Likelihood determination for a step-like pdf

Question

Suppose that random numbers x are generated on the computer using the following procedure:

1. Generate two numbers $x_1$, $x_2$ from a uniform distribution $\mathcal{U}$([0,1])
2. If $x_1$ > f, take x = $x_2$, otherwise x = m $x_2$,

where f and m are two real constants between 0 and 1.

The value of f=0.2 is known, while m is unknown and we want to infer its value from a list of N outputs using the Maximum Likelihood Estimator.

I somehow managed to write the pdf $p_X(x)$ for a single observation:

$p_X(x)=\frac{1}{m(1+f)+f}\mathcal{I}(0<x<m)+\frac{f}{m(1+f)-f}\mathcal{I}(m<x<1)$,

where $\mathcal{I}(a<x<b)$ is the indicator function.

The likelihood is defined as $\mathcal{L}=\prod_i p(x_i)$, and in order to find the MLE i have to solve the known equation $\frac{d}{dm}\log\mathcal{L}=0$.

I find some issues in writing the likelihood, because i find myself with

$\mathcal{L(m|x_1,...,x_N)}=\prod_i p(x_i)=\frac{1}{(m(1+f)+f)^a}+\frac{(1-f)^{N-a}}{(m(1+f)+f)^{N-a}}$, where $a$ is the number of observations $x_i$ for which $x_i<m$. But i added a further unknown parameter to my problem.

I think I am having some issues multiplying the indicator functions, or maybe I have determined the wrong pdf for my problem.

Many thanks in advance.

Answer 1

As $x_1$ and $x_2$ are independent, the PDF can be written as a mixture of two uniform distributions, one with probability $(1-f)$ and support $(0,1)$, and the other with probability $f$ and support $(0,m)$:

$$p(x) = {f \over m}1(x \leq m) + 1 - f$$

As a check, we show this integrates to one:

$$\int_0^1\left({f \over m}1(x \leq m) + 1 - f\right)dx = {f \over m}\int_0^mdx+(1-f)\int_0^1dx = f+(1-f) = 1$$

The multivariate version is:

$$\begin{eqnarray} p(x) &=& \Pi_{i=1}^n\left({f \over m}1(x_i \leq m) + 1 - f\right)\\ &=&\Pi_{i:x_i\leq m}\left({f \over m}+1-f\right)\Pi_{i:x_i>m}(1-f)\\ &\propto&\left({f \over m}+1-f\right)^{n(x \leq m)}(1-f)^{n - n(x \leq m)} \\ &\propto& \left(1+{f \over (1-f)m}\right)^{n(x \leq m)} \end{eqnarray}$$

where $n(x \leq m)$ is the number of observations for which $x \leq m$. The $\propto$ in the last two lines is because we have moved from evaluating the probability based on each $x_i$ to evaluating the probability based on $n(x \leq m)$; we need to find the constant of integration, which is straightforward:

$$p(x) = {\left(1+{f \over (1-f)m}\right)^{n(x \leq m)} \over \sum_{i=0}^n\left(1+{f \over (1-f)m}\right)^i}$$

Given that $m$ appears in the term $n(x \leq m)$ in the exponent as well as in the $(1 + f/((1-f)m))$ terms in both numerator and denominator and that $n(x \leq m)$ is a piecewise horizontal function with $n$ jumps, I suspect finding $\hat{m}$ by numerical methods is probably the way to go.