Let X be a random variable with the pdf of a regular case of...

Let $X$ be a random variable with the pdf of a regular case of the exponential class, given by $f(x ; \theta)=\exp [\theta K(x)+H(x)+q(\theta)], a<x<b, \gamma<\theta<\delta$. Show that $E[K(X)]=-q^{\prime}(\theta) / p^{\prime}(\theta)$, provided these derivatives exist, by differentiating both members of the equality $$ \int_{a}^{b} \exp [p(\theta) K(x)+H(x)+q(\theta)] d x=1 $$ with respect to $\theta .$ By a second differentiation, find the variance of $K(X)$.

Let $X$ be a random variable with the pdf of a regular case of the exponential class, given by $f(x ; \theta)=\exp [\theta K(x)+H(x)+q(\theta)], a<x<b, \gamma<\theta<\delta$. Show
that $E[K(X)]=-q^{\prime}(\theta) / p^{\prime}(\theta)$, provided these derivatives exist, by differentiating both members of the equality
$$
\int_{a}^{b} \exp [p(\theta) K(x)+H(x)+q(\theta)] d x=1
$$
with respect to $\theta .$ By a second differentiation, find the variance of $K(X)$.

Key Concepts

Cumulant Generating Function

In the exponential family, the logarithm of the partition function acts as the cumulant generating function for the sufficient statistic. The first derivative of this cumulant generating function with respect to the natural parameter gives the mean of the statistic, while the second derivative provides its variance. These properties allow one to obtain the moments of the sufficient statistic directly from the derivatives of the log-partition function.

Differentiation Under the Integral Sign

Differentiating under the integral sign is a technique that allows one to differentiate an integral whose limits or integrand depend on a parameter. In this context, applying this technique to the normalization equality yields equations relating the derivatives of the partition function to the expected values of the functions of the data (the sufficient statistic). This method is crucial for deriving moment identities without having to compute the integrals explicitly.

Moment Identification and Variance Computation

Once the first derivative of the normalization condition identifies the mean of the sufficient statistic, taking a second derivative allows one to derive an expression for the variance. By equating coefficients determined by differentiating the normalization equality twice, one obtains a formula that relates the second derivative of the partition function to the variance of the function appearing in the exponent, thereby facilitating the calculation of the variance of the sufficient statistic.

Exponential Family

The exponential family is a class of probability distributions whose densities or mass functions can be expressed in a canonical form, typically involving an exponent with a natural parameter, a sufficient statistic (or function thereof), a carrier measure, and a normalizing or partition function. This formulation makes many statistical properties, such as the existence of sufficient statistics and conjugacy in Bayesian inference, more apparent and accessible.

Normalization and Partition Function

For any probability density function, integrating over the entire support must yield one. In the context of the exponential family, this condition is given in terms of an integral that includes the partition function, which is often expressed as an exponential function involving the parameter. Differentiating the normalization condition with respect to the parameter helps reveal relationships between the partition function (or its logarithm) and the moments of the sufficient statistic.

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