally distributedsamples with unknown variance (see Student's t-test)

https://en.wikipedia.org/wiki/Student%27s_t-testhttps://en.wikipedia.org/wiki/Meanhttps://en.wikipedia.org/wiki/Chi_squared_distributionhttps://en.wikipedia.org/wiki/Standard_normalhttps://en.wikipedia.org/wiki/Student%27s_t_distributionhttps://en.wikipedia.org/wiki/Chi-squared_testhttps://en.wikipedia.org/wiki/Sample_variancehttps://en.wikipedia.org/wiki/Standard_normalhttps://en.wikipedia.org/wiki/Chi-squared_distributionhttps://en.wikipedia.org/wiki/Gamma_distributionhttps://en.wikipedia.org/wiki/Exponential_distributionhttps://en.wikipedia.org/wiki/Poisson_distributionhttps://en.wikipedia.org/wiki/Continuous_uniform_distributionhttps://en.wikipedia.org/wiki/Discrete_uniform_distributionhttps://en.wikipedia.org/wiki/Geometric_distributionhttps://en.wikipedia.org/wiki/Negative_binomial_distributionhttps://en.wikipedia.org/wiki/Independent_(statistics)https://en.wikipedia.org/wiki/Binomial_distributionhttps://en.wikipedia.org/wiki/Bernoulli_distributionhttps://en.wikipedia.org/wiki/Gaussian_distributionhttps://en.wikipedia.org/wiki/Normal_distributionhttps://en.wikipedia.org/wiki/Multivariate_normal_distributionhttps://en.wikipedia.org/wiki/Normal_distributionhttps://en.wikipedia.org/wiki/Hypergeometric_distributionhttps://en.wikipedia.org/wiki/Binomial_distributionhttps://en.wikipedia.org/wiki/Random_vectorhttps://en.wikipedia.org/wiki/Random_variablehttps://en.wikipedia.org/wiki/Multivariate_distributionhttps://en.wikipedia.org/wiki/Univariate_distributionhttps://en.wikipedia.org/wiki/Probability_measurehttps://en.wikipedia.org/wiki/Continuous_timehttps://en.wikipedia.org/wiki/Stochastic_processeshttps://en.wikipedia.org/wiki/Probability_density_functionhttps://en.wikipedia.org/wiki/Probability_mass_functionhttps://en.wikipedia.org/wiki/Random_variableshttps://en.wikipedia.org/wiki/Categorical_distributionhttps://en.wikipedia.org/wiki/Sample_spacehttps://en.wikipedia.org/wiki/Statistical_inferencehttps://en.wikipedia.org/wiki/Survey_methodologyhttps://en.wikipedia.org/wiki/Experiment_(probability_theory)https://en.wikipedia.org/wiki/Measure_(mathematics)https://en.wikipedia.org/wiki/Probabilityhttps://en.wikipedia.org/wiki/Probability_distributionhttps://en.wikipedia.org/wiki/Observational_studieshttps://en.wikipedia.org/wiki/Natural_experiments


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Beta distribution, for a single probability (real number between 0 and 1); conjugate to the Bernoullidistribution and binomial distribution

Statistical inferences

Statistical inference is the process of drawing conclusions from data that are subject to random variation,

for example, observational errors or sampling variation.[6] Initial requirements of such a system of

procedures for inference and induction are that the system should produce reasonable answers when applieto 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. Whereas

descriptive 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 befor

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, statistica

inference most often uses:

a statistical model of the random process that is supposed to generate the data, which is known whenrandomization has been used, anda particular realization of the random process; i.e., a set of data.

Regression

In statistics, regression analysis is a statistical process for estimating the relationships among variables. It

includes many techniques for modeling and analyzing several variables, when the focus is on therelationship between a dependent variable and one or more independent 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 the conditional expectation of the

dependent variable given the independent variables – that is, the average value of the dependent variable

when the independent variables are fixed. Less commonly, the focus is on a quantile, or other location

parameter of the conditional distribution of the dependent variable given the independent variables. In all

cases, the estimation target is a function of the independent variables called the regression 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 a probability distribution.

Many techniques for carrying out regression analysis have been developed. Familiar methods such as linea

regression and ordinary least squares regression are parametric, in that the regression function is defined in

terms of a finite number of unknown parameters that are estimated from the data. Nonparametric regressio

refers to techniques that allow the regression function to lie in a specified set of functions, which may be

infinite-dimensional.

Nonparametric statistics

https://en.wikipedia.org/wiki/Dimensionhttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Nonparametric_regressionhttps://en.wikipedia.org/wiki/Datahttps://en.wikipedia.org/wiki/Parameterhttps://en.wikipedia.org/wiki/Parametric_statisticshttps://en.wikipedia.org/wiki/Ordinary_least_squareshttps://en.wikipedia.org/wiki/Linear_regressionhttps://en.wikipedia.org/wiki/Probability_distributionhttps://en.wikipedia.org/wiki/Function_(mathematics)https://en.wikipedia.org/wiki/Location_parameterhttps://en.wikipedia.org/wiki/Quantilehttps://en.wikipedia.org/wiki/Average_valuehttps://en.wikipedia.org/wiki/Conditional_expectationhttps://en.wikipedia.org/wiki/Independent_variablehttps://en.wikipedia.org/wiki/Dependent_variablehttps://en.wikipedia.org/wiki/Statisticshttps://en.wikipedia.org/wiki/Statistical_modelhttps://en.wikipedia.org/wiki/Descriptive_statisticshttps://en.wikipedia.org/wiki/Inductive_reasoninghttps://en.wikipedia.org/wiki/Inferencehttps://en.wikipedia.org/wiki/Statistical_inferencehttps://en.wikipedia.org/wiki/Binomial_distributionhttps://en.wikipedia.org/wiki/Bernoulli_distributionhttps://en.wikipedia.org/wiki/Beta_distribution


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Nonparametric statistics are statistics not based on parameterized families of probability distributions.

They include both descriptive and inferential statistics. The typical parameters are the mean, variance, etc.

Unlike parametric statistics, nonparametric statistics make no assumptions about the probability

distributions of the variables being assessed.

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 da

have a ranking but no clear numerical interpretation, such as when assessing preferences. In terms of level

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 more robust.

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.

Statistics, mathematics, and mathematical statistics

Mathematical statistics has substantial overlap with the discipline of statistics. Statistical theorists study an

improve statistical procedures with mathematics, and statistical research often raises mathematical

questions. Statistical theory relies on probability and decision theory.

Mathematicians and statisticians like Gauss, Laplace, and C. S. Peirce used decision theory with probabilit

distributions and loss functions (or utility functions). The decision-theoretic approach to statistical inferenc

was reinvigorated by Abraham Wald and his successors,[7][8][9][10][11][12][13] and makes extensive use of

scientific computing, analysis, and optimization; for the design of experiments, statisticians use algebra an

combinatorics.

See also

Asymptotic theory (statistics)

References

1. Lakshmikantham,, ed. by D. Kannan,... V. (2002). Handbook of stochastic analysis and applications. New York

M. Dekker. ISBN 0824706609.

2. Schervish, Mark J. (1995). Theory of statistics (Corr. 2nd print. ed.). New York: Springer. ISBN 0387945466.

3. Freedman, D.A. (2005) Statistical Models: Theory and Practice, Cambridge University Press. ISBN 978-0-521

67105-7

4. Hogg, R. V., A. Craig, and J. W. McKean. "Intro to Mathematical Statistics." (2005).

5. Larsen, Richard J. and Marx, Morris L. "An Introduction to Mathematical Statistics and Its Applications"

(2012). Prentice Hall.

6. Upton, G., Cook, I. (2008) Oxford Dictionary of Statistics, OUP. ISBN 978-0-19-954145-4

7. Wald, Abraham (1947). Sequential analysis. New York: John Wiley and Sons. ISBN 0-471-91806-7. "See Dove

https://en.wikipedia.org/wiki/Special:BookSources/0486439127https://en.wikipedia.org/wiki/Special:BookSources/0-471-91806-7https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Abraham_Waldhttps://en.wikipedia.org/wiki/Special:BookSources/9780199541454https://en.wikipedia.org/wiki/Special:BookSources/9780521671057https://en.wikipedia.org/wiki/David_A._Freedman_(statistician)https://en.wikipedia.org/wiki/Special:BookSources/0387945466https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Special:BookSources/0824706609https://en.wikipedia.org/wiki/International_Standard_Book_Numberhttps://en.wikipedia.org/wiki/Asymptotic_theory_(statistics)https://en.wikipedia.org/wiki/Combinatorial_designhttps://en.wikipedia.org/wiki/Algebraic_statisticshttps://en.wikipedia.org/wiki/Design_of_experimentshttps://en.wikipedia.org/wiki/Optimization_(mathematics)https://en.wikipedia.org/wiki/Mathematical_analysishttps://en.wikipedia.org/wiki/Scientific_computinghttps://en.wikipedia.org/wiki/Abraham_Waldhttps://en.wikipedia.org/wiki/Utility_functionhttps://en.wikipedia.org/wiki/Loss_functionhttps://en.wikipedia.org/wiki/Probability_distributionhttps://en.wikipedia.org/wiki/Optimal_decisionhttps://en.wikipedia.org/wiki/Charles_Sanders_Peircehttps://en.wikipedia.org/wiki/Laplacehttps://en.wikipedia.org/wiki/Gausshttps://en.wikipedia.org/wiki/Optimal_decisionhttps://en.wikipedia.org/wiki/Probability_theoryhttps://en.wikipedia.org/wiki/Statisticianshttps://en.wikipedia.org/wiki/Statisticshttps://en.wikipedia.org/wiki/Robust_statistics#Introductionhttps://en.wikipedia.org/wiki/Level_of_measurementhttps://en.wikipedia.org/wiki/Preferenceshttps://en.wikipedia.org/wiki/Rankinghttps://en.wikipedia.org/wiki/Probability_distributionhttps://en.wikipedia.org/wiki/Parametric_statisticshttps://en.wikipedia.org/wiki/Statistical_inferencehttps://en.wikipedia.org/wiki/Descriptive_statisticshttps://en.wikipedia.org/wiki/Probability_distributionhttps://en.wikipedia.org/wiki/Parametrizationhttps://en.wikipedia.org/wiki/Statistics


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reprint, 2004: ISBN 0-486-43912-7"

8. Wald, Abraham (1950). Statistical Decision Functions. John Wiley and Sons, New York.

9. Lehmann, Erich (1997). Testing Statistical Hypotheses (2nd ed.). ISBN 0-387-94919-4.

10. Lehmann, Erich; Cassella, George (1998). Theory of Point Estimation (2nd ed.). ISBN 0-387-98502-6.

11. Bickel, Peter J.; Doksum, Kjell A. (2001). Mathematical Statistics: Basic and Selected Topics 1 (Second (updat

printing 2007) ed.). Pearson Prentice-Hall.

12. Le Cam, Lucien (1986). Asymptotic Methods in Statistical Decision Theory. Springer-Verlag. ISBN 0-387-

96307-3.

13. Liese, Friedrich and Miescke, Klaus-J. (2008). Statistical Decision Theory: Estimation, Testing, and Selection.

Springer.

Additional reading

Borovkov, A. A. (1999). Mathematical Statistics. CRC Press. ISBN 90-5699-018-7Virtual Laboratories in Probability and Statistics (Univ. of Ala.-Huntsville) (http://www.math.uah.edu/stat/)StatiBot (http://www.trigonella.ch/statibot/english/), interactive online expert system on statisticaltests.

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