[Boost-users] [math/distributions] invalid values for undefined moments

Subject: [Boost-users] [math/distributions] invalid values for undefined moments
From: Thomas Mang (thomasmang.ng_at_[hidden])
Date: 2012-07-25 19:52:08

Hi,

According to the documentation, in boost 1-50 non-member functions of 
moments (e.g. mean, variance) which are not defined for a given 
distribution shall issue compile-time errors, or throw a domain_error at 
runtime otherwise. For the Cauchy Distribution, that's what it does. For 
the t-distribution, that's not what it does - the non-member functions 
happily return some complete rubbish values for mean and variance (and 
hence I'd also suppose standard deviation). For kurtosis and skewness it 
seems to work correctly but are subject to mathematical definitions. I 
have not tested other non-member functions.

Note that there are different definitions floating around as to when a 
given moment is defined. The k-th moment is always defined if k < v, but 
in other cases opinions deviate. Wikipedia states it might be infinity 
under some circumstances -  obviously what boost seems to having taken 
over. In general however I am used to - and my gut feeling the clear 
majority of people would have this perception; indeed I am not aware of 
any other source than wikipedia stating these infinity things - that 
moments are undefined if k >= v (see e.g. Probability and Statistics, 
DeGroot & Schervish 2002, for a professional textbooks and heaps of 
other sources).
So while the boost output for variance and a t-2 distribution (or 
alternatively also the kurtosis of a t-4 distribution) might be 
considered ok by some, I'd guess to most people it would be fairly 
unexpected? It had been a surprise to me, at least.

Code snippet:

typedef  boost::math::students_t_distribution<double>  t_distr;

// this is ok
std::cout << "mean(t(2)): " << boost::math::mean( t_distr(2) ) << std::endl;

// this is NOT ok - first moment not defined for given distribution
std::cout << "mean(t(1)): " << boost::math::mean( t_distr(1) ) << std::endl;

// this is ok
std::cout << "var(t(3)): " << boost::math::variance( t_distr(3) ) << 
std::endl;

// this might be ok or not
// depending on point of view, either Inf or undefined
std::cout << "var(t(2)): " << boost::math::variance( t_distr(2) ) << 
std::endl;

// this is NOT ok - second moment not defined for the given distribution
std::cout << "var(t(1)): " << boost::math::variance( t_distr(1) ) << 
std::endl;

best,
Thomas