Question

Let the pdf \(f\left(x ; \theta_{1}, \theta_{2}\right)\) be of the form $$ \exp \left[p_{1}\left(\theta_{1}, \theta_{2}\right) K_{1}(x)+p_{2}\left(\theta_{1}, \theta_{2}\right) K_{2}(x)+H(x)+q_{1}\left(\theta_{1}, \theta_{2}\right)\right], \quad a

Short answer

The pdf \( f(x ; \theta_{1}, \theta_{2}) \) can be reduced to the form \[ f(x ; \theta_{1}, \theta_{2}) = \exp[p(\theta_1, \theta_2) K_2(x) + H(x) + q(\theta_1, \theta_2)]\] where \(p(\theta_1, \theta_2) = p_1(\theta_1, \theta_2)c + p_2(\theta_1, \theta_2)\) and \(q(\theta_1, \theta_2) = p_1(\theta_1, \theta_2) d+ q_1(\theta_1, \theta_2)\). This is due to the homogeneity relation between the derivatives of functions \(K_1\) and \(K_2\), which is a key dependency in finding sufficient statistics and parameters.

Step-by-step solution

Construct the Functions

The given function (pdf) is of the form: \[f(x;\theta_1, \theta_2) = \exp\left[p_1(\theta_1, \theta_2) K_1(x) + p_2(\theta_1, \theta_2) K_2(x) + H(x) + q_1(\theta_1, \theta_2)\right]\] where x is between a and b, and zero elsewhere. The assumption given is that: \(K_1'(x) = c K_2'(x)\)

Apply Assumption

Considering the derivate values and the given assumption, we can express \(K_1'(x)\) in terms of \(K_2'(x)\). Therefore, we can re-write \(K_1(x)\) as a function of \(K_2(x)\), which we denote by \(dK_2(x)\), because \(c\) represents the constant of proportionality between the derivatives of \(K_1(x)\) and \(K_2(x)\). So, we get the following relationship: \(K_1(x) = cK_2(x) + d\)

Substitute Back

Now substitute these relationships into the original function: \[ f(x ; \theta_{1}, \theta_{2}) = \exp\left[p_1(\theta_1, \theta_2) (cK_2(x) + d) + p_2(\theta_1,\ \theta_2) K_2(x) + H(x) + q_1(\theta_1, \theta_2)\right]\]

Simplify the Expression

Proceed to simplify the expression: \[f(x ; \theta_{1}, \theta_{2}) =\exp\left[ (p_1(\theta_1, \theta_2)c + p_2(\theta_1, \theta_2)) K_2(x) + p_1(\theta_1, \theta_2) d + H(x) + q_1(\theta_1, \theta_2)\right]\] This can now be represented as: \[f(x ; \theta_{1}, \theta_{2}) = \exp[p(\theta_1, \theta_2) K_2(x) + H(x) + q(\theta_1, \theta_2)]\] where \( p(\theta_1, \theta_2) = p_1(\theta_1, \theta_2)c + p_2(\theta_1, \theta_2)\) and \(q(\theta_1, \theta_2) = p_1(\theta_1, \theta_2) d+ q_1(\theta_1, \theta_2)\), and x is between a and b, zero elsewhere.