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In one book and a few papers, mostly on navigational tracking, I have found reference to the method of "non-random parameter estimation" but this term is not on the Wikipedia and not in a lot of standard texts. It seems to be similar, however, to maximum likelihood estimation which is a standard method found in every tracking and navigation text. Are these two terms referring to the same thing?

For example, in "Estimation with Applications to Tracking and Navigation" by Bar-Shalom, he writes "A common method of estimating nonrandom parameters is the maximum likelihood method that maximizes the likelihood function (2.2.2-2). This yields the maximum likelihood estimator (MLE)...."

So, from this language I am guessing when other authors speak of "non-random parameter estimation" they are talking about the MLE. Is that right?

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  • $\begingroup$ It would likely help to have the book and/or papers cited in your Question for a bit of a lead. It sounds to me like what is being said is that the parameter is not random, i.e. deterministic, which is the usual case. It might be useful in some applications (not navigational tracking as far as I know) so model random variables whose distribution depends on a random parameter (so-called compound distributions). But I suspect that here your situation is the straightforward kind. $\endgroup$
    – hardmath
    Jan 22, 2016 at 17:16
  • $\begingroup$ @hardmath I have added a quote from a relevant text. $\endgroup$ Jan 22, 2016 at 17:25
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    $\begingroup$ So you see there that "nonrandom" modifies "parameters" rather than being a modifier of the estimation methods. There are other ways to estimate (nonrandom) parameters. $\endgroup$
    – hardmath
    Jan 22, 2016 at 17:27
  • $\begingroup$ you should think to the general framework which is "optimization of some objective function related to a certain model for the data". that the model is probabilistic or not doesn't change fundamentally the tools you will use. $\endgroup$
    – reuns
    Jan 22, 2016 at 18:24

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First let's unpack the phrase "nonrandom parameter estimation". In context this means "estimating a parameter of a probability distribution for a random variable" where the parameter is a definite value which is only known through sampling of the random variable.

For example, many repeated measurements may be performed in order to estimate the true value of a distance. If the experimental errors are assumed to be normally distributed with mean zero, the measured distances will be samples of a normally distributed random variable whose mean is the true distance. In this case the average (arithmetic mean) of the samples is an estimator of the mean of the normal distribution of the measurement population.

Gauss, who analyzed geodetic survey data for the Kingdom of Hanover, justified this procedure on the basis of minimizing a sum of squares of errors. Today we may recast this justification in terms of the sample mean being a maximum likelihood estimator (MLE) for the normal population mean.

Although an MLE is often used, it is not always the best choice of an estimator. So we shouldn't think that "nonrandom parameter estimation" is the same as MLE. For example, if the distribution is not assumed to be normal, we may find outliers occur more often than in a normal distribution and bias the MLE. For these situations methods based on robust statistics may be better.

To give a simple example, one might replace the sample mean with a sample median (middle value of the ordered observations) because this will be less affected by one or a few very large excursions from the population mean (outliers) than just taking an average of observations would be.

Since the Question mentions "navigational tracking" it may be relevant to note the classic 1964 paper by Peter Huber, Robust Estimation of a Location Parameter:

What happens if the true distribution deviates slightly from the assumed normal one? As is now well known, the sample mean may then have a catastrophically bad performance: seemingly quite mild deviations may already explode its variance.

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[by] "non-random parameter estimation" they are talking about the MLE. Is that right?

They mean parameter as in parametric statistics, where the data are imagined to be random, but the type of random variation that led to that data is determined by some parameters such as the mean and variance of a Gaussian distribution. The parameters are thought of as non-random and MLE is one method for estimating the parameters in statistical models.

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