MLE Homework: Distance Measurement with Odometer Error

In summary, John is trying to measure the exact distance from his home to his office using his car's odometer. However, the odometer only records distance to the nearest mile and the readings can vary by up to 1 mile due to the lack of a reset button. To find the maximum likelihood estimator (m.l.e.) of the exact distance, we must make some assumptions and consider the distribution of the errors. However, it is challenging to determine the likelihood as we are dealing with a continuous variable (the error) and a discrete one (the odometer readout).
  • #1
Yagoda
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Homework Statement


John wants to measure the distance from his home to his office, so he drives to work several times and measures the distance on his car's odometer. Unfortunately, the odometer records distance only to the nearest mile. (Johns odometer changes abruptly from one digit to the next. When the odometer displays a digit d, it is not possible to infer how close it is to becoming d + 1.) John drives the route ten times and records the following data from the odometer: 3, 3, 3, 4, 3, 4, 3, 3, 4, 3. Find the m.l.e. of the exact distance.


Homework Equations





The Attempt at a Solution


The way I'm interpreting the problem, it seems the odometer doesn't have a button to reset it to zero at the beginning of the trip so he has to calculate the distance each time by subtracting the beginning readout from the end readout. This means he could have traveled up to 1 mile further than the odometer says he did.

We are allowed to make some assumptions so I am assuming that each error is independent and uniformly distributed on [0,1).

I'm stumped as to how to find the likelihood. I'm trying to come up with a distribution to find the likelihood, but don't know how to begin since it seems like we have this continuous variable, the error, that's begin added to the distance, but we are only looking at it discretely.

Thanks.
 
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  • #2
With a reset button, it would always show 3 miles.

We are allowed to make some assumptions so I am assuming that each error is independent and uniformly distributed on [0,1).
A distance of 3.8 miles can give 3 as measurement.
 
  • #3
Yeah, that's why I assumed there was no reset button, otherwise he would get the same readout each time.

A distance of 3.8 miles can give 3 as measurement.
Exactly, this what I'm having trouble encapsulating in the distribution of the odometer readout.
 
  • #4
The distribution of the error will depend on the actual distance. This is useful, as it will give different likelihoods later.
 

1. What is MLE and how does it relate to distance measurement with odometer error?

MLE stands for Maximum Likelihood Estimation. It is a statistical method used to estimate the parameters of a model by finding the values that maximize the likelihood of the observed data. In the context of distance measurement with odometer error, MLE can be used to estimate the actual distance traveled by a vehicle based on the data collected by an odometer.

2. How does odometer error affect distance measurement?

Odometer error occurs when the odometer of a vehicle does not accurately reflect the distance traveled. This can happen due to various factors such as wear and tear or calibration issues. Odometer error can significantly impact distance measurement as it can lead to overestimation or underestimation of the actual distance traveled.

3. What are some common sources of odometer error?

Some common sources of odometer error include wear and tear of the vehicle's wheels or tires, changes in tire pressure, and calibration issues with the odometer. Additionally, human error in recording the odometer readings can also contribute to odometer error.

4. How can MLE be used to correct for odometer error?

MLE can be used to correct for odometer error by estimating the true distance traveled based on the observed data. This is done by creating a statistical model that takes into account the factors that contribute to odometer error and using MLE to find the most likely values for these factors. The estimated true distance can then be used as a correction factor for the recorded distance from the odometer.

5. Are there any limitations to using MLE for distance measurement with odometer error?

While MLE can be a useful tool for correcting for odometer error, it does have some limitations. For example, it assumes that the error in the odometer readings follows a specific distribution, which may not always be the case. Additionally, MLE requires a large amount of data to accurately estimate the parameters of the model, so it may not be suitable for cases with limited data.

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