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What is the meaning of sufficient statistics?
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.
How to prove sufficient statistic?
Recall that an exponential family of random variables has its density of the form fX(x|θ) = c(θ)h(x) exp(ν(θ)T(x)). Thus, the sufficient statistic is sum of the observations T(x) = x1 + ··· + xn and the natural parameter ν(θ) = ln(θ/(1 − θ)), the log-odds, Example 6 (Gamma random variables).
Is a function of a sufficient statistic sufficient?
. Typically, the sufficient statistic is a simple function of the data, e.g. the sum of all the data points. The concept is equivalent to the statement that, conditional on the value of a sufficient statistic for a parameter, the joint probability distribution of the data does not depend on that parameter.
Which is an example of a sufficiently sufficient statistic?
For example, for a Gaussian distribution with unknown mean and variance, the jointly sufficient statistic, from which maximum likelihood estimates of both parameters can be estimated, consists of two functions, the sum of all data points and the sum of all squared data points (or equivalently, the sample mean and sample variance ).
When is a statistic sufficient for the underlying parameter?
Both the statistic and the underlying parameter can be vectors. A statistic t = T ( X) is sufficient for underlying parameter θ precisely if the conditional probability distribution of the data X, given the statistic t = T ( X ), does not depend on the parameter θ.
What do you call a jointly sufficient statistic?
In such a case, the sufficient statistic may be a set of functions, called a jointly sufficient statistic. Typically, there are as many functions as there are parameters.
When is a statistic sufficient for a family of probability distributions?
Sufficient statistic. In particular, a statistic is sufficient for a family of probability distributions if the sample from which it is calculated gives no additional information than does the statistic, as to which of those probability distributions is that of the population from which the sample was taken.