I have the following example code, creating a object collection.
How can I remove one of the objects? (e.g. $TestList would look as though the "delete me" item was never there. I've tried .remove, .splice, .delete etc but I'm told it's not a function.
Doing typeof($TestList) brings back object, and typeof($TestList[0]) also seems valid.
Surely I don't have to recreate the collection without one item?
(function($) { 
jQuery.QuickTest = {
    $TestList: {},
    build: function()
    {
        $TestList={};
        $TestList[0] = 
        {
            title: "part 1"
        };
        $TestList[1] = 
        {
            title: "delete me please"
        };
        $TestList[2] = 
        {
            title: "part 2"
        };
    }
}
jQuery.fn.QuickTest = jQuery.QuickTest.build;   
})(jQuery);
$(document).ready(function() {
$().QuickTest(
{
})
});
We're using jQuery 1.3.
Thanks!
Review
First of all, it's very non-obvious what your code is supposed to do, but here are some issues:
jQuery.QuickTest = {
    $TestList: {},
    build: function()
    {
        $TestList={};
You define jQuery.QuickTest.$TestList, but inside build() you declare a global object $TestList.
Functions declared under jQuery.fn are supposed to act on a matched set of elements (referenced by this) and return it as well; your function does neither.
Answers
An answer to some of your questions:
.remove() is a jQuery function that removes nodes from the DOM and needs to be called on a jQuery object.
.splice() only applies to Array and even though you're accessing $TestList as if it were one, it's still just an Object.
.delete() is not any function I know ;-)
Possible solution
To delete an entry from $TestList you could use the delete in this fashion:
delete $TestList[1];
                        use delete myObject, not myObject.delete
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