Question

Concept Check: Terminology

Suppose you observe iid samples ?₁,…,?_?∼? from some unknown distribution ? . Let F denote a parametric family of probability distributions (for example, F could be the family of normal distributions {N(?,?²)}_{?∈ℝ,?²>0} ).

In the topic of goodness of fit testing, our goal is to answer the question "Does ? belong to the family F , or is ? any distribution outside of F ?"

Parametric hypothesis testing is a particular case of goodness of fit testing (why?). However, in the context of parametric hypothesis testing, we assume that the data distribution ? comes from some parametric statistical model {??}?∈Θ , and we ask if the distribution ? belongs to a submodel {?_?}_?∈Θ0 or its complement {?_?}_?∈Θ1 . In parametric hypothesis testing, we allow only a small set of alternatives {?_?}_?∈Θ1 , where as in the goodness of fit testing, we allow the alternative to be anything.

Categorize the following problems as examples of parameter estimation, parametric hypothesis testing, or goodness of fit testing. (Choose all categories that apply.)

Problem 1: Estimate the bias of an unfair coin.

Parameter estimation
Parametric hypothesis testing
Goodness of fit testing unanswered

Problem 2: Decide if a 6-sided die is fair or not.

Parameter estimation
Parametric hypothesis testing
Goodness of fit testing unanswered

Problem 3: Decide if the heights of pine trees in Canada have a Gaussian distribution. Assume that the statistical model for this data is {ℝ,Π} where Π denotes the set of all probability distributions with sample space ℝ . (In particular, this model is non-parametric.)

Parameter estimation
Parametric hypothesis testing
Goodness of fit testing

Answer 1

ANSWER:

1)

The probability of heads (or tails) of a given coin is to be estimated here. Thus we are looking for a parameter 'BIAS'. Hence, it is an example of parameter estimation.

Option A is correct.

Parameter hypothesis testing is incorrect because we are not trying to determine if the value of the parameter lies in some specific range of the parametric space.

2)

Here we are interested in TESTING if the probability of each output from 1 to 6 is indeed 1/6 (fair) or not. Thus it is an example of parametric hypothesis testing. Our hypotheses will be that the dice is fair or not fair.

Also, if the dice is fair then the events of getting 1 to 6 will belong to the family of uniform distribution. Hence, it is also an example of goodness of fit testing as we are testing whether the probability distribution belongs to the family of uniform distribution or not.

Options B and C are correct.

3)

It has been specifically mentioned that the model is NON-PARAMETRIC. Thus it cannot belong to either parameter estimation or parametric hypothesis testing.

We just want to know whether the heights of pine trees are a good fit for some distribution of the Gaussian family.

Hence, option C is correct.

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