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V-statistics are a class of statistics named for Richard von Mises who developed their asymptotic distribution theory in a fundamental paper in 1947.^[1] V-statistics are closely related to U-statistics^[2]^[3] (U for “unbiased”) introduced by Wassily Hoeffding in 1948.^[4] A V-statistic is a statistical function (of a sample) defined by a particular statistical functional of a probability distribution.

Statistical functions

Statistics that can be represented as functionals $T (F_{n})$ of the empirical distribution function $(F_{n})$ are called statistical functions.^[5] Differentiability of the functional T plays a key role in the von Mises approach; thus von Mises considers differentiable statistical functionals.^[1]

Examples of statistical functions

The k-th central moment is the functional $T (F) = \int (x - μ)^{k} d F (x)$ , where $μ = E [X]$ is the expected value of X. The associated statistical function is the sample k-th central moment,
$T_{n} = m_{k} = T (F_{n}) = \frac{1}{n} \sum_{i = 1}^{n} (x_{i} - \overline{x})^{k} .$
The chi-squared goodness-of-fit statistic is a statistical function T(F_n), corresponding to the statistical functional
$T (F) = \sum_{i = 1}^{k} \frac{(\int_{A_{i}} d F - p_{i})^{2}}{p_{i}},$
where A_i are the k cells and p_i are the specified probabilities of the cells under the null hypothesis.
The Cramér–von-Mises and Anderson–Darling goodness-of-fit statistics are based on the functional
$T (F) = \int (F (x) - F_{0} (x))^{2} w (x; F_{0}) d F_{0} (x),$
where w(x; F₀) is a specified weight function and F₀ is a specified null distribution. If w is the identity function then T(F_n) is the well known Cramér–von-Mises goodness-of-fit statistic; if $w (x; F_{0}) = [F_{0} (x) (1 - F_{0} (x))]^{- 1}$ then T(F_n) is the Anderson–Darling statistic.

Representation as a V-statistic

Suppose x₁, ..., x_n is a sample. In typical applications the statistical function has a representation as the V-statistic

V_{m n} = \frac{1}{n^{m}} \sum_{i_{1} = 1}^{n} \dots \sum_{i_{m} = 1}^{n} h (x_{i_{1}}, x_{i_{2}}, \dots, x_{i_{m}}),

where h is a symmetric kernel function. Serfling^[6] discusses how to find the kernel in practice. V_mn is called a V-statistic of degree m.

A symmetric kernel of degree 2 is a function h(x, y), such that h(x, y) = h(y, x) for all x and y in the domain of h. For samples x₁, ..., x_n, the corresponding V-statistic is defined

V_{2, n} = \frac{1}{n^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} h (x_{i}, x_{j}) .

Example of a V-statistic

An example of a degree-2 V-statistic is the second central moment m₂. If h(x, y) = (x − y)²/2, the corresponding V-statistic is
$V_{2, n} = \frac{1}{n^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} \frac{1}{2} (x_{i} - x_{j})^{2} = \frac{1}{n} \sum_{i = 1}^{n} (x_{i} - \bar{x})^{2},$
which is the maximum likelihood estimator of variance. With the same kernel, the corresponding U-statistic is the (unbiased) sample variance:
$s^{2} = {(\binom{n}{2})}^{- 1} \sum_{i < j} \frac{1}{2} (x_{i} - x_{j})^{2} = \frac{1}{n - 1} \sum_{i = 1}^{n} (x_{i} - \bar{x})^{2}$ .

Asymptotic distribution

In examples 1–3, the asymptotic distribution of the statistic is different: in (1) it is normal, in (2) it is chi-squared, and in (3) it is a weighted sum of chi-squared variables.

Von Mises' approach is a unifying theory that covers all of the cases above.^[1] Informally, the type of asymptotic distribution of a statistical function depends on the order of "degeneracy," which is determined by which term is the first non-vanishing term in the Taylor expansion of the functional T. In case it is the linear term, the limit distribution is normal; otherwise higher order types of distributions arise (under suitable conditions such that a central limit theorem holds).

There are a hierarchy of cases parallel to asymptotic theory of U-statistics.^[7] Let A(m) be the property defined by:

A(m):

Var(h(X₁, ..., X_k)) = 0 for k < m, and Var(h(X₁, ..., X_k)) > 0 for k = m;
n^m/2R_mn tends to zero (in probability). (R_mn is the remainder term in the Taylor series for T.)

Case m = 1 (Non-degenerate kernel):

If A(1) is true, the statistic is a sample mean and the Central Limit Theorem implies that T(F_n) is asymptotically normal.

In the variance example (4), m₂ is asymptotically normal with mean $σ^{2}$ and variance $(μ_{4} - σ^{4}) / n$ , where $μ_{4} = E (X - E (X))^{4}$ .

Case m = 2 (Degenerate kernel):

Suppose A(2) is true, and $E [h^{2} (X_{1}, X_{2})] < \infty, E | h (X_{1}, X_{1}) | < \infty,$ and $E [h (x, X_{1})] \equiv 0$ . Then nV_2,n converges in distribution to a weighted sum of independent chi-squared variables:

n V_{2, n} \overset{d}{⟶} \sum_{k = 1}^{\infty} λ_{k} Z_{k}^{2},

where $Z_{k}$ are independent standard normal variables and $λ_{k}$ are constants that depend on the distribution F and the functional T. In this case the asymptotic distribution is called a quadratic form of centered Gaussian random variables. The statistic V_2,n is called a degenerate kernel V-statistic. The V-statistic associated with the Cramer–von Mises functional^[1] (Example 3) is an example of a degenerate kernel V-statistic.^[8]

Notes

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References

Template:Refbegin

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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

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Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

Template:Refend

↑ ^1.0 ^1.1 ^1.2 ^1.3 Template:Harvtxt
↑ Template:Harvtxt
↑ Template:Harvtxt
↑ Template:Harvtxt
↑ von Mises (1947), p. 309; Serfling (1980), p. 210.
↑ Serfling (1980, Section 6.5)
↑ Serfling (1980, Ch. 5–6); Lee (1990, Ch. 3)
↑ See Lee (1990, p. 160) for the kernel function.

[VM-1] 1.0 ^1.1 ^1.2 ^1.3 Template:Harvtxt

[2] Template:Harvtxt

[3] Template:Harvtxt

[4] Template:Harvtxt

[5] von Mises (1947), p. 309; Serfling (1980), p. 210.

[Serfling.a-6] Serfling (1980, Section 6.5)

[7] Serfling (1980, Ch. 5–6); Lee (1990, Ch. 3)

[8] See Lee (1990, p. 160) for the kernel function.

[1]

[2]

[3]

[4]

[5]

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[7]

[8]

@@ Line 1: / Line 1: @@
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+'''V-statistics''' are a class of statistics named for [[Richard von Mises]] who developed their [[asymptotic theory|asymptotic distribution theory]] in a fundamental paper in 1947.<ref name=VM>{{harvtxt|von Mises|1947}}</ref> V-statistics are closely related to [[U-statistic]]s<ref>{{harvtxt|Lee|1990}}</ref><ref>{{harvtxt|Koroljuk|Borovskich|1994}}</ref> (U for “[[Bias of an estimator|unbiased]]”) introduced by [[Wassily Hoeffding]] in 1948.<ref>{{harvtxt|Hoeffding|1948}}</ref>  A V-statistic is a statistical function (of a sample) defined by a particular statistical functional of a probability distribution.
+== Statistical functions ==
+Statistics that can be represented as functionals <math>T(F_n)</math> of the [[empirical distribution function]] <math>(F_n)</math> are called ''statistical functions''.<ref>von Mises (1947), p. 309; Serfling (1980), p. 210.</ref> [[Differentiable function|Differentiability]] of the functional ''T'' plays a key role in the von Mises approach; thus von Mises considers ''differentiable statistical functionals''.<ref name=VM/>
+=== Examples of statistical functions ===
+<ol>
+<li>
+The ''k''-th [[central moment]] is the ''functional'' <math>T(F)=\int(x-\mu)^k \, dF(x)</math>, where <math>\mu = E[X]</math> is the [[expected value]] of ''X''. The associated ''statistical function'' is the sample ''k''-th central moment,
+:<math>
+T_n=m_k=T(F_n) = \frac 1n \sum_{i=1}^n (x_i - \overline x)^k.
+</math>
+</li>
+<li>
+The [[Pearson's chi-squared test|chi-squared goodness-of-fit]] statistic is a statistical function ''T''(''F''<sub>''n''</sub>), corresponding to the statistical functional
+:<math>
+T(F) = \sum_{i=1}^k \frac{(\int_{A_i} \, dF - p_i)^2}{p_i},
+</math>
+where ''A''<sub>''i''</sub> are the ''k'' cells and ''p''<sub>''i''</sub> are the specified probabilities of the cells under the null hypothesis.
+</li>
+<li>
+The [[Cramér–von-Mises criterion|Cramér–von-Mises]] and [[Anderson–Darling]] goodness-of-fit statistics are based on the functional
+:<math>
+ T(F) = \int (F(x) - F_0(x))^2 \, w(x;F_0) \, dF_0(x),
+</math>
+where ''w''(''x'';&nbsp;''F''<sub>0</sub>) is a specified weight function and ''F''<sub>0</sub> is a specified null distribution. If ''w'' is the identity function then ''T''(''F''<sub>''n''</sub>) is the well known [[Cramér–von-Mises criterion|Cramér–von-Mises]] goodness-of-fit statistic; if <math>w(x;F_0)=[F_0(x)(1-F_0(x))]^{-1}</math> then ''T''(''F''<sub>''n''</sub>) is the [[Anderson–Darling]] statistic.
+</li>
+</ol>
+=== Representation as a V-statistic ===
+Suppose ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> is a sample. In typical applications the statistical function has a representation as the V-statistic
+:<math>
+V_{mn} = \frac{1}{n^m} \sum_{i_1=1}^n \cdots \sum_{i_m=1}^n h(x_{i_1}, x_{i_2}, \dots, x_{i_m}),
+</math>
+where ''h'' is a symmetric kernel function. Serfling<ref name=Serfling.a>Serfling (1980, Section 6.5)</ref> discusses how to find the kernel in practice.  ''V''<sub>''mn''</sub> is called a V-statistic of degree&nbsp;''m''.
+A symmetric kernel of degree 2 is a function ''h''(''x'',&nbsp;''y''), such that ''h''(''x'', ''y'') = ''h''(''y'', ''x'') for all ''x'' and ''y'' in the domain of h. For samples ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>, the corresponding V-statistic is defined
+:<math>
+  V_{2,n} = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n h(x_i, x_j).
+</math>
+=== Example of a V-statistic ===
+<ol start="4">
+<li>
+An example of a degree-2 V-statistic is the second [[central moment]] ''m''<sub>2</sub>.
+If ''h''(''x'', ''y'') = (''x'' &minus; ''y'')<sup>2</sup>/2, the corresponding V-statistic is
+:<math>
+V_{2,n} = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \frac{1}{2}(x_i - x_j)^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar x)^2,
+</math>
+which is the maximum likelihood estimator of [[Variance#Population variance and sample variance|variance]]. With the same kernel, the corresponding [[U-statistic]] is the (unbiased) sample variance:
+:<math>s^2=
+{n \choose 2}^{-1} \sum_{i < j} \frac{1}{2}(x_i - x_j)^2 =
+\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar x)^2</math>.
+</li>
+</ol>
+== Asymptotic distribution ==
+In examples 1–3, the [[asymptotic distribution]] of the statistic is different: in (1) it is [[Normal distribution|normal]], in (2) it is [[Chi-squared distribution|chi-squared]], and in (3) it is a weighted sum of chi-squared variables.
+Von Mises' approach is a unifying theory that covers all of the cases above.<ref name=VM/>  Informally, the type of [[asymptotic distribution]] of a statistical function depends on the order of "degeneracy," which is determined by which term is the first non-vanishing term in the [[Taylor series|Taylor expansion]] of the functional&nbsp;''T''. In case it is the linear term, the limit distribution is normal; otherwise higher order types of distributions arise (under suitable conditions such that a central limit theorem holds).
+There are a hierarchy of cases parallel to asymptotic theory of [[U-statistic]]s.<ref>Serfling (1980, Ch. 5–6); Lee (1990, Ch. 3)</ref> Let ''A''(''m'') be the property defined by:
+:''A''(''m''):
+<ol style="list-style-type:lower-roman">
+<li> Var(''h''(''X''<sub>1</sub>, ..., ''X''<sub>''k''</sub>)) = 0 for ''k'' < ''m'', and Var(''h''(''X''<sub>1</sub>, ..., ''X''<sub>''k''</sub>)) > 0  for ''k'' = ''m''; </li>
+<li> ''n''<sup>''m''/2</sup>''R''<sub>''mn''</sub> tends to zero (in probability). (''R''<sub>''mn''</sub> is the remainder term in the Taylor series for ''T''.)</li>
+</ol>
+'''Case ''m'' = 1''' (Non-degenerate kernel):
+If ''A''(1) is true, the statistic is a sample mean and the [[Central Limit Theorem]] implies that T(F<sub>n</sub>) is [[Asymptotic normality|asymptotically normal]].
+In the variance example (4), m<sub>2</sub> is asymptotically normal with mean <math>\sigma^2</math> and variance <math>(\mu_4 - \sigma^4)/n</math>, where <math>\mu_4=E(X-E(X))^4</math>.
+'''Case ''m'' = 2''' (Degenerate kernel):
+Suppose ''A''(2) is true, and <math>E[h^2(X_1,X_2)]<\infty, \, E|h(X_1,X_1)|<\infty, </math> and <math> E[h(x,X_1)]\equiv 0</math>. Then nV<sub>2,n</sub> converges in distribution to a weighted sum of independent chi-squared variables:
+:<math> n V_{2,n} {\stackrel d \longrightarrow} \sum_{k=1}^\infty \lambda_k Z^2_k,</math>
+where <math>Z_k</math> are independent [[standard normal]] variables and <math>\lambda_k</math> are constants that depend on the distribution ''F'' and the functional ''T''. In this case the [[asymptotic distribution]] is called a ''quadratic form of centered Gaussian random variables''. The statistic ''V''<sub>2,''n''</sub> is called a ''degenerate kernel V-statistic''. The V-statistic associated with the Cramer–von Mises functional<ref name=VM/> (Example 3) is an example of a degenerate kernel V-statistic.<ref>See Lee (1990, p. 160) for the kernel function.</ref>
+== See also ==
+* [[U-statistic]]
+* [[Asymptotic distribution]]
+* [[Asymptotic theory (statistics)]]
+== Notes ==
+{{Reflist}}
+== References ==
+{{refbegin}}
+* {{cite journal
+  | last = Hoeffding | first = W.
+  | year = 1948
+  | title = A class of statistics with asymptotically normal distribution
+  | journal = Annals of Mathematical Statistics
+  | volume = 19 | issue =   3
+  | pages = 293–325
+  | jstor = 2235637
+  | ref = harv
+  }}
+* {{cite book
+  | last1 = Koroljuk   | first1 = V.S.
+  | last2 = Borovskich | first2 = Yu.V.
+  | year = 1994
+  | title = Theory of ''U''-statistics
+  | edition = English translation by P.V.Malyshev and D.V.Malyshev from the 1989 Ukrainian
+  | publisher = Kluwer Academic Publishers | location = Dordrecht
+  | isbn = 0-7923-2608-3
+  | ref = harv
+  }}
+* {{cite book
+  | last = Lee | first = A.J.
+  | year = 1990
+  | title = ''U''-Statistics: theory and practice
+  | publisher = Marcel Dekker, Inc. | location = New York
+  | isbn = 0-8247-8253-4
+  | ref = harv
+  }}
+* {{cite journal
+  | last = Neuhaus | first = G.
+  | year = 1977
+  | title = Functional limit theorems for ''U''-statistics in the degenerate case
+  | journal = Journal of Multivariate Analysis
+  | volume = 7 | issue = 3
+  | pages = 424–439
+  | doi = 10.1016/0047-259X(77)90083-5
+  | ref = harv
+  }}
+* {{cite journal
+  | last = Rosenblatt | first = M.
+  | year = 1952
+  | title = Limit theorems associated with variants of the von Mises statistic
+  | journal = Annals of Mathematical Statistics
+  | volume = 23 | issue = 4
+  | pages = 617–623
+  | jstor = 2236587
+  | ref = harv
+  }}
+* {{cite book
+  | last = Serfling | first = R.J.
+  | year = 1980
+  | title = Approximation theorems of mathematical statistics
+  | publisher = John Wiley &amp; Sons | location = New York
+  | isbn = 0-471-02403-1
+  | ref = harv
+  }}
+* {{cite book
+  | last1 = Taylor    | first1 = R.L.
+  | last2 = Daffer    | first2 = P.Z.
+  | last3 = Patterson | first3 = R.F.
+  | year = 1985
+  | title = Limit theorems for sums of exchangeable random variables
+  | publisher = Rowman and Allanheld | location = New Jersey
+  | ref = harv
+  }}
+* {{cite journal
+  | last = von Mises | first = R.
+  | year = 1947
+  | title = On the asymptotic distribution of differentiable statistical functions
+  | journal = Annals of Mathematical Statistics
+  | volume = 18 | issue = 2
+  | pages = 309–348
+  | jstor = 2235734
+  | ref = harv
+  }}
+{{refend}}
+[[Category:Estimation theory]]
+[[Category:Asymptotic statistical theory]]

Jurin's law: Difference between revisions

Latest revision as of 14:09, 23 June 2013

Contents

Statistical functions

Examples of statistical functions

Representation as a V-statistic

Example of a V-statistic

Asymptotic distribution

See also

Notes

References

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Jurin's law: Difference between revisions

Latest revision as of 14:09, 23 June 2013

Statistical functions

Examples of statistical functions

Representation as a V-statistic

Example of a V-statistic

Asymptotic distribution

See also

Notes

References

Navigation menu

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