Programmatically infer categories of a specific Wordpress parent

In the example below, we will display links with the name of the categories of a certain parent rubric

<?php
   $categories = get_categories( array(
        'taxonomy'     => 'category',
        'type'         => 'post',
        'child_of'     => '',
        'parent'       => 2,
	'orderby'      => 'name',
	'order'        => 'ASC',
	'hide_empty'   => 1,
	'hierarchical' => 1,
	'exclude'      => '',
	'include'      => '',
	'number'       => 0,
	'pad_counts'   => false,
   ) );
   if( $categories ){
	foreach( $categories as $cat ){
		// Data in object $cat
		// $cat->term_id
	        // $cat->name (Rubric 1)
		// $cat->slug (rubrika-1)
		// $cat->term_group (0)
		// $cat->term_taxonomy_id (4)
		// $cat->taxonomy (category)
		// $cat->description (Description text)
		// $cat->parent (0)
		// $cat->count (14)
		// $cat->object_id (2743)
		// $cat->cat_ID (4)
		// $cat->category_count (14)
		// $cat->category_description (Description text)
		// $cat->cat_name (Rubric 1)
		// $cat->category_nicename (rubrika-1)
		// $cat->category_parent (0)
		echo '<p><a href="'.get_category_link( $cat->term_id ).'">'.$cat->name.'</a></p>';
	}
}
?>