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Suppose I draw $n$ samples of some random variable $X$. I repeat this process $k$ times so that I end up with $k \times n$ observations.

Each time I draw a random sample, my data is censored, meaning that I cannot observe values larger than some maximum value. Specifically, the values observed is bounded from above by $c_i$ for $i \in 1,…,k$, each time a sample is drawn, where $c_1$, $c_2$, … $c_k$ can be any positive real value.

We know that $X$ follows a lognormal distribution with parameters $\mu$ and $\sigma$.

How do I estimate $\mu$ and $\sigma$ with this censored data, with arbitrary upper bounds $\{c_i\}_{i=1}^k$?

We can suppose that we know the support of $X$.

I looked at this source about censored data, but its prescriptions are not clear, and I cannot access the source they cite:

If all observations are censored, the MLE is undefined when there is a single censoring limit and defined but extremely poorly estimated when there are multiple censoring limits. The best approach for such data is to report the median, calculated as the median of the censoring levels (Helsel 2012, p. 143-4)

I guess this suggests taking the median of $c_1,…,c_k$?

My intuition says that you can use information about how the estimated moments of the samples related to the limit $c_i$. Suppose the distribution of $X$ is a truncated lognormal and that we know that the support of $X$ is $[a,b]$, then. Then, as $c_i \rightarrow b$ we should expect the sample moments to be unbiased estimates of the population moments.

Hence, one thing I might try is the following. I could generate estimates for each $i$:

$$\{\hat{\mu}_i, \hat{\sigma}_i\, c_i\}_{i=1}^k$$

I could then model this as follows:

$$\hat{\mu}_i = \alpha_0 + \alpha_1 c_i $$ $$\hat{\sigma}_i = \beta_0 + \beta_1 c_i $$

which I could estimate using, say, OLS. Knowing the support of $X$, I can predict what the value of $\hat{\mu}_i$ and $\hat{\sigma}_i$ would be if $c_i \rightarrow b$. That is:

$$\underset{c_i \rightarrow b}{lim}[\hat{\mu}_i]= \hat{\alpha_0} + \hat{\alpha_1} b$$ $$\underset{c_i \rightarrow b}{lim}[\hat{\sigma}_i]= \hat{\beta_0} + \hat{\beta_1} b$$

Then, if $\hat{\mu}$ or $\hat{\sigma}$ relates to $c_i$ in this linear fashion, this should give unbiased estimates of the population moments. Of course, there's no reason to expect this linear relationship, but I don't currently have other ideas.

Suppose I draw $n$ samples of some random variable $X$. I repeat this process $k$ times so that I end up with $k \times n$ observations.

Each time I draw a random sample, my data is censored, meaning that I cannot observe values larger than some maximum value. Specifically, the values observed is bounded from above by $c_i$ for $i \in 1,…,k$, each time a sample is drawn, where $c_1$, $c_2$, … $c_k$ can be any positive real value.

We know that $X$ follows a lognormal distribution with parameters $\mu$ and $\sigma$.

How do I estimate $\mu$ and $\sigma$ with this censored data, with arbitrary upper bounds $\{c_i\}_{i=1}^k$?

We can suppose that we know the support of $X$.

I looked at this source about censored data, but its prescriptions are not clear, and I cannot access the source they cite:

If all observations are censored, the MLE is undefined when there is a single censoring limit and defined but extremely poorly estimated when there are multiple censoring limits. The best approach for such data is to report the median, calculated as the median of the censoring levels (Helsel 2012, p. 143-4)

I guess this suggests taking the median of $c_1,…,c_k$?

My intuition says that you can use information about how the estimated moments of the samples related to the limit $c_i$. Suppose we know that the support of $X$ is $[a,b]$, then, as $c_i \rightarrow b$ we should expect the sample moments to be unbiased estimates of the population moments.

Hence, one thing I might try is the following. I could generate estimates for each $i$:

$$\{\hat{\mu}_i, \hat{\sigma}_i\, c_i\}_{i=1}^k$$

I could then model this as follows:

$$\hat{\mu}_i = \alpha_0 + \alpha_1 c_i $$ $$\hat{\sigma}_i = \beta_0 + \beta_1 c_i $$

which I could estimate using, say, OLS. Knowing the support of $X$, I can predict what the value of $\hat{\mu}_i$ and $\hat{\sigma}_i$ would be if $c_i \rightarrow b$. That is:

$$\underset{c_i \rightarrow b}{lim}[\hat{\mu}_i]= \hat{\alpha_0} + \hat{\alpha_1} b$$ $$\underset{c_i \rightarrow b}{lim}[\hat{\sigma}_i]= \hat{\beta_0} + \hat{\beta_1} b$$

Then, if $\hat{\mu}$ or $\hat{\sigma}$ relates to $c_i$ in this linear fashion, this should give unbiased estimates of the population moments. Of course, there's no reason to expect this linear relationship, but I don't currently have other ideas.

Suppose I draw $n$ samples of some random variable $X$. I repeat this process $k$ times so that I end up with $k \times n$ observations.

Each time I draw a random sample, my data is censored, meaning that I cannot observe values larger than some maximum value. Specifically, the values observed is bounded from above by $c_i$ for $i \in 1,…,k$, each time a sample is drawn, where $c_1$, $c_2$, … $c_k$ can be any positive real value.

We know that $X$ follows a lognormal distribution with parameters $\mu$ and $\sigma$.

How do I estimate $\mu$ and $\sigma$ with this censored data, with arbitrary upper bounds $\{c_i\}_{i=1}^k$?

I looked at this source about censored data, but its prescriptions are not clear, and I cannot access the source they cite:

If all observations are censored, the MLE is undefined when there is a single censoring limit and defined but extremely poorly estimated when there are multiple censoring limits. The best approach for such data is to report the median, calculated as the median of the censoring levels (Helsel 2012, p. 143-4)

My intuition says that you can use information about how the estimated moments of the samples related to the limit $c_i$. Suppose the distribution of $X$ is a truncated lognormal and that we know that the support of $X$ is $[a,b]$. Then, as $c_i \rightarrow b$ we should expect the sample moments to be unbiased estimates of the population moments.

Hence, one thing I might try is the following. I could generate estimates for each $i$:

$$\{\hat{\mu}_i, \hat{\sigma}_i\, c_i\}_{i=1}^k$$

I could then model this as follows:

$$\hat{\mu}_i = \alpha_0 + \alpha_1 c_i $$ $$\hat{\sigma}_i = \beta_0 + \beta_1 c_i $$

which I could estimate using, say, OLS. Knowing the support of $X$, I can predict what the value of $\hat{\mu}_i$ and $\hat{\sigma}_i$ would be if $c_i \rightarrow b$. That is:

$$\underset{c_i \rightarrow b}{lim}[\hat{\mu}_i]= \hat{\alpha_0} + \hat{\alpha_1} b$$ $$\underset{c_i \rightarrow b}{lim}[\hat{\sigma}_i]= \hat{\beta_0} + \hat{\beta_1} b$$

Then, if $\hat{\mu}$ or $\hat{\sigma}$ relates to $c_i$ in this linear fashion, this should give unbiased estimates of the population moments. Of course, there's no reason to expect this linear relationship, but I don't currently have other ideas.

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Tamay
  • 505
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Suppose I draw $n$ samples of some random variable $X$. I repeat this process $k$ times so that I end up with $k \times n$ observations.

Each time I draw a random sample, my data is censored, meaning that I cannot observe values larger than some maximum value. Specifically, the values observed is bounded from above by $c_i$ for $i \in 1,…,k$, each time a sample is drawn, where $c_1$, $c_2$, … $c_k$ can be any positive real value.

We know that $X$ follows a lognormal distribution with parameters $\mu$ and $\sigma$.

How do I estimate $\mu$ and $\sigma$ with this censored data, with arbitrary upper bounds $\{c_i\}_{i=1}^k$?

We can suppose that we know the support of $X$.

I looked at this source about censored data, but its prescriptions are not clear, and I cannot access the source they cite:

If all observations are censored, the MLE is undefined when there is a single censoring limit and defined but extremely poorly estimated when there are multiple censoring limits. The best approach for such data is to report the median, calculated as the median of the censoring levels (Helsel 2012, p. 143-4)

I guess this suggests taking the median of $c_1,…,c_k$?

My intuition says that you can use information about how the estimated moments of the samples related to the limit $c_i$. Suppose we know that the support of $X$ is $[a,b]$, then, as $c_i \rightarrow b$ we should expect the sample moments to be unbiased estimates of the population moments.

Hence, one thing I might try is the following. I could generate estimates for each $i$:

$$\{\hat{\mu}_i, \hat{\sigma}_i\, c_i\}_{i=1}^k$$

I could then model this as follows:

$$\hat{\mu}_i = \alpha_0 + \alpha_1 c_i $$ $$\hat{\sigma}_i = \beta_0 + \beta_1 c_i $$

which I could estimate using, say, OLS. Knowing the support of $X$, I can predict what the value of $\hat{\mu}_i$ and $\hat{\sigma}_i$ would be if $c_i \rightarrow b$. That is:

$$\underset{c_i \rightarrow b}{lim}[\hat{\mu}_i]= \hat{\alpha_0} + \hat{\alpha_1} b$$ $$\underset{c_i \rightarrow b}{lim}[\hat{\sigma}_i]= \hat{\beta_0} + \hat{\beta_1} b$$

Then, if $\hat{\mu}$ or $\hat{\sigma}$ relates to $c_i$ in this linear fashion, this should give unbiased estimates of the population moments. Of course, there's no reason to expect this linear relationship, but I don't currently have other ideas.

Suppose I draw $n$ samples of some random variable $X$. I repeat this process $k$ times so that I end up with $k \times n$ observations.

Each time I draw a random sample, my data is censored, meaning that I cannot observe values larger than some maximum value. Specifically, the values observed is bounded from above by $c_i$ for $i \in 1,…,k$, each time a sample is drawn, where $c_1$, $c_2$, … $c_k$ can be any positive real value.

We know that $X$ follows a lognormal distribution with parameters $\mu$ and $\sigma$.

How do I estimate $\mu$ and $\sigma$ with this censored data, with arbitrary upper bounds $\{c_i\}_{i=1}^k$?

We can suppose that we know the support of $X$.

I looked at this source about censored data, but its prescriptions are not clear, and I cannot access the source they cite:

If all observations are censored, the MLE is undefined when there is a single censoring limit and defined but extremely poorly estimated when there are multiple censoring limits. The best approach for such data is to report the median, calculated as the median of the censoring levels (Helsel 2012, p. 143-4)

I guess this suggests taking the median of $c_1,…,c_k$?

My intuition says that you can use information about how the estimated moments of the samples related to the limit $c_i$. Suppose we know that the support of $X$ is $[a,b]$, then, as $c_i \rightarrow b$ we should expect the sample moments to be unbiased estimates of the population moments.

Suppose I draw $n$ samples of some random variable $X$. I repeat this process $k$ times so that I end up with $k \times n$ observations.

Each time I draw a random sample, my data is censored, meaning that I cannot observe values larger than some maximum value. Specifically, the values observed is bounded from above by $c_i$ for $i \in 1,…,k$, each time a sample is drawn, where $c_1$, $c_2$, … $c_k$ can be any positive real value.

We know that $X$ follows a lognormal distribution with parameters $\mu$ and $\sigma$.

How do I estimate $\mu$ and $\sigma$ with this censored data, with arbitrary upper bounds $\{c_i\}_{i=1}^k$?

We can suppose that we know the support of $X$.

I looked at this source about censored data, but its prescriptions are not clear, and I cannot access the source they cite:

If all observations are censored, the MLE is undefined when there is a single censoring limit and defined but extremely poorly estimated when there are multiple censoring limits. The best approach for such data is to report the median, calculated as the median of the censoring levels (Helsel 2012, p. 143-4)

I guess this suggests taking the median of $c_1,…,c_k$?

My intuition says that you can use information about how the estimated moments of the samples related to the limit $c_i$. Suppose we know that the support of $X$ is $[a,b]$, then, as $c_i \rightarrow b$ we should expect the sample moments to be unbiased estimates of the population moments.

Hence, one thing I might try is the following. I could generate estimates for each $i$:

$$\{\hat{\mu}_i, \hat{\sigma}_i\, c_i\}_{i=1}^k$$

I could then model this as follows:

$$\hat{\mu}_i = \alpha_0 + \alpha_1 c_i $$ $$\hat{\sigma}_i = \beta_0 + \beta_1 c_i $$

which I could estimate using, say, OLS. Knowing the support of $X$, I can predict what the value of $\hat{\mu}_i$ and $\hat{\sigma}_i$ would be if $c_i \rightarrow b$. That is:

$$\underset{c_i \rightarrow b}{lim}[\hat{\mu}_i]= \hat{\alpha_0} + \hat{\alpha_1} b$$ $$\underset{c_i \rightarrow b}{lim}[\hat{\sigma}_i]= \hat{\beta_0} + \hat{\beta_1} b$$

Then, if $\hat{\mu}$ or $\hat{\sigma}$ relates to $c_i$ in this linear fashion, this should give unbiased estimates of the population moments. Of course, there's no reason to expect this linear relationship, but I don't currently have other ideas.

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Tamay
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Estimating data distributionmoments of censored data with multiple bounds

Suppose I draw $n$ samples of some random variable $X$. I repeat this process $k$ times so that I end up with $k \times n$ observations.

Each time I draw a random sample, my data is censored, meaning that I cannot observe values larger than some maximum value. Specifically, the values observed is bounded from above by $c_i$ for $i \in 1,…,k$, each time a sample is drawn, where $c_1$, $c_2$, … $c_k$ can be any positive real value.

We know that $X$ follows a lognormal distribution with parameters $\mu$ and $\sigma$.

How do I estimate $\mu$ and $\sigma$ with this censored data, with arbitrary upper bounds $\{c_i\}_{i=1}^k$?

We can suppose that we know the support of $X$.

I looked at this source about censored data, but its prescriptions are not clear, and I cannot access the source they cite:

If all observations are censored, the MLE is undefined when there is a single censoring limit and defined but extremely poorly estimated when there are multiple censoring limits. The best approach for such data is to report the median, calculated as the median of the censoring levels (Helsel 2012, p. 143-4)

I guess this suggestsuggests taking the median of $c_1,…,c_k$?

My intuition says that you can use information about how the estimated moments of the samples related to the limit $c_i$. Suppose we know that the support of $X$ is $[a,b]$, then, as $c_i \rightarrow b$ we should expect the sample moments to be unbiased estimates of the population moments.

Estimating data distribution of censored data

Suppose I draw $n$ samples of some random variable $X$. I repeat this process $k$ times so that I end up with $k \times n$ observations.

Each time I draw a random sample, my data is censored, meaning that I cannot observe values larger than some maximum value. Specifically, the values observed is bounded from above by $c_i$ for $i \in 1,…,k$, each time a sample is drawn, where $c_1$, $c_2$, … $c_k$ can be any positive real value.

We know that $X$ follows a lognormal distribution with parameters $\mu$ and $\sigma$.

How do I estimate $\mu$ and $\sigma$ with this censored data, with arbitrary upper bounds $\{c_i\}_{i=1}^k$?

We can suppose that we know the support of $X$.

I looked at this source about censored data, but its prescriptions are not clear:

If all observations are censored, the MLE is undefined when there is a single censoring limit and defined but extremely poorly estimated when there are multiple censoring limits. The best approach for such data is to report the median, calculated as the median of the censoring levels (Helsel 2012, p. 143-4)

I guess this suggest taking the median of $c_1,…,c_k$?

Estimating moments of censored data with multiple bounds

Suppose I draw $n$ samples of some random variable $X$. I repeat this process $k$ times so that I end up with $k \times n$ observations.

Each time I draw a random sample, my data is censored, meaning that I cannot observe values larger than some maximum value. Specifically, the values observed is bounded from above by $c_i$ for $i \in 1,…,k$, each time a sample is drawn, where $c_1$, $c_2$, … $c_k$ can be any positive real value.

We know that $X$ follows a lognormal distribution with parameters $\mu$ and $\sigma$.

How do I estimate $\mu$ and $\sigma$ with this censored data, with arbitrary upper bounds $\{c_i\}_{i=1}^k$?

We can suppose that we know the support of $X$.

I looked at this source about censored data, but its prescriptions are not clear, and I cannot access the source they cite:

If all observations are censored, the MLE is undefined when there is a single censoring limit and defined but extremely poorly estimated when there are multiple censoring limits. The best approach for such data is to report the median, calculated as the median of the censoring levels (Helsel 2012, p. 143-4)

I guess this suggests taking the median of $c_1,…,c_k$?

My intuition says that you can use information about how the estimated moments of the samples related to the limit $c_i$. Suppose we know that the support of $X$ is $[a,b]$, then, as $c_i \rightarrow b$ we should expect the sample moments to be unbiased estimates of the population moments.

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