* Gauss-Markov theorem

Traditionally, statistical methods have relied on mean-unbiased estimators of treatment effects: Under the conditions of the Gauss-Markov theorem, least squares estimators have minimum variance among all mean-unbiased estimators.

which leads to the generalised least squares version of the Gauss-Markov theorem ( Chiles & Delfiner 1999, p. 159 ):

Both theories derive a best linear unbiased estimator, based on assumptions on covariances, make use of Gauss-Markov theorem to prove independence of the estimate and error, and make use of very similar formulae.

If the experimental errors,, are uncorrelated, have a mean of zero and a constant variance,, the Gauss-Markov theorem states that the least-squares estimator,, has the minimum variance of all estimators that are linear combinations of the observations.

If the conditions of the Gauss-Markov theorem apply, the arithmetic mean is optimal, whatever the distribution of errors of the measurements might be.

Ordinary least squares ( OLS ) is often used for estimation since it provides the BLUE or " best linear unbiased estimator " ( where " best " means most efficient, unbiased estimator ) given the Gauss-Markov assumptions.

In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of the excluded middle ( unlike in Martin-Löf type theory, where it does not ).

The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the set of real numbers and the set of natural numbers do not have the same cardinal number.

This incompleteness result is similar to Gödel's incompleteness theorem in that it shows that no consistent formal theory for arithmetic can be complete.

As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers.

Animation illustrating Pythagorean theorem | Pythagoras ' rule for a right-angle triangle, which shows the algebraic relationship between the triangle's hypotenuse, and the other two sides.

The Bohr – van Leeuwen theorem shows that magnetism cannot occur in purely classical solids.

Gödel's incompleteness theorem, another celebrated result, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics.

Therefore, just as Bayes ' theorem shows, the result of each trial comes down to the base probability of the fair coin:.

Shannon's entropy represents an absolute limit on the best possible lossless compression of any communication, under certain constraints: treating messages to be encoded as a sequence of independent and identically distributed random variables, Shannon's source coding theorem shows that, in the limit, the average length of the shortest possible representation to encode the messages in a given alphabet is their entropy divided by the logarithm of the number of symbols in the target alphabet.

The fundamental theorem of arithmetic guarantees that there is only one possible string that will be accepted ( providing the factors are required to be listed in order ), which shows that the problem is in both UP and co-UP.

However, even though it cannot be determined whether a particular file is incompressible, a simple theorem about incompressible strings shows that over 99 % of files of any given length cannot be compressed by more than one byte ( including the size of the decompressor ).

Gödel's second incompleteness theorem ( 1931 ) shows that no formal system extending basic arithmetic can be used to prove its own consistency.

While crystals, according to the classical crystallographic restriction theorem, can possess only two, three, four, and six-fold rotational symmetries, the Bragg diffraction pattern of quasicrystals shows sharp peaks with other symmetry orders, for instance five-fold.

For example, Rice's theorem shows that each of the following sets of computable functions is undecidable:

Gödel's first incompleteness theorem shows that for languages sufficient for doing a certain amount of arithmetic, there can be no effective deductive system that is complete with respect to the intended interpretation of the symbolism of that language.

The postulate is justified in part, for classical systems, by Liouville's theorem ( Hamiltonian ), which shows that if the distribution of system points through accessible phase space is uniform at some time, it remains so at later times.

For example, the Fitting lemma shows that the endomorphism ring of a finite length indecomposable module is a local ring, so that the strong Krull-Schmidt theorem holds and the category of finite length modules is a Krull-Schmidt category.

The Sonnenschein – Mantel – Debreu theorem shows that the standard model cannot be rigorously derived in general from general equilibrium theory.

Another theorem shows that there are problems solvable by Turing-complete languages that cannot be solved by any language with only finite looping abilities ( i. e., any language that guarantees every program will eventually finish to a halt ).

Gödel's incompleteness theorem shows that no consistent, recursively enumerable theory ( that is, one whose theorems form a recursively enumerable set ) in which the concept of natural numbers can be expressed, can include all true statements about them.

The theorem also shows that any group of prime order is cyclic and simple.

The Gauss – Markov theorem shows that, when this is so, is a best linear unbiased estimator ( BLUE ).

In statistics, the Gauss – Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear regression model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimator ( BLUE ) of the coefficients is given by the ordinary least squares estimator.

As used in describing simple linear regression analysis, one assumption of the fitted model ( to ensure that the least-squares estimators are each a best linear unbiased estimator of the respective population parameters, by the Gauss – Markov theorem ) is that the standard deviations of the error terms are constant and do not depend on the x-value.

The theorem states that any estimator which is unbiased for a given unknown quantity and which is based on only a complete, sufficient statistic ( and on no other data-derived values ) is the unique best unbiased estimator of that quantity.

In statistics, the Rao – Blackwell theorem, sometimes referred to as the Rao – Blackwell – Kolmogorov theorem, is a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria.

The Rao – Blackwell theorem states that if g ( X ) is any kind of estimator of a parameter θ, then the conditional expectation of g ( X ) given T ( X ), where T is a sufficient statistic, is typically a better estimator of θ, and is never worse.

The process of transforming an estimator using the Rao – Blackwell theorem is sometimes called Rao – Blackwellization.

The theorem seems very weak: it says only that the Rao – Blackwell estimator is no worse than the original estimator.

In fact, since S < sub > n </ sub > is complete and δ < sub > 0 </ sub > is unbiased, δ < sub > 1 </ sub > is the unique minimum variance unbiased estimator by the Lehmann – Scheffé theorem.

If the conditioning statistic is both complete and sufficient, and the starting estimator is unbiased, then the Rao – Blackwell estimator is the unique " best unbiased estimator ": see Lehmann – Scheffé theorem.

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.

Further, by the Lehmann – Scheffé theorem, an unbiased estimator that is a function of a complete, sufficient statistic is the UMVU estimator.