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Consider the example of estimating the number of items in the basket of a supermarket shopper. If we assume that a supermarket shopper will always buy at least one item, then a typical approach would be zero-truncated Poisson regression.

However, this has its drawbacks as zero-truncated Poisson regression makes it for example difficult to implement fixed effexts.

So why couldn't we model the number of additional items in the basket of the shopper? Then the dependent variable would start at 0 and we could fall back to standard Poisson regression.

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    $\begingroup$ Have you compared the probability mass function of your proposed shifted distribution with that of the truncated Poisson? $\endgroup$
    – Scortchi ♦
    Commented Jul 20, 2017 at 8:38
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    $\begingroup$ The answer at the indicated duplicate discusses the Poisson toward the end; see the plot there in particular. $\endgroup$
    – Glen_b
    Commented Jul 20, 2017 at 10:10
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    $\begingroup$ See also stats.stackexchange.com/q/179426/17230 - the means & variances of the shifted & truncated distributions are quite different for small rate parameters. $\endgroup$
    – Scortchi ♦
    Commented Jul 20, 2017 at 10:21

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