I am facing difficulty in identifying how the formula given by Eq(2) in the paper
Wen-Chi Tsai and Anirban DasGupta, On the Strong Consistency, Weak Limits and Practical Performance of the ML Estimate and Bayesian Estimates of a Symmetric Domain in $R^k$, Lecture Notes-Monograph Series Vol. 45, A Festschrift for Herman Rubin (2004), pp. 291-308
expressed as the log-likelihood $$L(p,r|x_1,\ldots,x_n) = \frac{1}{\lambda{(B_{p,r})}^n} \mathbf{1}_{(p,r):x_i \in B_{p,r}} \forall i =1,2,\ldots,n$$
comes. In the expression $\lambda(B)$ is the Lebesgue measure of a ball $B_{p,r}$. $B_{p,r}$ denotes the centered $L_p$ ball with radius $r$ w.r.t the metric induced from $p$- norms in the $k$ -dimensional Euclidean space, $R^k$. In general, the log-likelihood expression is expressed as the logarithm of the pdf. Here, where is the pdf and why the Lebesgue measure goes in the denominator and what is the numeral one ?
I checked the references given below the formula Table of Integrals Series and Products pg 647 but could not find anything related to the pdf or the formula. Can somebody please explain what is the meaning of the expression and what is the pdf?
The paper states directly before equation $2$ that $x_1, x_2, \cdots, x_n$ are uniformly drawn from the set $S$. The set $S$ was previously defined to be $B_{p, r}$, the $L^p$ ball of radius $r$ you mentioned above. The density of such as uniform random variable can be stated as: $$P(x_1, \cdots, x_n | p, r) = \frac{1}{\lambda(B_{p, r})}1_{(x_i \in B_{p,r}\,\forall i)}$$ Here $1_{(x_i \in B_{p,r}\,\forall i)}$ is the indicator function of the set $\{(x_i \in B_{p,r}\,\forall i)\}$. That is the set of all $n$ tuples of $x_i$ that all live within the ball $B_{p,r}$. The function is $1$ if all of the points are in the ball and $0$ otherwise. The volume of the ball is in the denominator to normalize the uniform density and ensure the total probability is $1$.
Now the likelihood function is just the same formula with the understand that $\{x_i\}$ have been observed and is a function of $p, r$.
$$L(p, r | x_1, \cdots, x_n) = \frac{1}{\lambda(B_{p, r})}1_{(x_i \in B_{p,r}\,\forall i)}$$