What is a sufficient population?

A sufficient statistic summarizes all of the information in a sample about a chosen parameter. For example, the sample mean, x̄, estimates the population mean, μ. x̄ is a sufficient statistic if it retains all of the information about the population mean that was contained in the original data points.

What is meant by sufficient estimator?

An estimator of a parameter θ which gives as much information about θ as is possible from the sample at hand is called a sufficient estimator. Sufficient estimators exist when one can reduce the dimensionality of the observed data without loss of information.

How do you know if a statistic is sufficient?

The mathematical definition is as follows. A statistic T = r(X1,X2,··· ,Xn) is a sufficient statistic if for each t, the conditional distribution of X1,X2, ···,Xn given T = t and θ does not depend on θ.

What is a complete sufficient statistic?

(Basu’s Lemma) If T(X) is complete and sufficient (for θ ∈ Θ), and S(X) is ancillary, then S(X) and T(X) are independent for all θ ∈ Θ. In other words, a complete sufficient statistic is independent of any ancillary statistic.

Why do we use sufficient statistics?

Short Answer: A sufficient statistics carries with it all the information needed to make inference about the population, excluding information that gives sample specific . So given the sufficient statistics, you can do the same inferential analysis without your entire data.

Is sufficient statistic unique?

Sufficient statistics are not unique: Any one-to-one transformation of a sufficient statistic is again a sufficient statistic.

What does jointly sufficient mean?

individually necessary
Frequently the terminology of “individually necessary” and “jointly sufficient” is used. One might say, for example, “each of the members of the foregoing set is individually necessary and, taken all together, they are jointly sufficient for x’s being a square.”

What is a scalar sufficient statistic?

In statistics, a statistic is sufficient with respect to a statistical model and its associated unknown parameter if “no other statistic that can be calculated from the same sample provides any additional information as to the value of the parameter”.

Does sufficient statistic always exist?

Under mild conditions, a minimal sufficient statistic does always exist.

Does sufficient statistics always exist?

Is sufficient statistic always exist?

Hence, a sufficient statistic always exists. We can compute the density of the sufficient statistics. Many statistical problems can be phrased in the language of decision theory. Suppose as usual that we have data X whose distribution depend on a parameter Θ.

Why is sufficiency important in statistics?