# Completeness (statistics)

**completecompletenessboundedly completecomplete class theoremscomplete statistic**

In statistics, completeness is a property of a statistic in relation to a model for a set of observed data.wikipedia

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### Statistic

**sample statisticempiricalmeasure**

In statistics, completeness is a property of a statistic in relation to a model for a set of observed data. Say T is statistic; that is, the composition of a measurable function with a random sample X 1,...,X n.

Important potential properties of statistics include completeness, consistency, sufficiency, unbiasedness, minimum mean square error, low variance, robustness, and computational convenience.

### Normal distribution

**normally distributednormalGaussian**

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

The statistic is complete and sufficient for μ, and therefore by the Lehmann–Scheffé theorem, is the uniformly minimum variance unbiased (UMVU) estimator.

### Basu's theorem

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

In statistics, Basu's theorem states that any boundedly complete sufficient statistic is independent of any ancillary statistic.

### Lehmann–Scheffé theorem

Completeness occurs in the Lehmann–Scheffé theorem,

The theorem states that any estimator which is unbiased for a given unknown quantity and that depends on the data only through a complete, sufficient statistic is the unique best unbiased estimator of that quantity.

### Minimum-variance unbiased estimator

**best unbiased estimatorminimum variance unbiased estimatoruniformly minimum variance unbiased**

See also minimum-variance unbiased estimator.

Using the Rao–Blackwell theorem one can also prove that determining the MVUE is simply a matter of finding a complete sufficient statistic for the family and conditioning any unbiased estimator on it.

### Sufficient statistic

**sufficient statisticssufficientsufficiency**

Bounded completeness occurs in Basu's theorem, which states that a statistic that is both boundedly complete and sufficient is independent of any ancillary statistic. 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. In the case where there exists at least one minimal sufficient statistic, a statistic which is sufficient and boundedly complete, is necessarily minimal sufficient.

If there exists a minimal sufficient statistic, and this is usually the case, then every complete sufficient statistic is necessarily minimal sufficient (note that this statement does not exclude the option of a pathological case in which a complete sufficient exists while there is no minimal sufficient statistic).

### Statistics

**statisticalstatistical analysisstatistician**

In statistics, completeness is a property of a statistic in relation to a model for a set of observed data.

### Identifiability

**identifiablenonidentifiableidentifiability condition**

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.

### Statistical theory

**statisticalstatistical theoriesmathematical statistics**

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.

### Random variable

**random variablesrandom variationrandom**

Consider a random variable X whose probability distribution belongs to a parametric model P θ parametrized by θ.

### Parametric model

**parametricregular parametric modelparameter**

Consider a random variable X whose probability distribution belongs to a parametric model P θ parametrized by θ.

### Measurable function

**measurableLebesgue measurableΣ-measurable**

Say T is statistic; that is, the composition of a measurable function with a random sample X 1,...,X n.

### Sampling (statistics)

**samplingrandom samplesample**

Let X be a random sample of size n such that each X i has the same Bernoulli distribution with parameter p.

### Bernoulli distribution

**BernoulliBernoulli random variableBernoulli random variables**

Let X be a random sample of size n such that each X i has the same Bernoulli distribution with parameter p.

### Binomial distribution

**binomialbinomial probability distributionbinomial random variable**

T is a statistic of X which has a binomial distribution with parameters (n,p). If the parameter space for p is (0,1), then T is a complete statistic.

### Positive real numbers

**positive realspositive real axislogarithmic measure**

First, observe that the range of r is the positive reals.

### Polynomial

**polynomial functionpolynomialsmultivariate polynomial**

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.

### Independence (probability theory)

**independentstatistically independentindependence**

Bounded completeness occurs in Basu's theorem, which states that a statistic that is both boundedly complete and sufficient is independent of any ancillary statistic. Suppose (X 1, X 2 ) are independent, identically distributed random variables, normally distributed with expectation θ and variance 1.

### Laplace transform

**Laplaces-domainℒ**

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.

### Raghu Raj Bahadur

**BahadurBahadur, Raghu Raj**

(A case in which there is no minimal sufficient statistic was shown by Bahadur in 1957.

### Convex function

**convexconvexitystrictly convex**

In other words, this statistic has a smaller expected loss for any convex loss function; in many practical applications with the squared loss-function, it has a smaller mean squared error among any estimators with the same expected value.

### Expected value

**expectationexpectedmean**

In other words, this statistic has a smaller expected loss for any convex loss function; in many practical applications with the squared loss-function, it has a smaller mean squared error among any estimators with the same expected value.

### Ancillary statistic

**conditions on ancillary information**

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

### Necessity and sufficiency

**necessary conditionnecessary and sufficient conditionsufficient condition**

In the case where there exists at least one minimal sufficient statistic, a statistic which is sufficient and boundedly complete, is necessarily minimal sufficient.

### List of statistics articles

**list of statistical topicslist of statistics topics**

Completeness (statistics)