Completeness (statistics)

Completeness (statistics)

In statistics, completeness is a property of a statistic in relation to a model for a set of observed data. In essence, it is a condition which ensures that the parameters of the probability distribution representing the model can all be estimated on the basis of the statistic: it ensures that the distributions corresponding to different values of the parameters are distinct.

It is closely related to the idea of identifiability, but in statistical theory it is often found as a condition imposed on a sufficient statistic from which certain optimality results are derived.



Consider a random variable X whose probability distribution belongs to a parametric family of probability distributions Pθ parametrized by θ.

Formally, a statistic s is a measurable function of X; thus, a statistic s is evaluated on a random variable X, taking the value s(X), which is itself a random variable. A given realization of the random variable X(ω) is a data-point (datum), on which the statistic s takes the value s(X(ω)).

The statistic s is said to be complete for the distribution of X if for every measurable function g the following implication holds:[citation needed]

E(g(s(X))) = 0 for all θ implies that Pθ(g(s(X)) = 0) = 1 for all θ.

The statistic s is said to be boundedly complete if the implication holds for all bounded functions g.

Example 1: Bernoulli model

The Bernoulli model admits a complete statistic.[1] Let X be a random sample of size n such that each Xi has the same Bernoulli distribution with parameter θ. Let T be the number of 1's observed in the sample. T is a statistic of X which has a Binomial distribution with parameters (n,θ). If the parameter space for θ is [0,1], then T is a complete statistic. To see this, note that

 \operatorname{E}(g(T)) = \sum_{t=1}^n {g(t){n \choose t}p^{t}(1-p)^{n-t}} = (1-p)^n \sum_{t=1}^n {g(t){n \choose t}\left(\frac{p}{1-p}\right)^t} .

Observe also that neither p nor 1 − p can be 0. Hence E(g(T)) = 0 if and only if:

\sum_{t=1}^n g(t){n \choose t}\left(\frac{p}{1-p}\right)^t = 0

One denoting p/(1 − p) by r one gets:

\sum_{t=1}^n g(t){n \choose t}r^t = 0 .

First, observe that the range of r is all reals except for 0. Also, E(g(T)) is a polynomial in r and, therefore, can only be identical to 0 if all coefficients are 0, that is, g(t) = 0 for all t.

It is important to notice that the result that all coefficients must be 0 was obtained because of the range of r. Had the parameter space been finite and with a number of elements smaller than n, it might be possible to solve the linear equations in g(t) obtained by substituting the values of r and get solutions different from 0. For example, if n = 1 and the parametric space is {0.5}, a single observation, T is not complete. Observe that, with the definition:

 g(t) = 2(t-0.5), \,

then, E(g(T)) = 0 although g(t) is not 0 for t = 0 nor for t = 1.

Example 2: Sum of normals

This example will show that, in a sample of size 2 from a normal distribution with known variance, the statistic X1+X2 is complete and sufficient. Suppose (X1, X2) are independent, identically distributed random variables, normally distributed with expectation θ and variance 1. The sum

s((X_1, X_2)) = X_1 + X_2\,\!

is a complete statistic for θ.[citation needed]

To show this, it is sufficient to demonstrate that there is no non-zero function g such that the expectation of

g(s(X_1, X_2)) = g(X_1+X_2)\,\!

remains zero regardless of the value of θ.

That fact may be seen as follows. The probability distribution of X1 + X2 is normal with expectation 2θ and variance 2. Its probability density function in x is therefore proportional to


The expectation of g above would therefore be a constant times

\int_{-\infty}^\infty g(x)\exp\left(-(x-2\theta)^2/4\right)\,dx.

A bit of algebra reduces this to

k(\theta) \int_{-\infty}^\infty h(x)e^{x\theta}\,dx\,\!

where k(θ) is nowhere zero and


As a function of θ this is a two-sided Laplace transform of h(X), and cannot be identically zero unless h(x) is zero almost everywhere.[citation needed] The exponential is not zero, so this can only happen if g(x) is zero almost everywhere.

Relation to sufficient statistics

For some parametric families, a complete sufficient statistic does not exist. Also, a minimal sufficient statistic need not exist. (A case in which there is no minimal sufficient statistic was shown by Bahadur 1957.[citation needed]) Under mild conditions, a minimal sufficient statistic does always exist. In particular, these conditions always hold if the random variables (associated with Pθ ) are all discrete or are all continuous.[citation needed]

Importance of completeness

The notion of completeness has many applications in statistics, particularly in the following two theorems of mathematical statistics.

Lehmann–Scheffé theorem

Completeness occurs in the Lehmann–Scheffé theorem,[citation needed] which states that if a statistic that is unbiased, complete and sufficient for some parameter θ, then it is the best mean-unbiased estimator for θ. In other words, this statistic has a smaller expected loss for any convex loss function; in many practice applications with the squared loss-function, it has a smaller mean squared error among any estimators with the same expected value.

See also minimum-variance unbiased estimator.

Basu's theorem

Bounded completeness occurs in Basu's theorem,[2] which states that a statistic which is both boundedly complete and sufficient is independent of any ancillary statistic.


  1. ^ Casella, G. and Berger, R. L. (2001). Statistical Inference. (pp. 285-286). Duxbury Press.
  2. ^ Casella, G. and Berger, R. L. (2001). Statistical Inference. (pp. 287). Duxbury Press.


  • Basu, D. (1988). J. K. Ghosh. ed. Statistical information and likelihood : A collection of critical essays by Dr. D. Basu. Lecture Notes in Statistics. 45. Springer. ISBN 0-387-96751-6. MR953081. 
  • Bickel, Peter J.; Doksum, Kjell A. (2001). Mathematical statistics, Volume 1: Basic and selected topics (Second (updated printing 2007) of the Holden-Day 1976 ed.). Pearson Prentice–Hall. ISBN 013850363X. MR443141. 
  • E. L., Lehmann; Romano, Joseph P. (2005). Testing statistical hypotheses. Springer Texts in Statistics (Third ed.). New York: Springer. pp. xiv+784. ISBN 0-387-98864-5. MR2135927. 
  • Lehmann, E.L.; Scheffé, H. (1950). "Completeness, similar regions, and unbiased estimation. I.". Sankhyā: the Indian Journal of Statistics 10 (4): 305–340. JSTOR 25048038. MR39201. 
  • Lehmann, E.L.; Scheffé, H. (1955). "Completeness, similar regions, and unbiased estimation. II". Sankhyā: the Indian Journal of Statistics 15 (3): 219–236. JSTOR 25048243. MR72410. 

Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • Completeness — In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields. Contents 1 Logical completeness 2 Mathematical completeness 3 Computing 4 …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • Official statistics — on Germany in 2010, published in UNECE Countries in Figures 2011. Official statistics are statistics published by government agencies or other public bodies such as international organizations. They provide quantitative or qualitative information …   Wikipedia

  • Church Statistics —     Ecclesiastical Statistics     † Catholic Encyclopedia ► Ecclesiastical Statistics     In dealing with statistics, both theoretically and practically, it is unimportant whether the men, matters, or actions subject to observation are… …   Catholic encyclopedia

  • Sufficiency (statistics) — In statistics, sufficiency is the property possessed by a statistic, with respect to a parameter, when no other statistic which can be calculated from the same sample provides any additional information as to the value of the parameter cite… …   Wikipedia

  • Glossary of probability and statistics — The following is a glossary of terms. It is not intended to be all inclusive. Concerned fields *Probability theory *Algebra of random variables (linear algebra) *Statistics *Measure theory *Estimation theory Glossary *Atomic event : another name… …   Wikipedia

  • List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia

  • Estimation theory — is a branch of statistics and signal processing that deals with estimating the values of parameters based on measured/empirical data. The parameters describe an underlying physical setting in such a way that the value of the parameters affects… …   Wikipedia

  • analysis — /euh nal euh sis/, n., pl. analyses / seez /. 1. the separating of any material or abstract entity into its constituent elements (opposed to synthesis). 2. this process as a method of studying the nature of something or of determining its… …   Universalium

  • dextro-Transposition of the great arteries — Classification and external resources ICD 10 Q20.3 ICD 9 745.10 …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”