Give examples of estimators or estimates which are a unbiase

Give examples of estimators (or estimates) which are (a) unbiased and efficient, (b) unbiased and inefficient,
(c) biased and inefficient.

Solution

(a) The sample mean X and the modified sample variance
s^2 = N /N-l * s^2 are two such examples.
(b) The sample median and the sample statistic 1/2 (Q1 + Q3). where Q1 and Q3 arc the lower and upper
sample quartiles, are two such examples. Both statistics are unbiased estimatcs of the population
mean, since the mean of their sampling distributions is the population mean.
(c) The sample standard deviation s. the modified standard deviation š, the mean deviation, and the
semi-interquartile range arc four such examples.