If X1, X2, …, Xn is a random sample from a distribution that...

If $X_{1}, X_{2}, \ldots, X_{n}$ is a random sample from a distribution that has a pdf which is a regular case of the exponential class, show that the pdf of $Y_{1}=\sum_{1}^{n} K\left(X_{i}\right)$ is of the form $f_{Y_{1}}\left(y_{1} ; \theta\right)=R\left(y_{1}\right) \exp \left[p(\theta) y_{1}+n q(\theta)\right]$. Hint: Let $Y_{2}=X_{2}, \ldots, Y_{n}=X_{n}$ be $n-1$ auxiliary random variables. Find the joint pdf of $Y_{1}, Y_{2}, \ldots, Y_{n}$ and then the marginal pdf of $Y_{1}$.

If $X_{1}, X_{2}, \ldots, X_{n}$ is a random sample from a distribution that has a pdf which is a regular case of the exponential class, show that the pdf of $Y_{1}=\sum_{1}^{n} K\left(X_{i}\right)$ is of the form $f_{Y_{1}}\left(y_{1} ; \theta\right)=R\left(y_{1}\right) \exp \left[p(\theta) y_{1}+n q(\theta)\right]$.
Hint: Let $Y_{2}=X_{2}, \ldots, Y_{n}=X_{n}$ be $n-1$ auxiliary random variables. Find the joint pdf of $Y_{1}, Y_{2}, \ldots, Y_{n}$ and then the marginal pdf of $Y_{1}$.

Key Concepts

Exponential Family

The exponential family of distributions is a broad class characterized by probability density or mass functions that can be expressed in the form f(x; ?) = h(x) exp{?(?)T(x) ? A(?)}. This form encapsulates many common distributions and endows them with convenient statistical properties such as the existence of sufficient statistics and conjugate priors, which simplify both theoretical analysis and practical inference.

Transformation of Random Variables

Changing variables in probability density functions involves defining new random variables as functions of the original ones and then determining the joint or marginal density of these new variables. This often requires computing the Jacobian of the transformation to correctly adjust the probability density, which is essential for deriving the density function of a sum or other functions of random variables.

Joint and Marginal Distributions

Joint distributions describe the probability behavior of multiple random variables simultaneously. When only one derived quantity is of interest, marginalization is used by integrating out the auxiliary variables from the joint distribution. This process is critical in deriving the distribution of a function of random variables, such as a sum, from a higher-dimensional joint density.

Sufficient Statistics

In the context of the exponential family, sufficient statistics summarize all the information in the sample about the parameter of interest. The fact that the sum of the functions of the sample values appears in the exponential part of the density underscores its role as a sufficient statistic, facilitating efficient parameter estimation and hypothesis testing.

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