I've got a general question regarding a certain sticking point I often encounter. When tackling questions where for example an UMVUE (uniformly minimum-variance unbiased estimator) has to found I get stuck at the part where an expectation has to be determined. Maybe I'm overlooking certain theorems.

Example 1. Consider a random sample of size $n$ with $X_i\tilde{}Poi(\mu)$.

I am able to find the UMVUE of $\mu$ and the MLE (Maximum Likelihood Estimator) of $\theta=e^{-\mu}$. But then for example follows the question: is this MLE unbiased for $\theta$?

If I want to determine the expectation, should I just write out the sum as follows from the definition of the expected value of a discrete distributed variable? This seems to lead to quite an ugly expression...

(For this particular problem I found that Jensen's Inequality could come in handy to show biasedness (or actually non-unbiasedness))

But if I could summarize the problem I've got: how to rewrite expected values of functions of stochast in general? Which theorems or properties should I use... for example I do use $E(aX)=aE(X)$ for $a$ a constant and $var(X)=E(X^2)-E(X)^2$ to rewrite certain expectations to a combination of simpler/known ones.

Example 2. Take a random sample $n$ of a distribution with pdf $f(x;\theta)=\theta x^{\theta -1}$ if $0<x<1$ and else $0$ and with $\theta > 0$.

Find the UMVUE of $\theta$

Here I get stuck again at the expectation. I've got a sufficient and complete statistic in $\sum ln(X_i)$.

The expected value, i'd figure, is equal to $nE(ln(X_i))=-n/ \theta$ $\,\,\,\,$(where $[-ln(X)]=\frac{1}{\theta}$ was given as a hint).

But what if I want to determine the expected value of $\frac{1}{\sum_{i}ln(X_i)}$ ?

In general: how to determine the expectations of functions of stochasts? For example $E[ln(X)]$or $E[e^{\bar{X}}]$...?


If $X$ has density $f_X$, then for every function $u$, $E[u(X)]=\displaystyle\int u(x)f_X(x)\mathrm dx$. For example, in your last case, $$ E_\theta\left[\frac1{\sum\limits_i\ln X_i}\right]=\iint\cdots\int\frac1{\sum\limits_i\ln x_i}\prod_if(x_i;\theta)\prod_i\mathrm dx_i. $$ If $X$ is discrete and $p_x=P[X=x]$ for every $x$, this should read $E[u(X)]=\displaystyle\sum\limits_xu(x)p_x$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.