Confidence Interval - Statistical Theory - Definition

Definition

Let X be a random sample from a probability distribution with statistical parameters θ, which is a quantity to be estimated, and φ, representing quantities that are not of immediate interest. A confidence interval for the parameter θ, with confidence level or confidence coefficient γ, is an interval with random endpoints (u(X), v(X)), determined by the pair of random variables u(X) and v(X), with the property:

The quantities φ in which there is no immediate interest are called nuisance parameters, as statistical theory still needs to find some way to deal with them. The number γ, with typical values close to but not greater than 1, is sometimes given in the form 1 − α (or as a percentage 100%·(1 − α)), where α is a small non-negative number, close to 0.

Here Prθ,φ indicates the probability distribution of X characterised by (θ, φ). An important part of this specification is that the random interval (u(X), v(X)) covers the unknown value θ with a high probability no matter what the true value of θ actually is.

Note that here Prθ,φ need not refer to an explicitly given parameterised family of distributions, although it often does. Just as the random variable X notionally corresponds to other possible realizations of x from the same population or from the same version of reality, the parameters (θ, φ) indicate that we need to consider other versions of reality in which the distribution of X might have different characteristics.

In a specific situation, when x is the outcome of the sample X, the interval (u(x), v(x)) is also referred to as a confidence interval for θ. Note that it is no longer possible to say that the (observed) interval (u(x), v(x)) has probability γ to contain the parameter θ. This observed interval is just one realization of all possible intervals for which the probability statement holds.

Read more about this topic:  Confidence Interval, Statistical Theory

Other articles related to "definition, definitions":

Sustainable Development - Definition
... the Brundtland Report, which included what is now one of the most widely recognised definitions "Sustainable development is development that meets the needs of the present without ... It offers an integrated vision and definition of strong sustainability ... It generates a more nuanced definition of sustainable development “the need to ensure a better quality of life for all, now and into the future, in a just and equitable manner, whilst living within the ...
ESPN - High Definition
... ESPN launched its 720p high-definition simulcast, originally branded as ESPNHD, on March 20, 2001 ... Live studio shows, along with most live events on ESPN, use high definition ... non-HD sources is presented in a standard definition, 43 format with stylized pillarboxes ...
Pashtun People - Pashtuns Defined
... These three definitions may be described as the ethno-linguistic definition, the religious-cultural definition, and the patrilineal definition, respectively ...
Definition of Dietary Fiber
... the components of plants that resist human digestive enzymes, a definition that includes lignin and polysaccharides ... The definition was later changed to also include resistant starches, along with inulin and other oligosaccharides ... Official definition of dietary fiber differs a little among different institutions Organization (reference) Definition Institute of Medicine Dietary fiber consists of nondigestible ...
Definition - In Medicine
... In medical dictionaries, definitions should to the greatest extent possible be simple and easy to understand, preferably even by the general public useful clinically or in related areas where the ...

Famous quotes containing the word definition:

    The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.
    Samuel Taylor Coleridge (1772–1834)

    No man, not even a doctor, ever gives any other definition of what a nurse should be than this—”devoted and obedient.” This definition would do just as well for a porter. It might even do for a horse. It would not do for a policeman.
    Florence Nightingale (1820–1910)

    Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.
    Willard Van Orman Quine (b. 1908)