Figure 2.4 on P32 looks different from what I learnt in CFA books.
Compared to Normal Distribution, isn’t Leptokurtic Distribution high-peak and fat-tailed?
Not sure what you are referring to since I do not have the books. Maybe you would consider uploading the figure somewhere.
But yes, Leptokurtic is kurtosis >3 => high peakedness.
Leptokurtosis doesn’t require a higher peak than a normal distribution; for example, a Student’s t-distribution with 10 degrees of freedom is leptokurtic (kurtosis = 4.0), but its peak is lower than that of a normal distribution. All Student’s t-distributions are leptokurtic, and all have a peak lower than that of a normal distribution.
A better description is _ narrow _-peaked and fat-tailed.
Yes leptokurtic distribution is of symmetrical ib shape and more peaked than on normal distribution or belled shape distribution and it has also fatter tails.
If by “more peaked” you mean that it has a higher peak, then you’re mistaken. All Student’s t-distributions are leptokurtic and all have a peak that is _ lower _ than a normal distribution. The key is that the peaks are _ narrower _, not that they’re higher.
Right, but the Student’s t-distribution is a special case. Most of the errors in understanding kurtosis stem from the fact that we are showered with illustrations showing a “leptokurtic” or “platykurtic” distribution superimposed upon a “normal” distribution – but the problem is that the textbook authors fail to explain that those pictures are based on distributions scaled to have the same variance.
If the two distributions (one normal and one with excess kurtosis) have the same scaled variance and the leptokurtic one satisfies the Dyson-Finucan condition, then the leptokurtic distribution most certainly will have a higher peak than the normal one.
But isn’t it better to have a definition (or characterization, or description) that covers _ all _ cases – even the special ones – than to have one that covers only the common cases? Isn’t this the essence of mathematical definitions?
This is certainly true, but hardly surprising, inasmuch as the Dyson-Finucan condition is that the density function of the subject distribution crosses that of the normal distribution four times (twice on each side of the mean); i.e., it has a higher peak. So all you’ve said is that if a function satisfies the condition of having a higher peak than the normal distribution, then it has a higher peak than the normal distribution. True, but not particularly profound.
Thank you for telling me what I already know about Dyson-Finucan. Tell me, do you disagree with the notion that two distributions, of the same scaled variance, where one is normal and the other is leptokurtic, that the leptokurtic one would have a higher peak, and not just be “narrower?” My point is simply that if we are looking at the same scale of data, then there is nothing wrong with the OP’s interpretation of how excess kurtosis looks graphically with two superimposed distributions.
In fact, you should look at DeCarlo (1997, “On the Meaning and Use of Kurtosis”) – this line sums it up best, on page 295: “…Figure 4 is also relevant to the illustration of the t distribution with varying degrees of freedom that is given in many textbooks. The typical illustration shows that, as degrees of freedom decrease, the t distribution appears flatter with heavier tails than the normal, and it is often described in this way. However, the t is actually more peaked than the normal…the apparent flatness in textbook illustrations arises because of the larger variance of the t as the degrees of freedom decrease.”
Good point: my mistake. I apologize.
Oh, no worries! I love me a good intellectual debate. Keeps my aging brain sharp…