The daily return of a stock (0700.HK) is assumed to follow normal distribution with parameters with ???? and ????2.To estimate the unknown parameters, we examine the historical return of the stock over the past 9 years period (1st Jan, 2011 – 31st Dec, 2019), 2212 sample data in total). It is found that Σ????????=2.881669,Σ????????2=0.849829
The probability that the daily price change of Tencent stock (0700.HK) exceeds 3%, denoted by ????, can be estimated as ????̂=???????????????????????? ???????? ???????????????? ????????????„Ž ???????????????????? ????„Ž???????????????? ???????????????????????????? ????„Ž???????? 3%???????????????????????? ???????? ????????????????.
We let {????1,????2,”¦,????????} be the sample data.
(a) Write down the mathematical formula of ????̂. Express your answer in terms of ???????? and ????.
(b) Determine if ????̂ is an unbiased estimator.
(c) Determine the mean square error (MSE) of the estimator. Express your answer in terms of ????.