Lihat sumber Statistika matematika

[[Berkas:Linear regression.svg|jmpl|ka|300px|Ilustrasi regresi linear pada suatu himpunan data. [[Analisis regresi]] adalah suatu bagian penting dalam statistika matematika.]]

'''Statistika matematika''' ({{lang-en|Mathematical statistics}}) adalah aplikasi [[matematika]] pada [[statistika]], yang asalnya dilahirkan sebagai suatu sains untuk negara (''state'')—kumpulan dan analisis fakta-fakta mengenai suatu negara: ekonomi, tanah, militer, populasi dan lain-lainnya. Teknik matematika yang digunakan di sini meliputi [[analisis matematis]], [[aljabar linear]], [[:en:stochastic analysis|analisis stokastik]], [[persamaan diferensial]], dan [[:en:measure-theoretic probability theory|teori probabilitas pengukuran-teoretis]].<ref>{{cite book|last=Lakshmikantham,|first=ed. by D. Kannan,... V.|title=Handbook of stochastic analysis and applications|date=2002|publisher=M. Dekker|location=New York|isbn=0824706609}}</ref><ref>{{cite book|last=Schervish|first=Mark J.|title=Theory of statistics|date=1995|publisher=Springer|location=New York|isbn=0387945466|edition=Corr. 2nd print.}}</ref>
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==Introduction==
Statistical science is concerned with the planning of studies, especially with the [[design of experiments|design of randomized experiments]] and with the planning of [[statistical survey|surveys]] using [[random 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 the data): For example, inferential statistics involves selecting a model for the data, checking whether the data fulfill the conditions of a particular model, and with quantifying the involved uncertainty (e.g. using [[confidence interval]]s).

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.<ref>[[David A. Freedman (statistician)|Freedman, D.A.]] (2005) ''Statistical Models: Theory and Practice'', Cambridge University Press. ISBN 978-0-521-67105-7</ref>
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Statistika matematika telah diilhami oleh [[:en:applied statistics|statistika terapan]] serta mengembangkan banyak prosedur pada penerapannya.

== Topik ==
Berikut adalah sejumlah topik penting dalam statistika matematika:<ref>Hogg, R. V., A. Craig, and J. W. McKean. "Intro to Mathematical Statistics." (2005).</ref><ref>Larsen, Richard J. and Marx, Morris L. "An Introduction to Mathematical Statistics and Its Applications" (2012). Prentice Hall.</ref>
=== Sebaran probabilitas ===
{{main|Sebaran probabilitas}}
[[Sebaran probabilitas]] atau "distribusi probabilitas" menggunakan [[peluang (matematika)|kemungkinan peluang atau probabilitas]] pada [[:en:measure (mathematics)|subset yang dapat diukur]] dari kemungkinan hasil suatu [[:en:Experiment (probability theory)|eksperimen]] yang bersifat acak (''random''), [[survei]], atau prosedur [[Statistika inferensi|inferensi statistik]].<!-- Examples are found in experiments whose [[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, such as those involving [[stochastic processes]] defined in [[continuous time]], may demand the use of more general [[probability measure]]s.

A probability distribution can either be [[Univariate distribution|univariate]] or [[Multivariate distribution|multivariate]]. A univariate distribution gives the probabilities of a single [[random variable]] taking on various alternative values; a multivariate distribution (a joint 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 distribution
*[[Bernoulli 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.) given a fixed total number of [[independent (statistics)|independent]] occurrences
*[[Negative binomial distribution]], for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs
*[[Geometric 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 of values (e.g. the outcome of a fair die)
*[[Continuous uniform distribution]], for continuously distributed values
*[[Poisson distribution]], for the number of occurrences of a Poisson-type event in a given period of time
*[[Exponential distribution]], for the time before the next Poisson-type event occurs
*[[Gamma 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 root of a scaled [[chi squared distribution|chi squared]] variable; useful for inference regarding the [[mean]] of normally distributed samples with unknown variance (see [[Student's t-test]])
*[[Beta distribution]], for a single probability (real number between 0 and 1); conjugate to the [[Bernoulli distribution]] and [[binomial distribution]]
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=== Inferensi statistik ===
{{main|Statistika inferensi}}
[[Statistika inferensi|Inferensi statistik]] adalah suatu proses untuk menarik kesimpulan dari data yang menjadi subjek variasi acak, misalnya, kesalahan observasi atau variasi pengambilan sampel.<ref name="Oxford">Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. ISBN 978-0-19-954145-4</ref><!-- Initial requirements of such a system of procedures for [[inference]] and [[Inductive reasoning|induction]] 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. 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 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:
* a [[statistical model]] of the random process that is supposed to generate the data, which is known when randomization has been used, and
* a particular realization of the random process; i.e., a set of data.
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=== Regresi ===
{{main|Analisis regresi}}

Dalam [[statistika]], '''analisis regresi''' adalah suatu proses statistik untuk memperkirakan hubungan antara variabel-variabel. Termasuk di dalamnya adalah teknik-tenik untuk pembuatan model dan analisis beberapa variabel, ketika fokusnya adalah hubungan antara suatu [[:en:dependent variable|variabel dependen]] dan satu atau lebih [[:en:independent variable|variabel independen]].<!-- 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 (mathematics)|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 [[linear regression]] and [[ordinary least squares]] regression are [[parametric statistics|parametric]], in that the regression function is defined in terms of a finite number of unknown [[parameter]]s that are estimated from the [[data]]. [[Nonparametric regression]] refers to techniques that allow the regression function to lie in a specified set of [[function (mathematics)|functions]], which may be [[dimension|infinite-dimensional]].
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=== Statistika non-parametrik ===
<!--{{main|Nonparametric statistics}}-->
'''[[Statistika nonparametrik]]''' adalah [[statistika]] yang tidak didasarkan atas familia [[:en:parametrization|berparameter]] dari [[sebaran probabilitas]]. Termasuk di dalamnya adalah baik [[:en:descriptive statistics|statistika deskriptif]] dan [[Statistika inferensi|inferensi]]. Parameter yang umum adalah rata-rata, variansi, dan lain-lain.<!--  [[parametric statistics]], nonparametric statistics make no assumptions about the [[probability distribution]]s 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 data have a [[ranking]] but no clear numerical interpretation, such as when assessing [[preferences]]. In terms of [[level of measurement|levels 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 statistics#Introduction|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.
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== Statistika, matematika, dan statistika matematika ==
Statistika matematika mempunyai ketumpangtindihan dengan bidang-bidang [[statistika]]. [[Statistikawan]] mempelajari dan memperbaiki prosedur statistika dengan matematika, dan riset statistika sering melahirkan pertanyaan matematis. Teori statistik tergantung pada [[Teori peluang|probabilitas atau peluang]] dan [[:en:optimal decision|teori keputusan]].

Matematikawan dan statistikawan seperti [[Gauss]], [[Laplace]], dan [[Charles Sanders Peirce|C. S. Peirce]] menggunakan [[:en:optimal decision|teori keputusan]] dengan [[sebaran probabilitas]] dan [[fungsi kerugian]] (atau [[:en:utility function|fungsi kegunaan]]). Pendekatan berdasarkan [[teori keputusan]] terhadap inferensi statistik dihidupkan kembali oleh [[Abraham Wald]] dan para penerusnya,<ref>{{Cite book
|first = Abraham
|last = Wald
|title = Sequential analysis
|url = https://archive.org/details/in.ernet.dli.2015.90255
|year = 1947
|publisher = John Wiley and Sons
|location = New York
|isbn = 0-471-91806-7
|quote = See Dover reprint, 2004: ISBN 0-486-43912-7
}}</ref><ref>{{cite book
|first=Abraham
|last=Wald
|title=Statistical Decision Functions
|year=1950
|publisher=John Wiley and Sons, New York
}}</ref><ref>{{cite book|last=Lehmann|first=Erich|authorlink=Erich Leo Lehmann
|title=Testing Statistical Hypotheses|url=https://archive.org/details/testingstatistic0000lehm_a7b0|year=1997|edition=2nd
|isbn=0-387-94919-4 }}</ref><ref>
{{cite book
|last1=Lehmann
|first1=Erich
|last2=Cassella
|first2=George
|title=Theory of Point Estimation
|year=1998|edition=2nd|isbn= 0-387-98502-6}}</ref><ref>
{{cite book
|last1=Bickel|first1= Peter J.|last2=Doksum|first2=Kjell A.
|authorlink1=Peter J. Bickel
|title=Mathematical Statistics: Basic and Selected Topics
|volume=1
|edition=Second (updated printing 2007)
|year=2001
|publisher=Pearson Prentice-Hall
}}</ref><ref>{{cite book
|first=Lucien
|last=Le Cam
|title=Asymptotic Methods in Statistical Decision Theory
|url=https://archive.org/details/asymptoticmethod0000leca
|year=1986
|publisher=Springer-Verlag|isbn=0-387-96307-3
}}</ref><ref>{{cite book
|author=Liese, Friedrich and Miescke, Klaus-J.
|title=Statistical Decision Theory: Estimation, Testing, and Selection
|year=2008
|publisher=Springer
}}
</ref> serta secara ekstensif menggunakan [[komputasi|komputasi ilmiah]], [[analisis matematis|analisis]], dan [[optimisasi]]; untuk [[perancangan percobaan]], statistikawan menggunakan [[:en:Algebraic statistics|aljabar]] dan [[:en:Combinatorial design|kombinatorika]].

== Lihat pula ==
* {{en}} [[:en:Asymptotic theory (statistics)|Teori asimptotik]]
* [[Statistika]]
* [[Ukuran pemusatan data]]
* [[Ukuran penyebaran data]]

== Referensi ==
<references/>

== Bacaan lebih lanjut ==
* {{cite book|last= Kurnianingsih|first= Sri|authorlink=|coauthors=Kuntarti, Sulistiyono|title=Matematika SMA dan MA 2A Untuk Kelas XI Semester 1 Program IPA|year= 2007|publisher= Esis/Erlangga|location= Jakarta|id= ISBN 979-734-502-5 }} {{id icon}}
* {{cite book|last= Kurnianingsih|first= Sri|authorlink=|coauthors=Kuntarti, Sulistiyono|title=Matematika SMA dan MA 2A Untuk Kelas XI Semester 1 Program IPS|year= 2007|publisher= Esis/Erlangga|location= Jakarta|id= ISBN 979-734-563-7 }} {{id icon}}

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

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