## What is a Parameter Space?

Statistical inference concerns an unknown parameter θ. Sometimes we know nothing about θ but most of the time we have an idea about what kinds of numbers θ can take on: real numbers, subsets of real numbers, complex numbers, even higher-dimensions. Those possible values for θ are called a **parameter space**, usually denoted by Ω. This space tells us everything that is known, or unknown, about the parameter. In other words, it tell us “to what degree” θ is unknown [1].

Parameter spaces are often used to describe probability distributions: the potential model is chosen based on a set of sampled experimental variables. These variables are input into the model combined with a set of real numbers or vectors to create a parameter space.

The elements of the parameter space are sometimes called “parameters”, but that term can get a little confusing because we often refer to an “unknown population parameter” in statistics as simply a *parameter*. The thing to remember here is that we are usually dealing with a large number of possible parameters (even an infinite number), but if we have specified the parameter space correctly, then our unknown parameter is definitely contained within that space. It narrows the field of candidates down, usually to a workable number.

## Parameter Space in Hypothesis Tests

In hypothesis testing, the total possible parameter space is all of the values that make up the null hypothesis and alternate hypotheses. Therefore, the null hypothesis is a subset of the total parameter space and so is the alternate hypothesis. The alternate hypothesis is the complement of the null hypothesis, with respect to the parameter space.

For example, let’s say the total possible parameter space for a mean value is the real number line (i.e., -∞ < μ < ∞).

- If your null hypothesis is H
_{0}: μ = 5, then the alternate hypothesis H_{1}is the complement of the null parameter space: μ ≠ 5. - If your null hypothesis is H
_{0}: μ > 5, then the alternate hypothesis H_{1}is the complement of the null parameter space: μ ≤ 5.

## References

[1] Parameters, estimation, likelihood function. Math 2710: A Second Course in Statistics. Retrieved November 17, 2021 from: http://pi.math.cornell.edu/~web2710/handouts/parameters-estimation.pdf

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