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

Let $X_{1}, X_{2}, \ldots, X_{n}$ be a random sample of size $n$ from a geometric distribution that has $\operatorname{pmf} f(x ; \theta)=(1-\theta)^{x} \theta, x=0,1,2, \ldots, 0<\theta<1$, zero elsewhere. Show that $\sum_{1}^{n} X_{i}$ is a sufficient statistic for $\theta$.

Key Concepts

Sufficient Statistic

A sufficient statistic is a function of the sample data that captures all the relevant information needed for inference about the parameter. Once this statistic is known, the rest of the data does not provide any additional information about the parameter, making it a key tool in reducing data complexity.

Factorization Theorem

The Factorization Theorem provides a method to determine if a statistic is sufficient for a parameter. It states that the joint probability function can be decomposed into two parts: one that depends on the sample data only through the statistic and the parameter, and another that does not depend on the parameter at all. This factorization directly indicates the sufficiency of the statistic.

Likelihood Function

The likelihood function is a fundamental concept in statistical inference that represents the probability of the observed data as a function of the parameter. By expressing the joint likelihood in terms of a sufficient statistic, one demonstrates that this statistic contains all the information needed for parameter estimation.

Transcript

00:01 Hi, here it has been given that x1, x2 up to xn be a random sample of size n from a geometric distribution that has pmf fx theta, that means probability mass function fx theta is equal to theta into 1 minus theta to the power x, where x is equal to 0, 1, 2 and so on, and 0 less than theta less than 1 and 0 otherwise as x i x1 x2 up to x n follows geometric distribution with parameter theta okay and what we have to show we have to show that summation irons from 1 to n x i is a sufficient statistic for the parameter theta in case of this geometric distribution at parameter theta okay so here we'll use name and factorization theorem name and factorization theorem nftiization theorem nfti fte okay so what does this theorem states this theorem states that suppose from a distribution all the random samples follows a distribution with pmf or pdf or probability mass function or probability density function f x theta then a statistic tx curl will be a sufficient statistic for the parameter theta if the joint probability mass function or joint probability function that means joint probability distribution if x -courl theta can be written as this is equals to g tx curl semicolon theta into h x curl okay now see these g tx curl semicolon this is a function of x1 x2 up to x n and theta only through small t small x curl okay small x curl is a realization of capitality capital x curl okay and hx curl is a function of only x1 x2 x n it is independent of the parameter theta if this the joint probability distribution can be written like this then we say that capital t x curl is a sufficient statistic for the parameter theta so we find out the joint probability distribution that is f x curl theta is equal to product of irons from one to n f x i theta, right? so, if x -i theta is equals to given in the question, that is the pmf of this distribution, that is theta into 1 minus theta to the power x -i.

03:06 Okay, so this is theta -to -the -power n into 1 -3 -to -the -per summation x -i, irons from 1 to n.

03:14 So this can be written as g, t -x -ccccccccccccccccccccccccccccccccc...

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