# Logarithm of Probability Density Function

Fréchet distribution logarithm of probability density function.

The probability density function for a Fréchet random variable is

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

## Usage

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


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

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

var y = logpdf( 10.0, 2.0, 3.0, 5.0 );
// returns ~-2.298

y = logpdf( -3.0, 1.0, 2.0, -4.0 );
// returns ~-1.307

y = logpdf( 0.0, 2.0, 1.0, -1.0 );
// returns ~-0.307


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

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


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

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

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

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

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


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

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

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


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

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

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


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

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

var mylogpdf = logpdf.factory( 3.0, 3.0, 5.0 );

var y = mylogpdf( 10.0 );
// returns ~-2.259

y = mylogpdf( 7.0 );
// returns ~-1.753


## 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 logpdf = require( '@stdlib/stats/base/dists/frechet/logpdf' );

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 = logpdf( 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 ) );
}