# Sufficient Statistic for Absolutely Continuous Distribution [duplicate]

The following is a homework problem. Please tell me if my solution is correct and if not please point out my mistakes.

Let $$x_{1}, x_{2},...,x_{M}$$ be i.i.d. samples from the absolute continuous distribution given by the p.d.f $$f(x| \theta) = \frac{ \theta}{(x+1)^{\theta +1}}$$

find a sufficient statistic for $$\theta$$

I assume the range of the function is $$[0, \inf)$$ otherwise the integral does not equal 1 so the function couldn't be a probability distribution.

$$f(x_{1}, x_{2},...,x_{n}) = \prod_{i=1}^n \frac{ \theta}{(x_{i}+1)^{\theta +1}}$$ $$= \prod_{i=1}^n \theta(x_{i}+1)^{-(\theta +1)} = \prod_{i=1}^nexp[ln(\theta)- ln (x_{i}+1)^{-(\theta +1)}]$$ $$= exp \hspace{1mm}[\hspace{1mm} \sum_{i=1}^n( ln(\theta)- (\theta +1)ln (x_{i}+1))]$$ $$= exp \hspace{1mm}[\hspace{1mm} n\hspace{1mm} ln(\theta)- (\theta +1) \sum_{i=1}^n ln (x_{i}+1)]$$ $$\therefore$$ using Neyman - Fisher Factorization $$T(x) = \sum_{i = 1}^n ln(x_{i} +1)$$ $$g(T(x), \theta) = exp [ \hspace{2mm} n \hspace{1mm}ln \hspace{1mm}\theta - ( \theta + 1) T(x)]$$ $$h(x) = 1$$

and T(x) is the sufficient statistic

Could someone please tell me if my solution is correct and if not what is a better solution?

Also as the samples $$x_{i}$$ only appear once in the joint probability function would the entire joint p.d.f $$\prod_{i=1}^n \frac{ \theta}{(x+1)^{\theta +1}}$$ be a sufficient statistic?

What does "absolute continuous" mean? What is an absolute continuous distribution?

– whuber
Sep 20, 2021 at 12:57

1. You are correct,

2. A sufficient statistic cannot be a function of the unknown parameters, so if you don't know the value of $$\theta$$, no, the joint PDF is not a sufficient statistic. If, on the other hand, you do know the value of $$\theta$$, then the joint PDF is in this case the sufficient statistic, although if you know the value of $$\theta$$ there's not much point in sampling...

3. "absolute continuous" should be "absolutely continuous", a definition of which you can find on the Wikipedia page https://en.wikipedia.org/wiki/Absolute_continuity.