Ask Question. Asked 9 years, 4 months ago. Active 7 months ago. Viewed k times. Improve this question. MathematicalOrchid MathematicalOrchid 2, 3 3 gold badges 13 13 silver badges 15 15 bronze badges. The figure you refer to claims that the estimator is consistent but biased, but doesn't explain why. The caption points out that each of the estimators in the sequence is biased and it also explains why the sequence is consistent.
Do you need an explanation of how the bias in these estimators is apparent from the figure? Show 1 more comment. Active Oldest Votes. Improve this answer. Community Bot 1. Macro Macro For proper consistency a few additional requirements, e. Examples of MLEs that aren't consistent are found in certain errors-in-variables models where the "maximum" turns out to be a saddle-point.
They're good examples of how the ML approach can fail though : I'm sorry that I can't give a relevant link right now - I'm on vacation. The necessary conditions were outlined in the link but that wasn't clear from the wording. The stated consistency result still holds, of course. Show 2 more comments. Michael R. Chernick Michael R. Chernick An estimator of a given parameter is said to be unbiased if its expected value is equal to the true value of the parameter.
In other words, an estimator is unbiased if it produces parameter estimates that are on average correct. The estimate is usually obtained by using a predefined rule a function that associates an estimate to each sample that could possibly be observed. The function is called an estimator.
This seems like a good property for an estimator to have. In many settings, natural estimators turn out to be unbiased. Let's look at some examples. We can do this by using the linear function rule and additivity. Why throw away the rest of the data?
We will show this later in the course. For now, just note that the same sample can be used to construct more than one unbiased estimator for the parameter. You know that the sum of a sequence of zeros and ones is equal to the number of ones in the sequence.
It follows that the average of a sequence of zeros and ones is the proportion of ones in the sequence. Suppose you roll a die 30 times and find the sample proportion of sixes.
The histogram below shows the results of 20, repetitions of this experiment. Germany had a seemingly never-ending fleet of Panzer tanks, and the Allies needed to estimate how many they had.
They decided to base their estimates on the serial numbers of the tanks that they saw. Notice the serial number on the top left. When tanks were disabled or destroyed, it was discovered that their parts had serial numbers too.
The ones from the gear boxes proved very useful. This is of course a very simplified model of reality, and we will make some additional simplifications. But estimates based on even such simple probabilistic models proved to be quite a bit more accurate than those based on the intelligence gathered by the Allies. For example, in August , intelligence estimates were that Germany was producing 1, tanks per month.
The prediction based on the probability model was per month. After the war, German records showed that the actual production rate was per month. But for even more simplicity, let's pretend that the draws were made with replacement. That is, if we saw the same tank twice then we would record it twice. But how much less?
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