Let X1, X2, …, Xn be a random sample of size n from a beta...

Key Concepts

Beta Distribution

The Beta distribution is a continuous probability distribution defined on the interval [0, 1] and characterized by two shape parameters. Understanding its form and properties is essential when analyzing samples drawn from it, as these properties influence how the likelihood function is constructed and how statistics, such as products of sample values, can serve as sufficient statistics.

Likelihood Function

The likelihood function represents the probability of the observed data as a function of the parameter(s). In the context of sufficiency, when the likelihood can be expressed in a way where the data influence the parameter only through a specific statistic, that statistic is deemed sufficient. This factorization of the likelihood is pivotal in establishing sufficiency.

Factorization Theorem

The factorization theorem provides a formal criterion for determining sufficiency. It states that a statistic is sufficient for a parameter if the joint probability density or mass function of the data can be factorized into a product where one factor depends on the data only through the statistic and the parameter, while the other factor depends solely on the data and not the parameter.

Sufficient Statistic

A sufficient statistic is a function of the observed data that captures all the information needed about a parameter of interest for inference purposes. When a statistic is sufficient, no additional insight about the parameter can be obtained from the sample beyond what this statistic provides.

Transcript

00:02 Hi, here x1, x2, xn be a random sample of size n from a beta distribution with parameters alpha equals to theta and beta equals to 5.

00:11 It has been given and we have to show that the product of x1 x2 xn, that means all the random samples is a sufficient statistic for the parameter theta.

00:21 So it has been given that x1, x2 up to xn, this follows beta distribution.

00:30 Of first kind with parameters alpha equals to theta and beta equals to 5 okay and we have to show that product of irons from 1 to n x i will be sufficient statistic for the parameter theta okay now the probability density function of this random variable we write f x alpha beta is equal to 1 by beta alpha beta that is beta function and this beta is parameter okay into x to the power alpha minus 1 into 1 minus x to the power beta minus 1 okay where 0 less than x less than 1 and alpha beta both parameters greater than 0 and 0 otherwise okay now this can be written as f x semicolon see alpha equals to theta and beta equals to 5 so beta that parameter beta is constant so we'll write only x semicolon theta so this is equals to one whole divided by beta theta 5 into x to the power theta minus 1 into 1 minus x to the power 5 minus 1 that is 4 so also 0 less than x less than 1 and theta than 0 and 0 otherwise this is the probiotic density function or pdf of the random variable x okay now to show that product of all the random samples is a sufficient statistic for the parameter theta here we use name and factorization theorem name and factorization theorem denoted by nmfd these theorem states that suppose x1 x2 xn follows a distribution with probability density function or probability mass function fx theta with the parameter of with the parameter of interest theta then a statistic px curl is said to be a sufficient statistic for the parameter theta if the joint probability density function or joint probability mass function that means joint probability distribution is equals to g tx curl semicolon theta into h x curl okay if the joint probability distribution can be written as g t x -cull semicolon theta into h -x -cull.

03:03 Now what are these? this is this is a function of x -1 x2 x -n, that means realizations of all the random samples and theta only through tx curl.

03:23 Okay so this is a function of x -1 x2 x -n and theta only through p x -cull and this t x -cull is realization of capital t -c -c -ccccull.

03:33 The sufficient statistic, okay, and hx curl, this is a function of only x1, x2 up to xn.

03:39 It does not contain theta, the parameter of interest...

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