# Logarithm of Cumulative Distribution Function

Fréchet distribution logarithm of cumulative distribution function.

The cumulative distribution function for a Fréchet random variable is

where alpha > 0 is the shape, s > 0 the scale, and m the location parameter.

## Usage

var logcdf = require( '@stdlib/stats/base/dists/frechet/logcdf' );


#### logcdf( x, alpha, s, m )

Evaluates the natural logarithm of the cumulative distribution function (CDF) for a Fréchet distribution with shape alpha, scale s, and location m at a value x.

var y = logcdf( 10.0, 2.0, 3.0, 5.0 );
// returns ~-0.36

y = logcdf( -3.4, 1.0, 2.0, -4.0 );
// returns ~-3.333

y = logcdf( 0.0, 2.0, 1.0, -1.0 );
// returns -1.0


If provided x <= m, the function returns -Infinity.

y = logcdf( -2.0, 2.0, 1.0, -1.0 );
// returns -Infinity


If provided NaN as any argument, the function returns NaN.

var y = logcdf( NaN, 1.0, 1.0, 0.0 );
// returns NaN

y = logcdf( 0.0, NaN, 1.0, 0.0 );
// returns NaN

y = logcdf( 0.0, 1.0, NaN, 0.0);
// returns NaN

y = logcdf( 0.0, 1.0, 1.0, NaN );
// returns NaN


If provided alpha <= 0, the function returns NaN.

var y = logcdf( 2.0, -0.1, 1.0, 1.0 );
// returns NaN

y = logcdf( 2.0, 0.0, 1.0, 1.0 );
// returns NaN


If provided s <= 0, the function returns NaN.

var y = logcdf( 2.0, 1.0, -1.0, 1.0 );
// returns NaN

y = logcdf( 2.0, 1.0, 0.0, 1.0 );
// returns NaN


#### logcdf.factory( alpha, s, m )

Returns a function for evaluating the natural logarithm of the cumulative distribution function of a Fréchet distribution with shape alpha, scale s, and location m.

var mylogcdf = logcdf.factory( 3.0, 3.0, 5.0 );

var y = mylogcdf( 10.0 );
// returns ~-0.216

y = mylogcdf( 7.0 );
// returns ~-3.375


## Notes

• In virtually all cases, using the logpdf or logcdf functions is preferable to manually computing the logarithm of the pdf or cdf, respectively, since the latter is prone to overflow and underflow.

## Examples

var randu = require( '@stdlib/random/base/randu' );
var logcdf = require( '@stdlib/stats/base/dists/frechet/logcdf' );

var alpha;
var m;
var s;
var x;
var y;
var i;

for ( i = 0; i < 100; i++ ) {
alpha = randu() * 10.0;
x = randu() * 10.0;
s = randu() * 10.0;
m = randu() * 10.0;
y = logcdf( x, alpha, s, m );
console.log( 'x: %d, α: %d, s: %d, m: %d, ln(F(x;α,s,m)): %d', x.toFixed( 4 ), alpha.toFixed( 4 ), s.toFixed( 4 ), m.toFixed( 4 ), y.toFixed( 4 ) );
}