Statistic test to determine whether $f(x)$ is biased towards positive value using samples of $x$ from an identical unknown distribution

Question

Some background about myself: I have sufficient knowledge in probability but not in statistics.

Suppose I have $n$ i.i.d samples, $\{x_i\}$, sampled from an unknown distribution $X$. Given a function $f: \mathcal X \rightarrow \mathbb R$, I want to conduct a statistical test using this sample to determine if $f(X)$ is "skewed towards positive" or negative value. Specifically, $f(X)$ is said to be "skewed towards positive" if $f(X) > 0$ for more than 50% of the whole population of $X$. What is the appropriate statistic test for this case? Answers with some assumptions on $X$ and/or $f$ are also welcome.

I think, I need to first define a null hypothesis that is $f(X)$ is "skewed towards positive". Then, I evaluate $\{f(x_i)\}$ for all samples and decide whether number of observation of positive values surpasses the negative values. Is this a correct statistical test or is there a better test?

Answer 1

This is a median test --- it is usually done using nonparametric methods

To put your problem in standard statistical terminology, let $Y = f(X)$ and let $\phi_Y$ denote the median of the distribution of $Y$. You want to test whether $\phi$ is positive, which is a one-sided test of the hypotheses:

$$H_0: \phi_Y = 0 \quad \quad \quad \quad \quad H_A: \phi_Y > 0.$$

(It is probably more sensible to describe things in these terms, since "skewness" has a specific meaning in probability and statistics that is different to what you are looking at here.) This is a one sample median test where you observe data of the form $y_1,...,y_n$ and you test the unknown median of the distribution of this data against a hypothesised value.

This type of testing is usually done using nonparametric methods similar to a test of distribution, but instead we now focus on only a single quantile. There are a few nonparametric tests of this kind, which are usually variants of the Pearson chi-squared test. The specific test you want to use will depend on whether or not you can have outcomes that occur exactly at the median in the null hypothesis, which depends on the details of the function you are using and the type of underlying data. In particular, if $X$ is a continuous random variable and $f$ is a continuous strictly monotonic function then $Y$ will also be a continuous function and so you will not get any data points exactly on the hypothesised median. In other cases there may be a non-zero probability of data occurring on the hypothesised median. I recommend you have a look at nonparametric one-sided median tests and use one of these to perform your hypothesis test.

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Simplification in a special case: One simple case of interest occurs when $f$ is a known monotonic function and $X$ is a continuous random variable. Consider the case where $f$ is montonically increasing and compute the input value:

$$x_* = \inf \{ x \in \mathbb{R} | f(x) > 0 \}.$$

We then have $f(x) > 0$ for all $x > x_*$ and $f(x) \leqslant 0$ for $x < 0$, which means that $\phi_Y > 0$ is equivalent to $\phi_X > x_*$. Your hypothesis test can then be framed directly in terms of the distribution of the underlying values without reference to the function:

$$H_0: \phi_X = x_* \quad \quad \quad \quad \quad H_A: \phi_X > x_*.$$

In this case you can perform your one-sided hypothesis test directly on the underlying data for $X$ using your knowledge of the structure of the function $f$. For most median tests this will be equivalent to testing on $Y$ but it takes things back a step to your underlying data before it is inputted to the function.