# 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/math/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 0.

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


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 ~-3.381


## Examples

var randu = require( '@stdlib/random/base/randu' );
var logcdf = require( '@stdlib/math/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 ) );
}