Estimates the location and scaling parameters of the latent variables from existing survey data.

## Arguments

- data
survey data with columns representing individual items. Apart from this,

`data`

can be of almost any class such as "data.frame" "matrix" or "array".- n_levels
number of response categories, a vector or a number.

- skew
marginal skewness of latent variables, defaults to 0.

## Details

The relationship between the continuous random variable \(X\) and the discrete probability distribution \(p_k\), for \(k = 1, \dots, K\), can be described by a system of non-linear equations: $$ p_{k} = F_{X}\left( \frac{x_{k - 1} - \xi}{\omega} \right) - F_{X}\left( \frac{x_{k} - \xi}{\omega} \right) \quad \text{for} \ k = 1, \dots, K $$ where:

- \(F_{X}\)
is the cumulative distribution function of \(X\),

- \(K\)
is the number of possible response categories,

- \(x_{k}\)
are the endpoints defining the boundaries of the response categories,

- \(p_{k}\)
is the probability of the \(k\)-th response category,

- \(\xi\)
is the location parameter of \(X\),

- \(\omega\)
is the scaling parameter of \(X\).

The endpoints \(x_{k}\) are calculated by discretizing a random variable \(Z\) with mean 0 and standard deviation 1 that follows the same distribution as \(X\). By solving the above system of non-linear equations iteratively, we can find the parameters that best fit the observed discrete probability distribution \(p_{k}\).

The function `estimate_params`

:

Computes the proportion table of the responses for each item.

Estimates the probabilities \(p_{k}\) for each item.

Computes the estimates of \(\xi\) and \(\omega\) for each item.

Combines the estimated parameters for all items into a table.

## See also

`discretize_density`

for details on calculating
the endpoints, and `part_bfi`

for example of the survey data.