The Poisson distribution truncated at zero has probability function given by f (x; θ) = exp(−θ)θx A(θ)x! , x = 1, 2, ... (θ > 0), where A(θ) = 1 − exp(−θ). Let X1, . . . , Xn be a random sample of size n from this truncated Poisson distribution. (i) Show that this distribution belongs to the regular exponential distri- bution and give its canonical parameter c(θ) and the corresponding sufficient statistic T (X1, ..., Xn)

The regular exponential family of distributions has the probability density function (pdf) or probability mass function (pmf) of the form:

f(x; θ) = h(x) * exp{η(θ) * T(x) - A(θ)}

where:

• h(x) is the base measure,
• η(θ) is the natural parameter or canonical parameter,
• T(x) is the sufficient statistic,
• A(θ) is the log partition function.

Given the truncated Poisson distribution:

f(x; θ) = exp(−θ) * θ^x / (x! * A(θ)), x = 1, 2, ... (θ > 0),

where A(θ) = 1 - exp(−θ).

We can rewrite this in the form of the regular exponential family:

f(x; θ) = [1 / (x! * A(θ))] * exp{x * log(θ) - θ}

Comparing this with the general form of the regular exponential family, we can identify:

• h(x) = 1 / (x! * A(θ))
• η(θ) = log(θ)
• T(x) = x
• A(θ) = θ

Therefore, the canonical parameter c(θ) is log(θ) and the corresponding sufficient statistic T(X1, ..., Xn) is ∑xi (the sum of the sample).

Note: This is a simplified explanation and the actual proof may require more rigorous mathematical steps.