# What may be an inefficient estimator of the population mean?

If the sample mean is an efficient estimator of the population mean, what may be an example of an inefficient such estimator?

• It is not that simple. For example sample mean is an efficient estimator for population mean for data coming from Gaussian, but not from Laplace distribution. Similarly sample median will be efficient to estimate population mean for Laplace but will be inefficient to estimate population mean for Gaussian. Aug 14 '19 at 12:00
• Pick one element of the sample randomly as your estimator.
– whuber
Aug 14 '19 at 12:36
• OK, let's be more explicit. Your setting is a sample of values $x_1, x_2, \ldots, x_n.$ Roll an n-sided die. Call its outcome $i.$ Let $x_i$ estimate the mean. I hope it's obvious that for $n\gt 1$ this is an inefficient procedure.
– whuber
Aug 14 '19 at 12:56
• What @whuber is telling you is not using your entire data also leads to an inefficient estimator. Estimator is a function of your data such as $f(x_1,x_2,...,x_n)$. This function doesn't have to be in the form of adding elements or doing symmetric/commutative operations. It can be anything. And if this function doesn't use some of its input arguments that will (always, I think) lead to inefficiency. Aug 14 '19 at 13:59
• Lars, since my comments aren't making sense to you, we had better back up a bit: could you explain what you understand an "estimator" to be and what your definition of "efficiency" is?
– whuber
Aug 14 '19 at 14:03

Consider a sample of size $$N$$ drawn from a normal distribution. The sample median $$\tilde{X}$$ is an unbiased and consistent estimator for $$\mu$$. For large $$N$$ the sample median is approximately normally distributed with mean $$\mu$$ and variance $$\pi/2N$$. The efficiency for large $$N$$ is thus $$2/\pi \approx 0.64$$. This is the asymptotic efficiency, that is the efficiency in the limit as sample size $$N$$ tends to infinity.