Show that the n th order statistic of a random sample of...

Show that the $n$ th order statistic of a random sample of size $n$ from the uniform distribution having pdf $f(x ; \theta)=1 / \theta, 0<x<\theta, 0<\theta<\infty$, zero elsewhere, is a sufficient statistic for $\theta$. Generalize this result by considering the pdf $f(x ; \theta)=Q(\theta) M(x), 0<x<\theta, 0<\theta<\infty$, zero elsewhere. Here, of course, $$ \int_{0}^{\theta} M(x) d x=\frac{1}{Q(\theta)} $$

Show that the $n$ th order statistic of a random sample of size $n$ from the uniform distribution having pdf $f(x ; \theta)=1 / \theta, 0<x<\theta, 0<\theta<\infty$, zero elsewhere, is a sufficient statistic for $\theta$. Generalize this result by considering the pdf $f(x ; \theta)=Q(\theta) M(x), 0<x<\theta, 0<\theta<\infty$, zero elsewhere. Here, of course,
$$
\int_{0}^{\theta} M(x) d x=\frac{1}{Q(\theta)}
$$

Key Concepts

Order Statistics

Order statistics are the sorted values of a sample, and the maximum (or nth order statistic) in a sample often provides crucial information about the range or support of a distribution. In many cases, the largest observation contains all the information about the upper limit of the distribution, making it a natural candidate for serving as a sufficient statistic for parameters determining the bound.

Sufficient Statistic

A sufficient statistic is a function of the sample data that captures all the information necessary for inferring the unknown parameter. Once the sufficient statistic is known, the data itself provides no additional information about the parameter. This is a central concept in estimation and decision theory as it allows for data reduction without loss of information regarding the parameter.

Factorization Theorem

The factorization theorem provides a formal method to identify sufficient statistics. It states that a statistic is sufficient for a parameter if the joint probability density (or mass) function can be factored into a product: one factor that depends on the data only through this statistic and the parameter, and another factor that does not depend on the parameter. This theorem offers a methodological approach to prove sufficiency in various statistical models.

Parametric Models with Parameter-Dependent Support

In distributions where the support depends on the parameter, such as the uniform distribution on (0, ?) or other similarly defined densities, the extremal observations (like the maximum) often play a key role. They summarize the boundary of the support and, via factorization, can be shown to be sufficient statistics. This general principle extends to other families of distributions with similar structure, where the density is expressed in a way that the integration constant is defined by the parameter.

Transcript

00:01 Hi, here we have to show that the nth order statistic of a random sample of size n from the uniform distribution having ptf fx3 equals to 1 by theta is a sufficient statistic for the parameter theta which is our parameter of interest and we have to generalize this result with the pdf which has some form okay so it has been given that we have n random samples x1 x2 up to xn which follows uniform distribution with a parameter 0 theta and we have to show that x order n equals to maximum of x i where one less than equals to i less than equals to n this is a sufficient statistic for the parameter theta we have to show that okay so first we start this by writing the covert density function of this random variable which is f x theta is equals to 1 by theta where 0 less than x less than theta is also theta is greater than 0 and 0 otherwise okay now it has also be given that if x theta is equals to q theta into mx where q theta is equal as to 1 by theta and mx is equal to 1 such that indication 0 to theta m x d x is equal to theta so you can write 1 whole divided by q theta.

01:43 Okay.

01:44 Now, here we use a name and factorization theorem, which is nft.

01:50 Ft.

01:53 So what does this theorem state? these theorem states that suppose we have n random samples, x1, x2, xn, which follows some distribution with the probability distribution fx theta, that is either probability mass function or probability density function.

02:10 Then a statistic, tx curl is said to be a sufficient statistic for the parameter theta if the joint probability distribution if x curl theta that means the joint pdf poverty density function or joint poverty mass function this is equals to g tx culled semicolon theta into hx curl okay if the joint probability distribution can be written as a product of two functions so what are these functions these functions g tx called chemical of theta is a function of x1 x2 x n and theta only through t xx called small t small x k that means it is a function of x1 x2 x and these are the realizations of the random sample and the parameter of interest theta only through small t small x girl where this small t small x curl is a realization of the sufficient statistic okay now now, what is hx squared? it is a function of x1, x2, and xn on a.

03:24 That means it is a function of early the realizations of the random samples...

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