A way to see if Shakespeare actually wrote what he claimed to is to use Stylometry. Stylometry was invented to investigate questions of literary authorship. Stylometry would go over the syntax, semantics, vocabulary, and grammar of his works.
Each author's writing will have different characteristics. Whether it's the average sentence length, a most commonly used word, or even the arrangement of words. If you take some of these characteristic examples and apply them to Shakespeare's writing, you will be able to see if his characteristics remain true, and if he really wrote what he claimed to.
If you look at all the times Shakespeare has used the word "Thee," "Gentle," or "Virtue" in his writing, you will see that they are common words that he uses around the same amount of times within all of his works.
Using PCI, you can reduce the multidimensional space into simple principle components that collectively measure the variance in Shakespeare's works. You can then test the works of the believed REAL authors of Shakespeare's works. So, if enough works of a different author fall within Shakespeare's variance, it would give strong evidence that Shakespeare did not really write what he claimed to.
According to the stylometrists who carried this out, they concluded that Shakespeare was not fictional, and really did write what he claimed to. His style of writing just didn't match up with his candidates, even past the compelling evidence.
Shakespeare is one of the greatest English writers of all time, which of course brought speculations from readers. Although the evidence was compelling, nothing could fully prove that it was not him who wrote his works. Using Stylometry and Principle Component Analysis, the speculations were proved wrong and he is still known as one of the greatest writers.