Maximum spacing estimation: Difference between revisions

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The MSE method was derived independently by Russel Cheng and Nik Amin at the [[Cardiff University|University of Wales Institute of Science and Technology]], and Bo Ranneby at the [[Swedish University of Agricultural Sciences]].<ref name="R84" /> The authors explained that due to the [[probability integral transform]] at the true parameter, the “spacing” between each observation should be uniformly distributed. This would imply that the difference between the values of the [[cumulative distribution function]] at consecutive observations should be equal. This is the case that maximizes the [[geometric mean]] of such spacings, so solving for the parameters that maximize the geometric mean would achieve the “best” fit as defined this way. {{harvtxt|Ranneby|1984}} justified the method by demonstrating that it is an estimator of the [[Kullback–Leibler divergence]], similar to [[maximum likelihood estimation]], but with more robust properties for some classes of problems.

There are certain distributions, especially those with three or more parameters, whose [[Likelihood#LikelihoodsRelationship forbetween continuousthe distributionslikelihood and probability density functions|likelihoods]] may become infinite along certain paths in the [[parameter space]]. Using maximum likelihood to estimate these parameters often breaks down, with one parameter tending to the specific value that causes the likelihood to be infinite, rendering the other parameters inconsistent. The method of maximum spacings, however, being dependent on the difference between points on the cumulative distribution function and not individual likelihood points, does not have this issue, and will return valid results over a much wider array of distributions.<ref name="CA83" />

The distributions that tend to have likelihood issues are often those used to model physical phenomena. {{harvtxt|Hall|al.|2004}} seek to analyze flood alleviation methods, which requires accurate models of river flood effects. The distributions that better model these effects are all three-parameter models, which suffer from the infinite likelihood issue described above, leading to Hall's investigation of the maximum spacing procedure. {{harvtxt|Wong|Li|2006}}, when comparing the method to maximum likelihood, use various data sets ranging from a set on the oldest ages at death in Sweden between 1905 and 1958 to a set containing annual maximum wind speeds.

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Calculating the geometric mean and then taking the logarithm, statistic ''S''_''n'' will be equal to

<math display="block">

S_n(a,b) = \tfrac{1}{n+1}\ln(x_{(1)}-a)}{n+1} + \tfrac{\sum_{i=2}^n \ln(x_{(i)}-x_{(i-1)}) + \tfrac{1}{n+1} + \tfrac{\ln(b-x_{(n)})}{n+1} - \ln(b-a)

</math>

Here only three terms depend on the parameters ''a'' and ''b''. Differentiating with respect to those parameters and solving the resulting linear system, the maximum spacing estimates will be

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The distribution can also be approximated by that of <math>A</math>, where

<math display="block"> A = C_1 + C_2\chi^2_n \,, </math>

A = C_1 + C_2\chi^2_n \,

</math>,

in which

<math display="block">\begin{align}

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C_2 &= {\sqrt\frac{\sigma^2_M}{2n}},\\

\end{align}</math>

and where <math>\chi^2_n</math> follows a [[chi-squared distribution]] with <math>n</math> [[Degrees of freedom (statistics)|degrees of freedom]]. Therefore, to test the hypothesis <math>H_0</math> that a random sample of <math>n</math> values comes from the distribution <math>F(x,\theta)</math>, the statistic <math>T(\theta)= \frac{M(\theta)-C_1}{C_2}</math> can be calculated. Then <math>H_0</math> should be rejected with [[Statistical significance|significance]] <math>\alpha</math> if the value is greater than the [[critical value (statistics)|critical value]] of the appropriate chi-squared distribution.<ref name="CS89" />

Where ''θ''₀ is being estimated by <math>\hat\theta</math>, {{harvtxt|Cheng|Stephens|1989}} showed that <math>S_n(\hat\theta) = M_n(\hat\theta)</math> has the same asymptotic mean and variance as in the known case. However, the test statistic to be used requires the addition of a bias correction term and is:

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{{refbegin}}

* {{cite journal

| last1 = Anatolyev
| first1 = Stanislav

| last2 = Kosenok
| first2 = Grigory

| year = 2005

| title = An alternative to maximum likelihood based on spacings

| journal = Econometric Theory

| volume = 21
| issue = 2
| pages = 472–476

| doi = 10.1017/S0266466605050255

| url = http://fir.nes.ru/~gkosenok/MPS.pdf

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| citeseerx = 10.1.1.494.7340

| s2cid = 123004317

| accessarchive-date = 20082011-1208-3116

| archive-url = https://web.archive.org/web/20110816101736/http://fir.nes.ru/~gkosenok/MPS.pdf

| url-status = dead

}}

* {{cite journal

| last1 = Beirlant | first1 = J.Beirlant

| last2 = Dudewicz first1 | first2 = E.J.

| last3 = Györfi last2 | first3 = L.Dudewicz

|first2 last4 = van der Meulen | first4 = E.CJ.

|last3 year = 1997Györfi

|first3 = L.

| title = Nonparametric entropy estimation: an overview

|last4 = van der Meulen

| journal = International Journal of Mathematical and Statistical Sciences

|first4 volume = 6 | issue = 1 | pages = 17–40E.C.

| year = 20041997

| issn = 1055-7490

| title = Nonparametric entropy estimation: an overview

| url = http://www.menem.com/ilya/digital_library/entropy/beirlant_etal_97.pdf

| journal = International Journal of Mathematical and Statistical Sciences

| access-date = 2008-12-31

|volume = 6

| ref = CITEREFBeirlantal.2001

|issue = 1

|archive-url = https://web.archive.org/web/20050505044534/http://www.menem.com/ilya/digital_library/entropy/beirlant_etal_97.pdf |archive-date = May 5, 2005}} <small>''Note: linked paper is an updated 2001 version.''</small>

|pages = 17–40

| issn = 1055-7490

| url = http://www.menem.com/ilya/digital_library/entropy/beirlant_etal_97.pdf

| access-date = 2008-12-31

| ref = CITEREFBeirlantal.2001

|archive-url = https://web.archive.org/web/20050505044534/http://www.menem.com/ilya/digital_library/entropy/beirlant_etal_97.pdf |archive-date = May 5, 2005}} <small>''Note: linked paper is an updated 2001 version.''</small>

|archive-date = May 5, 2005

}} <small>''Note: linked paper is an updated 2001 version.''</small>

* {{cite journal

| last1 = Cheng | first1 = R.C.H.

Line 222 ⟶ 235:

}}

* {{cite journal

| last = Ekström

| first = Magnus

| year = 1997

| title = Generalized maximum spacing estimates

| journal = University of Umeå, Department of Mathematics

Line 231 ⟶ 244:

| url = http://www.matstat.umu.se/varia/reports/rep9706.ps.gz

| access-date = 2008-12-30

| archive-url = https://web.archive.org/web/20070214143052/http://www.matstat.umu.se/varia/reports/rep9706.ps.gz
| archive-date = February 14, 2007}}

* {{cite journal

| last1 = Hall | first1 = M.J.

| last2 = van den Boogaard | first2 = H.F.P.

| last3 = Fernando | first3 = R.C.

| last4 = Mynett | first4 = A.E.

| year = 2004

| title = The construction of confidence intervals for frequency analysis using resampling techniques

| journal = Hydrology and Earth System Sciences

| volume = 8 | issue = 2 | pages = 235–246

| issn = 1027-5606

| ref = CITEREFHallal.2004 | doi=10.5194/hess-8-235-2004

| url = https://hal.archives-ouvertes.fr/hal-00304907/document

| doi-access = free

}}

* {{cite journal

|last1 = Hall

|first1 = M.J.

| last2 = van den Boogaard | first2 = H.F.P.

|first2 = H.F.P.

|last3 = Fernando

|first3 = R.C.

|last4 = Mynett

|first4 = A.E.

|year = 2004

| title = The construction of confidence intervals for frequency analysis using resampling techniques

| journal = Hydrology and Earth System Sciences

|volume = 8

|issue = 2

|pages = 235–246

| issn = 1027-5606

|ref = CITEREFHallal.2004

|doi ref = CITEREFHallal.2004 | doi= 10.5194/hess-8-235-2004

| url = https://hal.archives-ouvertes.fr/hal-00304907/document

| doi-access = free

}}

* {{cite conference

| last1 = Pieciak
| first1 = Tomasz

| year = 2014

| title = The maximum spacing noise estimation in single-coil background MRI data

| conference = IEEE International Conference on Image Processing

| pages = 1743–1747

| ___location = Paris

| doi = 10.1109/icip.2014.7025349

| url = http://home.agh.edu.pl/pieciak/publikacje_pieciak/2014_ICIP_Pieciak.pdf

| url = https://scholar.archive.org/work/e2l3rb6s3va7pd3kf6oioymgza

| access-date = 2015-07-07

}}

* {{cite journal

Line 277 ⟶ 300:

}}

* {{cite journal

| last1 = Ranneby
| first1 = Bo

| last2 = Ekström
| first2 = Magnus

| year = 1997

| title = Maximum spacing estimates based on different metrics

| journal = University of Umeå, Department of Mathematics

| volume = 5

| issn = 0345-3928

| url = http://www.matstat.umu.se/varia/reports/rep9705.ps.gz

| access-date = 2008-12-30

|archive-url = https://web.archive.org/web/20070214143042/http://www.matstat.umu.se/varia/reports/rep9705.ps.gz
|archive-date = February 14, 2007
}}

* {{cite journal

| last1 = Ranneby | first1 = BoRanneby

|first1 last2 = Jammalamadakab | first2 = S. RaoBo

|last2 = Jammalamadakab

| last3 = Teterukovskiy | first3 = Alex

|first2 year = 2005S. Rao

|last3 = Teterukovskiy

| title = The maximum spacing estimation for multivariate observations

|first3 = Alex

| journal = Journal of Statistical Planning and Inference

|year volume = 129 | issue = 1–2 | pages = 427–4462005

| title = The maximum spacing estimation for multivariate observations

| doi = 10.1016/j.jspi.2004.06.059

| journal = Journal of Statistical Planning and Inference

| url = http://www.pstat.ucsb.edu/faculty/jammalam/html/research%20publication_files/MSP2.pdf

|volume = 129

| access-date = 2008-12-31

|issue = 1–2

| ref = CITEREFRannebyal.2005

|pages = 427–446

}}

| doi = 10.1016/j.jspi.2004.06.059

| url = http://www.pstat.ucsb.edu/faculty/jammalam/html/research%20publication_files/MSP2.pdf

| access-date = 20152008-0712-0731

| ref = CITEREFRannebyal.2005

}}

* {{cite book

| last1 = Wong | first1 = T.S.T