364:(Burgin 1982). The axiomatic approach encompasses other approaches in the algorithmic information theory. It is possible to treat different measures of algorithmic information as particular cases of axiomatically defined measures of algorithmic information. Instead of proving similar theorems, such as the basic invariance theorem, for each particular measure, it is possible to easily deduce all such results from one corresponding theorem proved in the axiomatic setting. This is a general advantage of the axiomatic approach in mathematics. The axiomatic approach to algorithmic information theory was further developed in the book (Burgin 2005) and applied to software metrics (Burgin and Debnath, 2003; Debnath and Burgin, 2003).
3757:
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383:"description language" in which the "descriptions" are given), the collection of random strings does depend on the choice of fixed universal machine. Nevertheless, the collection of random strings, as a whole, has similar properties regardless of the fixed machine, so one can (and often does) talk about the properties of random strings as a group without having to first specify a universal machine.
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424:âon the space of infinite binary sequences) is random. Also, since it can be shown that the Kolmogorov complexity relative to two different universal machines differs by at most a constant, the collection of random infinite sequences does not depend on the choice of universal machine (in contrast to finite strings). This definition of randomness is usually called
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249:, but any choice gives identical asymptotic results because the Kolmogorov complexity of a string is invariant up to an additive constant depending only on the choice of universal Turing machine. For this reason the set of random infinite sequences is independent of the choice of universal machine.)
209:
From this point of view, a 3000-page encyclopedia actually contains less information than 3000 pages of completely random letters, despite the fact that the encyclopedia is much more useful. This is because to reconstruct the entire sequence of random letters, one must know what every single letter
533:
The major drawback of AC and AP are their incomputability. Time-bounded "Levin" complexity penalizes a slow program by adding the logarithm of its running time to its length. This leads to computable variants of AC and AP, and universal "Levin" search (US) solves all inversion problems in optimal
537:
AC and AP also allow a formal and rigorous definition of randomness of individual strings to not depend on physical or philosophical intuitions about non-determinism or likelihood. Roughly, a string is algorithmic "Martin-Löf" random (AR) if it is incompressible in the sense that its algorithmic
382:
of the string is at least the length of the string. A simple counting argument shows that some strings of any given length are random, and almost all strings are very close to being random. Since
Kolmogorov complexity depends on a fixed choice of universal Turing machine (informally, a fixed
210:
is. On the other hand, if every vowel were removed from the encyclopedia, someone with reasonable knowledge of the
English language could reconstruct it, just as one could likely reconstruct the sentence "Ths sntnc hs lw nfrmtn cntnt" from the context and consonants present.
99:
does, as in classical information theory; randomness is incompressibility; and, within the realm of randomly generated software, the probability of occurrence of any data structure is of the order of the shortest program that generates it when running on a universal machine.
63:. In other words, it is shown within algorithmic information theory that computational incompressibility "mimics" (except for a constant that only depends on the chosen universal programming language) the relations or inequalities found in
436:
to distinguish it from other stronger notions of randomness (2-randomness, 3-randomness, etc.). In addition to Martin-Löf randomness concepts, there are also recursive randomness, Schnorr randomness, and Kurtz randomness etc.
90:
Besides the formalization of a universal measure for irreducible information content of computably generated objects, some main achievements of AIT were to show that: in fact algorithmic complexity follows (in the
484:
Algorithmic information theory (AIT) is the information theory of individual objects, using computer science, and concerns itself with the relationship between computation, information, and randomness.
541:
AC, AP, and AR are the core sub-disciplines of AIT, but AIT spawns into many other areas. It serves as the foundation of the
Minimum Description Length (MDL) principle, can simplify proofs in
796:
Paper from conference on "Cerebral
Systems and Computers", California Institute of Technology, February 8â11, 1960, cited in "A Formal Theory of Inductive Inference, Part 1, 1964, p. 1
1420:
Zvonkin, A.K. and Levin, L. A. (1970). "The
Complexity of Finite Objects and the Development of the Concepts of Information and Randomness by Means of the Theory of Algorithms".
333:
in 1960, and in a report, February 1960, "A Preliminary Report on a
General Theory of Inductive Inference." Algorithmic information theory was later developed independently by
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when its input is supplied by flips of a fair coin (sometimes thought of as the probability that a random computer program will eventually halt). Although
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also contributed significantly to the information theory of infinite sequences. An axiomatic approach to algorithmic information theory based on the
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Informally, from the point of view of algorithmic information theory, the information content of a string is equivalent to the length of the most-
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The information content or complexity of an object can be measured by the length of its shortest description. For instance the string
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978:
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119:. One of the main motivations behind AIT is the very study of the information carried by mathematical objects as in the field of
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123:, e.g., as shown by the incompleteness results mentioned below. Other main motivations came from surpassing the limitations of
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151:). In this way, AIT is known to be basically founded upon three main mathematical concepts and the relations between them:
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or, for the mutual algorithmic information, informing the algorithmic complexity of the input along with the input itself.
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256:, appear to challenge common mathematical and philosophical intuitions. Most notable among these is the construction of
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There are several variants of
Kolmogorov complexity or algorithmic information; the most widely used one is based on
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1956:
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1989:
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636:
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possible self-contained representation of that string. A self-contained representation is essentially a
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574:
257:
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263:, a real number that expresses the probability that a self-delimiting universal Turing machine will
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1999:
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601:
546:
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1399:"Algorithmic Information Content, Church-Turing Thesis, physical entropy, and Maxwell's demon, in"
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2049:
1971:
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1347:
1266:
1187:
Kolmogorov, A.N. (1965). "Three approaches to the definition of the quantity of information".
1173:
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974:
925:
Burgin, M. (1982). "Generalized
Kolmogorov complexity and duality in theory of computations".
754:
647:
429:
361:
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17:
1398:
1344:
Algorithmic
Probability: Theory and Applications, Information Theory and Statistical Learning
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1994:
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1086:
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903:
878:
421:
195:
938:
Burgin, M. (1990). "Generalized
Kolmogorov Complexity and other Dual Complexity Measures".
809:", Report V-131, Zator Co., Cambridge, Ma., (November Revision of February 4, 1960 report.)
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3353:
3215:
3142:
2817:
2691:
2664:
2641:
2610:
2237:
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1916:
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1029:. Texts in Theoretical Computer Science. An EATCS Series (2nd ed.). Springer-Verlag.
522:
A closely related notion is the probability that a universal computer outputs some string
432:, to distinguish it from other similar notions of randomness. It is also sometimes called
338:
264:
120:
68:
1044:
Chaitin, G.J. (1966). "On the Length of Programs for Computing Finite Binary Sequences".
1433:
1075:"On the Simplicity and Speed of Programs for Computing Definite Sets of Natural Numbers"
321:, who published the basic ideas on which the field is based as part of his invention of
234:
3558:
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2016:
1946:
1592:
1362:
326:
318:
180:
108:
72:
60:
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of strings, it can be used to study a wide variety of mathematical objects, including
183:). Because most mathematical objects can be described in terms of strings, or as the
115:
of strings, it can be used to study a wide variety of mathematical objects, including
3801:
3715:
3682:
3545:
3506:
3317:
3286:
2750:
2704:
2309:
2011:
1838:
1602:
1597:
1449:
1389:
1227:"Laws of information (nongrowth) and aspects of the foundation of probability theory"
1166:
1065:
959:
820:
416:. It can be shown that almost every sequence (from the point of view of the standard
306:
214:
111:). Because most mathematical objects can be described in terms of strings, or as the
1868:
1441:
1217:
1135:
1100:
917:
545:, has been used to define a universal similarity metric between objects, solves the
3657:
3590:
3567:
3482:
2812:
2108:
2006:
1941:
1883:
1805:
1760:
846:
595:
562: â mathematical method of assigning a prior probability to a given observation
349:
144:
360:(Blum 1967) was introduced by Mark Burgin in a paper presented for publication by
1260:
1024:
968:
748:
530:
is key in addressing the old philosophical problem of induction in a formal way.
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1936:
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presumably has no simple description other than writing down the string itself.
438:
357:
345:
92:
80:
48:
785:
583: â In computer science, relationship between two families of distributions
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1502:
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272:
238:
1209:
2699:
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1966:
1878:
1863:
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1823:
786:
Obituary: Ray Solomonoff, Founding Father of Algorithmic Information Theory"
1244:"Various Measures of Complexity for Finite Objects (Axiomatic Description)"
1126:
1109:
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1074:
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908:
891:
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is defined as the length of the shortest program that computes or outputs
213:
Unlike classical information theory, algorithmic information theory gives
2215:
1833:
1710:
1705:
1700:
1672:
610: â Determining the probability of future events based on past events
1156:
3720:
3421:
951:
519:, where the program is run on some fixed reference universal computer.
392:
An infinite binary sequence is said to be random if, for some constant
330:
325:âa way to overcome serious problems associated with the application of
188:
116:
103:
AIT principally studies measures of irreducible information content of
534:
time (apart from some unrealistically large multiplicative constant).
3642:
2623:
2597:
2577:
1828:
1619:
1281:
1243:
806:
218:
1226:
1321:
Solomonoff, R.J. (1964). "A Formal Theory of Inductive Inference".
1110:"A Theory of Program Size Formally Identical to Information Theory"
892:"A Machine-independent Theory of Complexity of Recursive Functions"
644: â Term used in theoretical computer science and cryptography
1197:
500:"1100100001100001110111101110110011111010010000100101011110010110"
492:"0101010101010101010101010101010101010101010101010101010101010101"
444:(Related definitions can be made for alphabets other than the set
329:
in statistics. He first described his results at a Conference at
289:, providing an absolute limit on knowledge that is reminiscent of
275:
589: â sequence of probability distributions or random variables
305:
and thus its binary digits are evenly distributed (in fact it is
1562:
3531:
3098:
2845:
2144:
1914:
1531:
1475:
1342:
Solomonoff, R.J. (2009). Emmert-Streib, F.; Dehmer, M. (eds.).
1283:
A Preliminary Report on a General Theory of Inductive Inference
807:
A Preliminary Report on a General Theory of Inductive Inference
252:
Some of the results of algorithmic information theory, such as
633: â Formal information theory restatement of Occam's Razor
1471:
1262:
An Introduction to Kolmogorov Complexity and its Applications
1198:"Logical basis for information theory and probability theory"
1289:(Technical report). Cambridge, Mass: Zator Company. ZTB-138.
441:
showed that all of these randomness concepts are different.
659: â Establishes the limits to possible data compression
496:
has the short description "32 repetitions of '01'", while
241:. (The set of random strings depends on the choice of the
577: â Halting probability of a random computer program
127:
for single and fixed objects, formalizing the concept of
95:
case) the same inequalities (except for a constant) that
1143:
Chaitin, G.J. (1977). "Algorithmic information theory".
1026:
Information and Randomness: An Algorithmic Perspective
450:
3384:
Autoregressive conditional heteroskedasticity (ARCH)
652:
Pages displaying wikidata descriptions as a fallback
591:
Pages displaying wikidata descriptions as a fallback
564:
Pages displaying wikidata descriptions as a fallback
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1980:
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171:Algorithmic information theory principally studies
43:that concerns itself with the relationship between
27:
Subfield of information theory and computer science
1403:Complexity, Entropy and the Physics of Information
1165:
1114:Journal of the Association for Computing Machinery
1079:Journal of the Association for Computing Machinery
1046:Journal of the Association for Computing Machinery
468:
202:âin some fixed but otherwise irrelevant universal
747:Downey, Rodney G.; Hirschfeldt, Denis R. (2010).
229:that do not depend on physical or philosophical
87:into a cocktail shaker and shaking vigorously."
2932:Multivariate adaptive regression splines (MARS)
990:"Algorithmic information theory: Open problems"
526:when fed with a program chosen at random. This
51:of computably generated objects (as opposed to
317:Algorithmic information theory was founded by
206:âthat, when run, outputs the original string.
1487:
281:one can only compute finitely many digits of
8:
1462:: CS1 maint: multiple names: authors list (
973:. Monographs in computer science. Springer.
463:
451:
378:A binary string is said to be random if the
682:
680:
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1924:
1911:
1550:
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1494:
1480:
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731:
729:
663:Solomonoff's theory of inductive inference
1311:
1125:
1090:
907:
773:
621: â Measure of algorithmic complexity
528:algorithmic "Solomonoff" probability (AP)
449:
297:cannot be determined, many properties of
1296:"A Formal Theory of Inductive Inference"
598: â Philosophical study of knowledge
1145:IBM Journal of Research and Development
852:Chaitin's account of the history of AIT
686:
676:
141:independent and identically distributed
3458:KaplanâMeier estimator (product limit)
1455:
735:
750:Algorithmic Randomness and Complexity
7:
3768:
3468:Accelerated failure time (AFT) model
1231:Problems of Information Transmission
1189:Problems of Information Transmission
3780:
3063:Analysis of variance (ANOVA, anova)
869:(1967). "On the Size of Machines".
604: â Method of logical reasoning
538:complexity is equal to its length.
3158:CochranâMantelâHaenszel statistics
1784:Pearson product-moment correlation
1405:. Addison-Wesley. pp. 73â89.
581:Computational indistinguishability
25:
1265:(2nd ed.). Springer-Verlag.
627: â Model selection principle
404:of the initial segment of length
301:are known; for example, it is an
3779:
3767:
3755:
3742:
3741:
1363:"Algorithmic Information Theory"
828:(PhD). University of Heidelberg.
699:"Algorithmic Information Theory"
254:Chaitin's incompleteness theorem
3417:Least-squares spectral analysis
1442:10.1070/RM1970v025n06ABEH001269
657:Shannon's source coding theorem
569:Algorithmically random sequence
543:computational complexity theory
388:Algorithmically random sequence
303:algorithmically random sequence
291:Gödel's incompleteness theorems
135:without prior knowledge of the
71:, it is "the result of putting
3808:Algorithmic information theory
2398:Mean-unbiased minimum-variance
1172:. Cambridge University Press.
1168:Algorithmic Information Theory
843:Algorithmic Information Theory
33:Algorithmic information theory
18:Algorithmic Information Theory
1:
3711:Geographic information system
2927:Simultaneous equations models
1335:10.1016/S0019-9958(64)90131-7
1313:10.1016/S0019-9958(64)90223-2
883:10.1016/S0019-9958(67)90546-3
665: â A mathematical theory
2894:Coefficient of determination
2505:Uniformly most powerful test
1422:Russian Mathematical Surveys
1259:Li, M.; Vitanyi, P. (2013).
1003:(5): 439â441. Archived from
408:of the sequence is at least
125:classical information theory
41:theoretical computer science
3463:Proportional hazards models
3407:Spectral density estimation
3389:Vector autoregression (VAR)
2823:Maximum posterior estimator
2055:Randomized controlled trial
509:algorithmic complexity (AC)
131:, and finding a meaningful
3834:
3223:Multivariate distributions
1643:Average absolute deviation
970:Super-recursive algorithms
625:Minimum description length
549:problem, and many others.
385:
371:
271:is easily defined, in any
3737:
3540:
3527:
3211:Structural equation model
3119:
3094:
2865:
2841:
2573:
2547:Score/Lagrange multiplier
2153:
2140:
1962:Sample size determination
1923:
1910:
1540:
1527:
1509:
1382:10.1017/S0022481200041153
1370:Journal of Symbolic Logic
1294:Solomonoff, R.J. (1964).
1280:Solomonoff, R.J. (1960).
1196:Kolmogorov, A.N. (1968).
822:Randomness and Complexity
293:. Although the digits of
285:, so it is in some sense
3706:Environmental statistics
3228:Elliptical distributions
3021:Generalized linear model
2950:Simple linear regression
2720:HodgesâLehmann estimator
2177:Probability distribution
2086:Stochastic approximation
1648:Coefficient of variation
1210:10.1109/TIT.1968.1054210
650: â cognitive theory
346:self-delimiting programs
243:universal Turing machine
227:random infinite sequence
137:probability distribution
3366:Cross-correlation (XCF)
2974:Non-standard predictors
2408:LehmannâScheffĂ© theorem
2081:Adaptive clinical trial
1323:Information and Control
1300:Information and Control
1202:IEEE Trans. Inf. Theory
871:Information and Control
571: â Binary sequence
560:Algorithmic probability
469:{\displaystyle \{0,1\}}
323:algorithmic probability
161:algorithmic probability
133:probabilistic inference
3762:Mathematics portal
3583:Engineering statistics
3491:NelsonâAalen estimator
3068:Analysis of covariance
2955:Ordinary least squares
2879:Pearson product-moment
2283:Statistical functional
2194:Empirical distribution
2027:Controlled experiments
1756:Frequency distribution
1534:Descriptive statistics
1164:Chaitin, G.J. (1987).
1108:Chaitin, G.J. (1975).
1073:Chaitin, G.J. (1969).
642:Pseudorandom generator
631:Minimum message length
470:
157:algorithmic randomness
153:algorithmic complexity
3818:Randomized algorithms
3678:Population statistics
3620:System identification
3354:Autocorrelation (ACF)
3282:Exponential smoothing
3196:Discriminant analysis
3191:Canonical correlation
3055:Partition of variance
2917:Regression validation
2761:(JonckheereâTerpstra)
2660:Likelihood-ratio test
2349:Frequentist inference
2261:Locationâscale family
2182:Sampling distribution
2147:Statistical inference
2114:Cross-sectional study
2101:Observational studies
2060:Randomized experiment
1889:Stem-and-leaf display
1691:Central limit theorem
1397:Zurek, W.H. (2018) .
1361:Van Lambagen (1989).
1225:Levin, L. A. (1974).
1127:10.1145/321892.321894
1092:10.1145/321526.321530
1058:10.1145/321356.321363
1023:Calude, C.S. (2013).
988:Calude, C.S. (1996).
909:10.1145/321386.321395
819:Wang, Yongge (1996).
774:Li & Vitanyi 2013
637:Pseudorandom ensemble
619:Kolmogorov complexity
608:Inductive probability
587:Distribution ensemble
471:
402:Kolmogorov complexity
380:Kolmogorov complexity
374:Kolmogorov complexity
348:and is mainly due to
247:Kolmogorov complexity
139:(e.g., whether it is
3601:Probabilistic design
3186:Principal components
3029:Exponential families
2981:Nonlinear regression
2960:General linear model
2922:Mixed effects models
2912:Errors and residuals
2889:Confounding variable
2791:Bayesian probability
2769:Van der Waerden test
2759:Ordered alternative
2524:Multiple comparisons
2403:RaoâBlackwellization
2366:Estimating equations
2322:Statistical distance
2040:Factorial experiment
1573:Arithmetic-Geometric
1242:Levin, L.A. (1976).
1204:. IT-14 (5): 662â4.
1010:on November 28, 2021
448:
204:programming language
85:computability theory
55:generated), such as
3673:Official statistics
3596:Methods engineering
3277:Seasonal adjustment
3045:Poisson regressions
2965:Bayesian regression
2904:Regression analysis
2884:Partial correlation
2856:Regression analysis
2455:Prediction interval
2450:Likelihood interval
2440:Confidence interval
2432:Interval estimation
2393:Unbiased estimators
2211:Model specification
2091:Up-and-down designs
1779:Partial correlation
1735:Index of dispersion
1653:Interquartile range
1434:1970RuMaS..25...83Z
1157:10.1147/rd.214.0350
967:Burgin, M. (2005).
705:on January 23, 2016
602:Inductive reasoning
507:More formally, the
368:Precise definitions
185:limit of a sequence
113:limit of a sequence
3813:Information theory
3693:Spatial statistics
3573:Medical statistics
3473:First hitting time
3427:Whittle likelihood
3078:Degrees of freedom
3073:Multivariate ANOVA
3006:Heteroscedasticity
2818:Bayesian estimator
2783:Bayesian inference
2632:KolmogorovâSmirnov
2517:Randomization test
2487:Testing hypotheses
2460:Tolerance interval
2371:Maximum likelihood
2266:Exponential family
2199:Density estimation
2159:Statistical theory
2119:Natural experiment
2065:Scientific control
1982:Survey methodology
1668:Standard deviation
952:10.1007/BF01068189
896:Journal of the ACM
614:Invariance theorem
575:Chaitin's constant
466:
428:randomness, after
258:Chaitin's constant
77:information theory
65:information theory
3795:
3794:
3733:
3732:
3729:
3728:
3668:National accounts
3638:Actuarial science
3630:Social statistics
3523:
3522:
3519:
3518:
3515:
3514:
3450:Survival function
3435:
3434:
3297:Granger causality
3138:Contingency table
3113:Survival analysis
3090:
3089:
3086:
3085:
2942:Linear regression
2837:
2836:
2833:
2832:
2808:Credible interval
2777:
2776:
2560:
2559:
2376:Method of moments
2245:Parametric family
2206:Statistical model
2136:
2135:
2132:
2131:
2050:Random assignment
1972:Statistical power
1906:
1905:
1902:
1901:
1751:Contingency table
1721:
1720:
1588:Generalized/power
1353:978-0-387-84815-0
1248:Soviet Math. Dokl
927:Soviet Math. Dokl
890:Blum, M. (1967).
805:Solomonoff, R., "
760:978-0-387-68441-3
648:Simplicity theory
480:Specific sequence
362:Andrey Kolmogorov
335:Andrey Kolmogorov
221:definitions of a
39:) is a branch of
16:(Redirected from
3825:
3783:
3782:
3771:
3770:
3760:
3759:
3745:
3744:
3648:Crime statistics
3542:
3529:
3446:
3412:Fourier analysis
3399:Frequency domain
3379:
3326:
3292:Structural break
3252:
3201:Cluster analysis
3148:Log-linear model
3121:
3096:
3037:
3011:Homoscedasticity
2867:
2843:
2762:
2754:
2746:
2745:(KruskalâWallis)
2730:
2715:
2670:Cross validation
2655:
2637:AndersonâDarling
2584:
2571:
2542:Likelihood-ratio
2534:Parametric tests
2512:Permutation test
2495:1- & 2-tails
2386:Minimum distance
2358:Point estimation
2354:
2305:Optimal decision
2256:
2155:
2142:
2124:Quasi-experiment
2074:Adaptive designs
1925:
1912:
1789:Rank correlation
1551:
1542:
1529:
1496:
1489:
1482:
1473:
1467:
1461:
1453:
1416:
1393:
1376:(4): 1389â1400.
1367:
1357:
1338:
1317:
1315:
1290:
1288:
1276:
1255:
1238:
1221:
1192:
1183:
1171:
1160:
1139:
1129:
1104:
1094:
1069:
1040:
1019:
1017:
1015:
1009:
994:
984:
963:
934:
921:
911:
886:
830:
829:
827:
816:
810:
803:
797:
794:
788:
782:
776:
771:
765:
764:
744:
738:
733:
724:
721:
715:
714:
712:
710:
701:. Archived from
695:
689:
684:
653:
592:
565:
501:
493:
475:
473:
472:
467:
422:Lebesgue measure
420:â"fair coin" or
21:
3833:
3832:
3828:
3827:
3826:
3824:
3823:
3822:
3798:
3797:
3796:
3791:
3754:
3725:
3687:
3624:
3610:quality control
3577:
3559:Clinical trials
3536:
3511:
3495:
3483:Hazard function
3477:
3431:
3393:
3377:
3340:
3336:BreuschâGodfrey
3324:
3301:
3241:
3216:Factor analysis
3162:
3143:Graphical model
3115:
3082:
3049:
3035:
3015:
2969:
2936:
2898:
2861:
2860:
2829:
2773:
2760:
2752:
2744:
2728:
2713:
2692:Rank statistics
2686:
2665:Model selection
2653:
2611:Goodness of fit
2605:
2582:
2556:
2528:
2481:
2426:
2415:Median unbiased
2343:
2254:
2187:Order statistic
2149:
2128:
2095:
2069:
2021:
1976:
1919:
1917:Data collection
1898:
1810:
1765:
1739:
1717:
1677:
1629:
1546:Continuous data
1536:
1523:
1505:
1500:
1470:
1454:
1419:
1413:
1396:
1365:
1360:
1354:
1341:
1320:
1293:
1286:
1279:
1273:
1258:
1241:
1224:
1195:
1186:
1180:
1163:
1142:
1107:
1072:
1043:
1037:
1022:
1013:
1011:
1007:
992:
987:
981:
966:
937:
924:
889:
865:
861:
859:Further reading
839:
834:
833:
825:
818:
817:
813:
804:
800:
795:
791:
783:
779:
772:
768:
761:
746:
745:
741:
734:
727:
722:
718:
708:
706:
697:
696:
692:
685:
678:
673:
668:
651:
590:
563:
555:
499:
491:
482:
446:
445:
390:
376:
370:
341:, around 1966.
339:Gregory Chaitin
315:
245:used to define
181:data structures
169:
121:metamathematics
109:data structures
69:Gregory Chaitin
67:. According to
28:
23:
22:
15:
12:
11:
5:
3831:
3829:
3821:
3820:
3815:
3810:
3800:
3799:
3793:
3792:
3790:
3789:
3777:
3765:
3751:
3738:
3735:
3734:
3731:
3730:
3727:
3726:
3724:
3723:
3718:
3713:
3708:
3703:
3697:
3695:
3689:
3688:
3686:
3685:
3680:
3675:
3670:
3665:
3660:
3655:
3650:
3645:
3640:
3634:
3632:
3626:
3625:
3623:
3622:
3617:
3612:
3603:
3598:
3593:
3587:
3585:
3579:
3578:
3576:
3575:
3570:
3565:
3556:
3554:Bioinformatics
3550:
3548:
3538:
3537:
3532:
3525:
3524:
3521:
3520:
3517:
3516:
3513:
3512:
3510:
3509:
3503:
3501:
3497:
3496:
3494:
3493:
3487:
3485:
3479:
3478:
3476:
3475:
3470:
3465:
3460:
3454:
3452:
3443:
3437:
3436:
3433:
3432:
3430:
3429:
3424:
3419:
3414:
3409:
3403:
3401:
3395:
3394:
3392:
3391:
3386:
3381:
3373:
3368:
3363:
3362:
3361:
3359:partial (PACF)
3350:
3348:
3342:
3341:
3339:
3338:
3333:
3328:
3320:
3315:
3309:
3307:
3306:Specific tests
3303:
3302:
3300:
3299:
3294:
3289:
3284:
3279:
3274:
3269:
3264:
3258:
3256:
3249:
3243:
3242:
3240:
3239:
3238:
3237:
3236:
3235:
3220:
3219:
3218:
3208:
3206:Classification
3203:
3198:
3193:
3188:
3183:
3178:
3172:
3170:
3164:
3163:
3161:
3160:
3155:
3153:McNemar's test
3150:
3145:
3140:
3135:
3129:
3127:
3117:
3116:
3099:
3092:
3091:
3088:
3087:
3084:
3083:
3081:
3080:
3075:
3070:
3065:
3059:
3057:
3051:
3050:
3048:
3047:
3031:
3025:
3023:
3017:
3016:
3014:
3013:
3008:
3003:
2998:
2993:
2991:Semiparametric
2988:
2983:
2977:
2975:
2971:
2970:
2968:
2967:
2962:
2957:
2952:
2946:
2944:
2938:
2937:
2935:
2934:
2929:
2924:
2919:
2914:
2908:
2906:
2900:
2899:
2897:
2896:
2891:
2886:
2881:
2875:
2873:
2863:
2862:
2859:
2858:
2853:
2847:
2846:
2839:
2838:
2835:
2834:
2831:
2830:
2828:
2827:
2826:
2825:
2815:
2810:
2805:
2804:
2803:
2798:
2787:
2785:
2779:
2778:
2775:
2774:
2772:
2771:
2766:
2765:
2764:
2756:
2748:
2732:
2729:(MannâWhitney)
2724:
2723:
2722:
2709:
2708:
2707:
2696:
2694:
2688:
2687:
2685:
2684:
2683:
2682:
2677:
2672:
2662:
2657:
2654:(ShapiroâWilk)
2649:
2644:
2639:
2634:
2629:
2621:
2615:
2613:
2607:
2606:
2604:
2603:
2595:
2586:
2574:
2568:
2566:Specific tests
2562:
2561:
2558:
2557:
2555:
2554:
2549:
2544:
2538:
2536:
2530:
2529:
2527:
2526:
2521:
2520:
2519:
2509:
2508:
2507:
2497:
2491:
2489:
2483:
2482:
2480:
2479:
2478:
2477:
2472:
2462:
2457:
2452:
2447:
2442:
2436:
2434:
2428:
2427:
2425:
2424:
2419:
2418:
2417:
2412:
2411:
2410:
2405:
2390:
2389:
2388:
2383:
2378:
2373:
2362:
2360:
2351:
2345:
2344:
2342:
2341:
2336:
2331:
2330:
2329:
2319:
2314:
2313:
2312:
2302:
2301:
2300:
2295:
2290:
2280:
2275:
2270:
2269:
2268:
2263:
2258:
2242:
2241:
2240:
2235:
2230:
2220:
2219:
2218:
2213:
2203:
2202:
2201:
2191:
2190:
2189:
2179:
2174:
2169:
2163:
2161:
2151:
2150:
2145:
2138:
2137:
2134:
2133:
2130:
2129:
2127:
2126:
2121:
2116:
2111:
2105:
2103:
2097:
2096:
2094:
2093:
2088:
2083:
2077:
2075:
2071:
2070:
2068:
2067:
2062:
2057:
2052:
2047:
2042:
2037:
2031:
2029:
2023:
2022:
2020:
2019:
2017:Standard error
2014:
2009:
2004:
2003:
2002:
1997:
1986:
1984:
1978:
1977:
1975:
1974:
1969:
1964:
1959:
1954:
1949:
1947:Optimal design
1944:
1939:
1933:
1931:
1921:
1920:
1915:
1908:
1907:
1904:
1903:
1900:
1899:
1897:
1896:
1891:
1886:
1881:
1876:
1871:
1866:
1861:
1856:
1851:
1846:
1841:
1836:
1831:
1826:
1820:
1818:
1812:
1811:
1809:
1808:
1803:
1802:
1801:
1796:
1786:
1781:
1775:
1773:
1767:
1766:
1764:
1763:
1758:
1753:
1747:
1745:
1744:Summary tables
1741:
1740:
1738:
1737:
1731:
1729:
1723:
1722:
1719:
1718:
1716:
1715:
1714:
1713:
1708:
1703:
1693:
1687:
1685:
1679:
1678:
1676:
1675:
1670:
1665:
1660:
1655:
1650:
1645:
1639:
1637:
1631:
1630:
1628:
1627:
1622:
1617:
1616:
1615:
1610:
1605:
1600:
1595:
1590:
1585:
1580:
1578:Contraharmonic
1575:
1570:
1559:
1557:
1548:
1538:
1537:
1532:
1525:
1524:
1522:
1521:
1516:
1510:
1507:
1506:
1501:
1499:
1498:
1491:
1484:
1476:
1469:
1468:
1417:
1411:
1394:
1358:
1352:
1339:
1329:(2): 224â254.
1318:
1291:
1277:
1271:
1256:
1239:
1222:
1193:
1184:
1178:
1161:
1140:
1120:(3): 329â340.
1105:
1085:(3): 407â412.
1070:
1052:(4): 547â569.
1041:
1035:
1020:
985:
979:
964:
946:(4): 481â490.
935:
922:
902:(2): 322â336.
887:
877:(3): 257â265.
862:
860:
857:
856:
855:
849:
838:
837:External links
835:
832:
831:
811:
798:
789:
777:
766:
759:
739:
725:
716:
690:
675:
674:
672:
669:
667:
666:
660:
654:
645:
639:
634:
628:
622:
616:
611:
605:
599:
593:
584:
578:
572:
566:
556:
554:
551:
547:Maxwell daemon
481:
478:
465:
462:
459:
456:
453:
430:Per Martin-Löf
386:Main article:
372:Main article:
369:
366:
354:Per Martin-Löf
337:, in 1965 and
319:Ray Solomonoff
314:
311:
235:nondeterminism
168:
165:
93:self-delimited
61:data structure
53:stochastically
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3830:
3819:
3816:
3814:
3811:
3809:
3806:
3805:
3803:
3788:
3787:
3778:
3776:
3775:
3766:
3764:
3763:
3758:
3752:
3750:
3749:
3740:
3739:
3736:
3722:
3719:
3717:
3716:Geostatistics
3714:
3712:
3709:
3707:
3704:
3702:
3699:
3698:
3696:
3694:
3690:
3684:
3683:Psychometrics
3681:
3679:
3676:
3674:
3671:
3669:
3666:
3664:
3661:
3659:
3656:
3654:
3651:
3649:
3646:
3644:
3641:
3639:
3636:
3635:
3633:
3631:
3627:
3621:
3618:
3616:
3613:
3611:
3607:
3604:
3602:
3599:
3597:
3594:
3592:
3589:
3588:
3586:
3584:
3580:
3574:
3571:
3569:
3566:
3564:
3560:
3557:
3555:
3552:
3551:
3549:
3547:
3546:Biostatistics
3543:
3539:
3535:
3530:
3526:
3508:
3507:Log-rank test
3505:
3504:
3502:
3498:
3492:
3489:
3488:
3486:
3484:
3480:
3474:
3471:
3469:
3466:
3464:
3461:
3459:
3456:
3455:
3453:
3451:
3447:
3444:
3442:
3438:
3428:
3425:
3423:
3420:
3418:
3415:
3413:
3410:
3408:
3405:
3404:
3402:
3400:
3396:
3390:
3387:
3385:
3382:
3380:
3378:(BoxâJenkins)
3374:
3372:
3369:
3367:
3364:
3360:
3357:
3356:
3355:
3352:
3351:
3349:
3347:
3343:
3337:
3334:
3332:
3331:DurbinâWatson
3329:
3327:
3321:
3319:
3316:
3314:
3313:DickeyâFuller
3311:
3310:
3308:
3304:
3298:
3295:
3293:
3290:
3288:
3287:Cointegration
3285:
3283:
3280:
3278:
3275:
3273:
3270:
3268:
3265:
3263:
3262:Decomposition
3260:
3259:
3257:
3253:
3250:
3248:
3244:
3234:
3231:
3230:
3229:
3226:
3225:
3224:
3221:
3217:
3214:
3213:
3212:
3209:
3207:
3204:
3202:
3199:
3197:
3194:
3192:
3189:
3187:
3184:
3182:
3179:
3177:
3174:
3173:
3171:
3169:
3165:
3159:
3156:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3133:Cohen's kappa
3131:
3130:
3128:
3126:
3122:
3118:
3114:
3110:
3106:
3102:
3097:
3093:
3079:
3076:
3074:
3071:
3069:
3066:
3064:
3061:
3060:
3058:
3056:
3052:
3046:
3042:
3038:
3032:
3030:
3027:
3026:
3024:
3022:
3018:
3012:
3009:
3007:
3004:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2986:Nonparametric
2984:
2982:
2979:
2978:
2976:
2972:
2966:
2963:
2961:
2958:
2956:
2953:
2951:
2948:
2947:
2945:
2943:
2939:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2915:
2913:
2910:
2909:
2907:
2905:
2901:
2895:
2892:
2890:
2887:
2885:
2882:
2880:
2877:
2876:
2874:
2872:
2868:
2864:
2857:
2854:
2852:
2849:
2848:
2844:
2840:
2824:
2821:
2820:
2819:
2816:
2814:
2811:
2809:
2806:
2802:
2799:
2797:
2794:
2793:
2792:
2789:
2788:
2786:
2784:
2780:
2770:
2767:
2763:
2757:
2755:
2749:
2747:
2741:
2740:
2739:
2736:
2735:Nonparametric
2733:
2731:
2725:
2721:
2718:
2717:
2716:
2710:
2706:
2705:Sample median
2703:
2702:
2701:
2698:
2697:
2695:
2693:
2689:
2681:
2678:
2676:
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2663:
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2658:
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2596:
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2587:
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2387:
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2367:
2364:
2363:
2361:
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2350:
2346:
2340:
2337:
2335:
2332:
2328:
2325:
2324:
2323:
2320:
2318:
2315:
2311:
2310:loss function
2308:
2307:
2306:
2303:
2299:
2296:
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2276:
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2264:
2262:
2259:
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2188:
2185:
2184:
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2178:
2175:
2173:
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2165:
2164:
2162:
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2152:
2148:
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2139:
2125:
2122:
2120:
2117:
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2112:
2110:
2107:
2106:
2104:
2102:
2098:
2092:
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2084:
2082:
2079:
2078:
2076:
2072:
2066:
2063:
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2058:
2056:
2053:
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2048:
2046:
2043:
2041:
2038:
2036:
2033:
2032:
2030:
2028:
2024:
2018:
2015:
2013:
2012:Questionnaire
2010:
2008:
2005:
2001:
1998:
1996:
1993:
1992:
1991:
1988:
1987:
1985:
1983:
1979:
1973:
1970:
1968:
1965:
1963:
1960:
1958:
1955:
1953:
1950:
1948:
1945:
1943:
1940:
1938:
1935:
1934:
1932:
1930:
1926:
1922:
1918:
1913:
1909:
1895:
1892:
1890:
1887:
1885:
1882:
1880:
1877:
1875:
1872:
1870:
1867:
1865:
1862:
1860:
1857:
1855:
1852:
1850:
1847:
1845:
1842:
1840:
1839:Control chart
1837:
1835:
1832:
1830:
1827:
1825:
1822:
1821:
1819:
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1813:
1807:
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1800:
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1428:(6): 83â124.
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1412:9780429982514
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1301:
1297:
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1272:9781475726060
1268:
1264:
1263:
1257:
1253:
1249:
1245:
1240:
1237:(3): 206â210.
1236:
1232:
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1223:
1219:
1215:
1211:
1207:
1203:
1199:
1194:
1190:
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1179:9780521343060
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1123:
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1111:
1106:
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1080:
1076:
1071:
1067:
1063:
1059:
1055:
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1038:
1036:9783662049785
1032:
1028:
1027:
1021:
1006:
1002:
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980:9780387955698
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957:
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923:
919:
915:
910:
905:
901:
897:
893:
888:
884:
880:
876:
872:
868:
864:
863:
858:
853:
850:
848:
844:
841:
840:
836:
824:
823:
815:
812:
808:
802:
799:
793:
790:
787:
784:Vitanyi, P. "
781:
778:
775:
770:
767:
762:
756:
752:
751:
743:
740:
737:
732:
730:
726:
720:
717:
704:
700:
694:
691:
688:
683:
681:
677:
670:
664:
661:
658:
655:
649:
646:
643:
640:
638:
635:
632:
629:
626:
623:
620:
617:
615:
612:
609:
606:
603:
600:
597:
594:
588:
585:
582:
579:
576:
573:
570:
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558:
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552:
550:
548:
544:
539:
535:
531:
529:
525:
520:
518:
514:
510:
505:
502:
497:
494:
489:
486:
479:
477:
460:
457:
454:
442:
440:
435:
431:
427:
423:
419:
415:
412: â
411:
407:
403:
399:
395:
389:
384:
381:
375:
367:
365:
363:
359:
355:
351:
347:
342:
340:
336:
332:
328:
324:
320:
312:
310:
308:
304:
300:
296:
292:
288:
284:
280:
277:
276:axiomatizable
274:
270:
266:
262:
259:
255:
250:
248:
244:
240:
236:
232:
228:
224:
223:random string
220:
216:
211:
207:
205:
201:
197:
192:
190:
186:
182:
178:
174:
166:
164:
162:
158:
154:
150:
146:
142:
138:
134:
130:
126:
122:
118:
114:
110:
106:
101:
98:
94:
88:
86:
82:
78:
74:
70:
66:
62:
59:or any other
58:
54:
50:
46:
42:
38:
34:
30:
19:
3784:
3772:
3753:
3746:
3658:Econometrics
3608: /
3591:Chemometrics
3568:Epidemiology
3561: /
3534:Applications
3376:ARIMA model
3323:Q-statistic
3272:Stationarity
3168:Multivariate
3111: /
3107: /
3105:Multivariate
3103: /
3043: /
3039: /
2813:Bayes factor
2712:Signed rank
2624:
2598:
2590:
2578:
2273:Completeness
2109:Cohort study
2007:Opinion poll
1942:Missing data
1929:Study design
1884:Scatter plot
1806:Scatter plot
1799:Spearman's Ï
1761:Grouped data
1458:cite journal
1425:
1421:
1402:
1373:
1369:
1346:. Springer.
1343:
1326:
1322:
1303:
1299:
1282:
1261:
1251:
1247:
1234:
1230:
1201:
1188:
1167:
1151:(4): 350â9.
1148:
1144:
1117:
1113:
1082:
1078:
1049:
1045:
1025:
1012:. Retrieved
1005:the original
1000:
996:
969:
943:
939:
930:
926:
899:
895:
874:
870:
847:Scholarpedia
821:
814:
801:
792:
780:
769:
753:. Springer.
749:
742:
719:
707:. Retrieved
703:the original
693:
687:Chaitin 1975
596:Epistemology
540:
536:
532:
523:
521:
516:
512:
511:of a string
506:
503:
498:
495:
490:
487:
483:
443:
434:1-randomness
433:
425:
413:
409:
405:
397:
393:
391:
377:
350:Leonid Levin
343:
327:Bayes' rules
316:
298:
294:
286:
282:
268:
260:
251:
212:
208:
193:
175:measures on
170:
102:
89:
36:
32:
31:
29:
3786:WikiProject
3701:Cartography
3663:Jurimetrics
3615:Reliability
3346:Time domain
3325:(LjungâBox)
3247:Time-series
3125:Categorical
3109:Time-series
3101:Categorical
3036:(Bernoulli)
2871:Correlation
2851:Correlation
2647:JarqueâBera
2619:Chi-squared
2381:M-estimator
2334:Asymptotics
2278:Sufficiency
2045:Interaction
1957:Replication
1937:Effect size
1894:Violin plot
1874:Radar chart
1854:Forest plot
1844:Correlogram
1794:Kendall's Ï
1306:(1): 1â22.
940:Cybernetics
933:(3): 19â23.
736:Calude 2013
439:Yongge Wang
358:Blum axioms
49:information
45:computation
3802:Categories
3653:Demography
3371:ARMA model
3176:Regression
2753:(Friedman)
2714:(Wilcoxon)
2652:Normality
2642:Lilliefors
2589:Student's
2465:Resampling
2339:Robustness
2327:divergence
2317:Efficiency
2255:(monotone)
2250:Likelihood
2167:Population
2000:Stratified
1952:Population
1771:Dependence
1727:Count data
1658:Percentile
1635:Dispersion
1568:Arithmetic
1503:Statistics
1254:: 522â526.
1191:(1): 3â11.
671:References
426:Martin-Löf
396:, for all
287:unknowable
273:consistent
239:likelihood
231:intuitions
196:compressed
179:(or other
173:complexity
149:stationary
147:, or even
129:randomness
107:(or other
3034:Logistic
2801:posterior
2727:Rank sum
2475:Jackknife
2470:Bootstrap
2288:Bootstrap
2223:Parameter
2172:Statistic
1967:Statistic
1879:Run chart
1864:Pie chart
1859:Histogram
1849:Fan chart
1824:Bar chart
1706:L-moments
1593:Geometric
1450:250850390
1390:250348327
1066:207698337
960:121736453
145:Markovian
3748:Category
3441:Survival
3318:Johansen
3041:Binomial
2996:Isotonic
2583:(normal)
2228:location
2035:Blocking
1990:Sampling
1869:QâQ plot
1834:Box plot
1816:Graphics
1711:Skewness
1701:Kurtosis
1673:Variance
1603:Heronian
1598:Harmonic
1218:11402549
1136:14133389
1101:12584692
1014:June 30,
918:15710280
867:Blum, M.
553:See also
352:(1974).
219:rigorous
189:integers
167:Overview
117:integers
3774:Commons
3721:Kriging
3606:Process
3563:studies
3422:Wavelet
3255:General
2422:Plug-in
2216:L space
1995:Cluster
1696:Moments
1514:Outline
1430:Bibcode
418:measure
331:Caltech
313:History
200:program
177:strings
105:strings
97:entropy
73:Shannon
57:strings
3643:Census
3233:Normal
3181:Manova
3001:Robust
2751:2-way
2743:1-way
2581:-test
2252:
1829:Biplot
1620:Median
1613:Lehmer
1555:Center
1448:
1409:
1388:
1350:
1269:
1216:
1176:
1134:
1099:
1064:
1033:
997:J. UCS
977:
958:
916:
757:
709:May 3,
400:, the
307:normal
279:theory
233:about
225:and a
215:formal
159:, and
81:Turing
3267:Trend
2796:prior
2738:anova
2627:-test
2601:-test
2593:-test
2500:Power
2445:Pivot
2238:shape
2233:scale
1683:Shape
1663:Range
1608:Heinz
1583:Cubic
1519:Index
1446:S2CID
1386:S2CID
1366:(PDF)
1287:(PDF)
1214:S2CID
1132:S2CID
1097:S2CID
1062:S2CID
1008:(PDF)
993:(PDF)
956:S2CID
914:S2CID
826:(PDF)
3500:Test
2700:Sign
2552:Wald
1625:Mode
1563:Mean
1464:link
1407:ISBN
1348:ISBN
1267:ISBN
1174:ISBN
1031:ISBN
1016:2019
975:ISBN
755:ISBN
711:2010
265:halt
79:and
47:and
2680:BIC
2675:AIC
1438:doi
1426:256
1378:doi
1331:doi
1308:doi
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1153:doi
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.