Using a small paring knife, remove sections from grapefruit. If sections are too large, cut in half crosswise. Set aside.
Slice avocado into 1/8-inch-thick slices; cut slices into 1-inch lengths. Toss in bowl with lemon juice, and set aside.
Cut lobster into bite-size pieces; set aside.
Trim bottoms from endives; separate spears. Place a grapefruit section on top of each endive spear. Top with piece of avocado, then a piece of lobster. Season with salt and pepper; garnish with tarragon. Repeat with remaining ingredients. Serve immediately.