inline RealType quantile(const inverse_chi_squared_distribution<RealType, Policy>& dist, const RealType& p)
{
using boost::math::gamma_q_inv;
RealType df = dist.degrees_of_freedom();
RealType scale = dist.scale();
static const char* function = "boost::math::quantile(const inverse_chi_squared_distribution<%1%>&, %1%)";
// Error check:
RealType error_result;
if(false == detail::check_df(
function, df, &error_result, Policy())
&& detail::check_probability(
function, p, &error_result, Policy()))
{
return error_result;
}
if(false == detail::check_probability(
function, p, &error_result, Policy()))
{
return error_result;
}
// RealType shape = df /2; // inv_gamma shape,
// RealType scale = df * scale/2; // inv_gamma scale,
// result = scale / gamma_q_inv(shape, p, Policy());
RealType result = gamma_q_inv(df /2, p, Policy());
if(result == 0)
return policies::raise_overflow_error<RealType, Policy>(function, "Random variable is infinite.", Policy());
result = df * (scale / 2) / result;
return result;
} // quantile
inline RealType cdf(const inverse_chi_squared_distribution<RealType, Policy>& dist, const RealType& x)
{
static const char* function = "boost::math::cdf(const inverse_chi_squared_distribution<%1%>&, %1%)";
RealType df = dist.degrees_of_freedom();
RealType scale = dist.scale();
RealType error_result;
if(false ==
detail::check_inverse_chi_squared(function, df, scale, &error_result, Policy())
)
{ // Bad distribution.
return error_result;
}
if((x < 0) || !(boost::math::isfinite)(x))
{ // Bad x.
return policies::raise_domain_error<RealType>(
function, "inverse Chi Square parameter was %1%, but must be >= 0 !", x, Policy());
}
if (x == 0)
{ // Treat zero as a special case.
return 0;
}
// RealType shape = df /2; // inv_gamma shape,
// RealType scale = df * scale/2; // inv_gamma scale,
// result = boost::math::gamma_q(shape, scale / x, Policy()); // inverse_gamma code.
return boost::math::gamma_q(df / 2, (df * (scale / 2)) / x, Policy());
} // cdf
inline RealType mean(const inverse_chi_squared_distribution<RealType, Policy>& dist)
{ // Mean of inverse Chi-Squared distribution.
RealType df = dist.degrees_of_freedom();
RealType scale = dist.scale();
static const char* function = "boost::math::mean(const inverse_chi_squared_distribution<%1%>&)";
if(df <= 2)
return policies::raise_domain_error<RealType>(
function,
"inverse Chi-Squared distribution only has a mode for degrees of freedom > 2, but got degrees of freedom = %1%.",
df, Policy());
return (df * scale) / (df - 2);
} // mean
inline RealType variance(const inverse_chi_squared_distribution<RealType, Policy>& dist)
{ // Variance of inverse Chi-Squared distribution.
RealType df = dist.degrees_of_freedom();
RealType scale = dist.scale();
static const char* function = "boost::math::variance(const inverse_chi_squared_distribution<%1%>&)";
if(df <= 4)
{
return policies::raise_domain_error<RealType>(
function,
"inverse Chi-Squared distribution only has a variance for degrees of freedom > 4, but got degrees of freedom = %1%.",
df, Policy());
}
return 2 * df * df * scale * scale / ((df - 2)*(df - 2) * (df - 4));
} // variance
inline RealType mode(const inverse_chi_squared_distribution<RealType, Policy>& dist)
{ // mode is not defined in Mathematica.
// See Discussion section http://en.wikipedia.org/wiki/Talk:Scaled-inverse-chi-square_distribution
// for origin of the formula used below.
RealType df = dist.degrees_of_freedom();
RealType scale = dist.scale();
static const char* function = "boost::math::mode(const inverse_chi_squared_distribution<%1%>&)";
if(df < 0)
return policies::raise_domain_error<RealType>(
function,
"inverse Chi-Squared distribution only has a mode for degrees of freedom >= 0, but got degrees of freedom = %1%.",
df, Policy());
return (df * scale) / (df + 2);
}
inline RealType kurtosis_excess(const inverse_chi_squared_distribution<RealType, Policy>& dist)
{
RealType df = dist.degrees_of_freedom();
static const char* function = "boost::math::kurtosis(const inverse_chi_squared_distribution<%1%>&)";
if(df <= 8)
return policies::raise_domain_error<RealType>(
function,
"inverse Chi-Squared distribution only has a kurtosis excess for degrees of freedom > 8, but got degrees of freedom = %1%.",
df, Policy());
return 12 * (5 * df - 22) / ((df - 6 )*(df - 8)); // Not a function of scale.
}
inline RealType skewness(const inverse_chi_squared_distribution<RealType, Policy>& dist)
{
BOOST_MATH_STD_USING // For ADL
RealType df = dist.degrees_of_freedom();
static const char* function = "boost::math::skewness(const inverse_chi_squared_distribution<%1%>&)";
if(df <= 6)
return policies::raise_domain_error<RealType>(
function,
"inverse Chi-Squared distribution only has a skewness for degrees of freedom > 6, but got degrees of freedom = %1%.",
df, Policy());
return 4 * sqrt (2 * (df - 4)) / (df - 6); // Not a function of scale.
}
RealType pdf(const inverse_chi_squared_distribution<RealType, Policy>& dist, const RealType& x)
{
BOOST_MATH_STD_USING // for ADL of std functions.
RealType df = dist.degrees_of_freedom();
RealType scale = dist.scale();
RealType error_result;
static const char* function = "boost::math::pdf(const inverse_chi_squared_distribution<%1%>&, %1%)";
if(false == detail::check_inverse_chi_squared
(function, df, scale, &error_result, Policy())
)
{ // Bad distribution.
return error_result;
}
if((x < 0) || !(boost::math::isfinite)(x))
{ // Bad x.
return policies::raise_domain_error<RealType>(
function, "inverse Chi Square parameter was %1%, but must be >= 0 !", x, Policy());
}
if(x == 0)
{ // Treat as special case.
return 0;
}
// Wikipedia scaled inverse chi sq (df, scale) related to inv gamma (df/2, df * scale /2)
// so use inverse gamma pdf with shape = df/2, scale df * scale /2
// RealType shape = df /2; // inv_gamma shape
// RealType scale = df * scale/2; // inv_gamma scale
// RealType result = gamma_p_derivative(shape, scale / x, Policy()) * scale / (x * x);
RealType result = df * scale/2 / x;
if(result < tools::min_value<RealType>())
return 0; // Random variable is near enough infinite.
result = gamma_p_derivative(df/2, result, Policy()) * df * scale/2;
if(result != 0) // prevent 0 / 0, gamma_p_derivative -> 0 faster than x^2
result /= (x * x);
return result;
} // pdf