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Let $A$ and $B$ be random variables and $f(A,B)=\frac{A}{B}$. How should I approximate $E(f(A,B))$? I think a Taylor expansion may be in order, but I am not sure how to fire it off in this function.

My question comes from a practical problem in survey statistics. It may be discussed in textbooks, but I would not know where. Let a sample of size $n$ be taken from an (infinite) population. Not every sample unit may reply to the survey. Let $S$ indicate response ($S=1$) or non-response ($S=0$). The mean estimator $\hat{\mu}=\frac{1}{\sum{S_i}}\sum{S_iy_i}$ thus may be biased due to non-response from the population mean $\mu$. We may be inclined to specify this bias as $$B(\hat{\mu})=E(\frac{1}{\sum{S_i}}\sum{S_iy_i})-\mu$$ for realizations $y_i$ on random variable $Y$. Let $n_r$ be the realized sample size (response sample). I believe $$E(\frac{1}{\sum{S_i}})=\frac{1}{n_r}$$ is only true approximately and relates to the more general problem introduced above, where $A=1$. My specific question is why this equation holds true (approximately).

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$1/\sum{S_i}$ is a convex function in $\sum{S_i}$. Then by Jensen's inequality

$$E\left(\frac{1}{\sum{S_i}}\right)>\left(\frac{1}{E[\sum{S_i}]}\right) =\frac{1}{n\cdot P(S_i=1)}$$

the last equality if we assume that each respondent has an equal probability to respond or not. An estimator of this probability is the sample proportion $\hat P(S_i=1) =n_r/n$ so we get

$$est.\left(\frac{1}{E[\sum{S_i}]}\right) =\frac 1{n_r}$$

So "between" $E\left(\frac{1}{\sum{S_i}}\right)$ and $1/n_r$ there exists both the distance due to the non-linearity, as well as the estimation error. The estimation error can go either way, so we cannot conclude on the final relation between the two.

Nevertheless appealing to asymptotics, $\hat P(S_i=1) \xrightarrow{p} P(S_i=1)$ and $E\left(\frac{1}{\sum{S_i}}\right)\rightarrow \left(\frac{1}{E[\sum{S_i}]}\right)$, so for "large samples" we accept $1/n_r$ as an approximation to $E\left(\frac{1}{\sum{S_i}}\right)$.

But for the general case, the situation changes. Write $w_i \equiv S_i/\sum S_i$ and so $\sum w_i =1$, and we have

$$\hat \mu = \sum w_iY_i$$

If we assume that
a) $S_i$ and $Y_i$ are independent,(i.e. that whether somebody responds or not does not depend on his own value of $Y$ -and this is not always the case, for example think of a survey that asks something "sensitive", say "what is your monthly income"? People with high income may choose not to respond rather than record a true or false statement), and taking into account that

b) all members of the population are identically distributed as random variables, and have the common mean $E(Y_i) =\mu, \; \forall i$, then

$$E(\hat \mu) = \sum E(w_i)E(Y_i) = \mu\cdot\sum E(w_i) = \mu\cdot E\left(\sum w_i\right) = \mu\cdot E(1) = \mu$$

so the estimator is, after all, unbiased. This depends crucially on the independence assumption between $S_i$ and $Y_i$ because it implies that the sub-sample of those that responded remains a random sample from, a "representative" sample of, the population, and so its sample average is still an unbiased estimator.

ADDENDUM

Regarding the Taylor series expansion, for the function $1/Z$ it is, around some center $z_0$,

$$E(1/Z) = \frac 1{z_0} - \frac 1{z_0^2}[E(Z) - z_0] + \frac 1{z_0^3}E[Z - z_0]^2 + E(R_2)$$

$\sum S_i$ is a binomial random variable. So centering on $z_0=E(\sum S_i) = np_s$ we have

$$E\left(\frac{1}{\sum{S_i}}\right) = \frac 1{np_s}-\frac 1{n^2p_s^2}(E(S_i)-np_s)+\frac 1{n^3p_s^3}\text{Var}(S_i) +E(R_2)$$

$$=\frac 1{np_s} -0+\frac {np_s(1-p_s)}{n^3p_s^3} + E(R_2) = \frac 1{np_s} + O(n^{-2})$$

Since $\hat p_s = n_r/n$ we arrive at

$$\hat E\left(\frac{1}{\sum{S_i}}\right) \approx \frac 1{n_r}$$

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  • $\begingroup$ Thank you - I once heard that Taylor expension plays a role, too. In particular I think we may be able to write $E(\frac{1}{\sum{S_i}})$ as a Taylor series. I believe it should be shown that $\frac{1}{n_r}$ is the major term, whereas the remainder is negligible (maybe in particular in large samples?). Are you aware of anything like this? $\endgroup$
    – tomka
    Oct 22 '14 at 11:53
  • 1
    $\begingroup$ Added the case. $\endgroup$ Oct 22 '14 at 12:21

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