# Does Maximizing Likelihood Over Truncated Support Increase Probability?

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• kullbach_liebler
kullbach_liebler
Suppose ##\mathbf{X}## is a random variable with a finite support ##\Omega## and with some pdf ##f(\cdot; \mathbf{v}_0)## where ##\mathbf{v}_0## is the parameter vector. Define, ##\mathcal{A}:= \{\mathbf{x}:S(\mathbf{x}) \geq \gamma\} \subset \Omega## and ##\tilde{\mathbf{x}}:=S(\tilde{\mathbf{x}}) = \gamma##, ##\mathcal{B}:=\{\mathbf{x} \geq \tilde{\mathbf{x}}\}## where ##S:\mathbf{x} \mapsto \mathbb{R}^n##. Moreover, suppose that,

$$\#(\mathcal{B} \cap \mathcal{A}) > \#(\neg \mathcal{B} \cap \mathcal{A})$$
and
$$!\exists \mathbf{x}^*>\tilde{\mathbf{x}}:=\arg \max_{\mathbf{x}}S(\mathbf{x}).$$

Then, it implies that,

\begin{align}
\max_{\mathbf{v}}\sum_{\mathbf{x} :S(\mathbf{x}) \geq \gamma} f(\mathbf{x};\mathbf{v}) > \sum_{\mathbf{x} :S(\mathbf{x}) \geq \gamma} f(\mathbf{x};\mathbf{v}_0)
\end{align}

My questions:

1) Is the statement true?;
2) How could I improve the presentation of this proposition?;
3) What are the mildest possible conditions under which (1) will hold?

Last edited:
kullbach_liebler said:
2) How could I improve the presentation of this proposition?;
I don't know what a statistical journal would want, but for the purpose of getting an answer on an internet forum, you could give a more verbal statement of the proposition before presenting it using only notation.

\begin{align}
\max_{\mathbf{v}}\sum_{\mathbf{x} :S(\mathbf{x}) \geq \gamma} f(\mathbf{x};\mathbf{v}) > \sum_{\mathbf{x} :S(\mathbf{x}) \geq \gamma} f(\mathbf{x};\mathbf{v}_0)
\end{align}

This appears to say the maximum value of a function ( which is a summation rather than an integration) when taken over a set is greater than the value of that function evaluated at the particular element ##v_o## in that set. Is that the general idea?

kullbach_liebler said:
1) Is the statement true?
Let ## \mathbf{v_max} ## be a value of ## \mathbf{v} ## which maximizes the sum. What happens when ## \mathbf{v_0} = \mathbf{v_max} ##?

## 1. What does "maximizing likelihood over truncated support" mean?

Maximizing likelihood over truncated support refers to the process of optimizing a likelihood function where the domain (support) of the variable is restricted or truncated. This is often done in cases where the data or the variable of interest is only observed within certain bounds, and the likelihood function is adjusted accordingly to reflect this truncated support.

## 2. How does truncation affect the likelihood function?

Truncation modifies the likelihood function by limiting the range of values that the variable can take. This changes the normalization constant of the probability distribution, as the total probability mass must now be redistributed over the truncated range. As a result, the likelihood function is adjusted to account for the fact that observations outside the truncated range are impossible or not considered.

## 3. Does maximizing the likelihood over truncated support lead to a higher probability for observed data?

Maximizing the likelihood over truncated support does not necessarily lead to a higher probability for the observed data in an absolute sense. Instead, it leads to the best possible fit given the constraints of the truncated range. The likelihood is maximized within the feasible region, which means the model parameters are adjusted to best explain the observed data within the truncated support.

## 4. How is the process of maximum likelihood estimation (MLE) affected by truncation?

In the presence of truncation, the MLE process must account for the restricted support by modifying the likelihood function accordingly. This often involves integrating the probability density function over the truncated range and adjusting the normalization constant. The resulting MLE estimates are those that maximize the truncated likelihood function, which may differ from the estimates obtained without truncation.

## 5. Are there any practical applications where maximizing likelihood over truncated support is particularly useful?

Yes, maximizing likelihood over truncated support is particularly useful in fields such as survival analysis, econometrics, and reliability engineering, where data may be censored or truncated. For example, in survival analysis, the time to an event (such as failure or death) may only be observed within a certain period, leading to truncated data. Similarly, in econometrics, income data might be truncated at certain thresholds due to reporting limits. In these cases, adjusting the likelihood function for truncation allows for more accurate parameter estimation and better model fitting.

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