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The aim of robust estimation is to derive estimators with variance near that of the sample mean when the distribution is standard normal while having the variance remain relatively stable as δ increases. Compare the variances as the value of δ increases. In order to more easily identify with this value is so 80 confidence over two is such that negatively off over two and positively also virtues with the tales of the normal current into area 20.1 each something to point to 1 980 thus far from our table. For various values of δ, say 0.0, 0.01, 0.05, 0.1, 0.2, 0.3, and 0.4, create a table of variances of sample mean and sample variance. (a)Ĭonduct a simulation study with sample size n that takes, say, 5000 random samples of 100 observations each.

A less effective alternative would be the sample median.

We already know that the MVUE of the mean μ of an uncontaminated normal distribution is the sample mean. That is, for 0 ≤ δ ≤ 1, (1 − δ)100% of the observations come from an N(0, 1) distribution and the remaining (δ)100% of observations come from an N(0, 5) distribution. Suppose the population actually follows a contaminated normal distribution. Let X 1, …, X n be a random sample from a standard normal distribution. The following illustrates how the variance of an estimator can be affected by deviations from the presumed underlying population model.Ĭonsider estimating the mean of a standard normal distribution. We call such estimators robust estimators. Hence, it is desirable for the derived estimators to have small variance over a range of distributions. The standard normal distribution table is a compilation of areas from the standard normal distribution, more commonly known as a bell curve, which provides the area of the region located under the bell curve and to the left of a given z- score to represent probabilities of occurrence in a given population. If the behavior of an estimator is taken as its variance, a given estimator may have minimum variance for the distribution used, but it may not be very good for the actual distribution. Formally, a statistical procedure is robust if its behavior is relatively insensitive to deviations from the assumptions on which it is based. Calculating the Confidence Interval X is the mean Z is the chosen Z-value from the table above s is the standard deviation n is the number of observations. However, if the choice of the underlying family of distributions is based on past experience, there is a possibility that the true population will be slightly different from the model used to derive the estimators. The estimators derived in this chapter are for particular parameters of a presumed underlying family of distributions. Tsokos, in Mathematical Statistics with Applications in R (Third Edition), 2021 5.10.2 Robust estimation