IRI distribution assessment plan

Author

Boris Lebedenko

Problem definition

  • The primary objective of this project is to elucidate the factors which affect Inventory Record Inaccuracy (IRI).

  • In order to achieve this objective via regression modeling, the distributions of the IRI’s need to be studied.

  • We focus on parametric models previously suggested in the literature, both continuous and discrete.

Data

  • The data will be collected from \(M=86\) branches about \(n=15\) items.

  • For each item there are 6 types of IRI, each to be examined separately.

Table 1: Types of IRI and their notation
IRI type Notation Suggested distributions
Shelf s
Backroom shelf b
Checkout w Normal
Data capture c Log-normal
Visual complexity v Log-normal
Total T

  • Denote by \(\varepsilon^{(k)}_{j,i}\) the k-th type of IRI for branch \(j\) \((j = 1,2,\dots, M)\) and item type \(i\) \((i=1,2,\dots,n)\).

    • We omit the upper index \(^{(k)}\) for notation brevity.

Assumption 1: The errors for each item are independent and identically distributed within a branch, i.e., \(\varepsilon_{j,1} \sim \varepsilon_{j,2} \sim \dots \sim \varepsilon_{j,15} \sim \mathcal{F}_{\theta_j}\).

Assumption 2: The errors are independent and identically distributed across all branches and items, i.e., \(\varepsilon_{j_1,i_1} \sim \varepsilon_{j2,i2} \sim \mathcal{F}_\theta\) for every \(j_1 \neq j_2\) and \(i_1 \neq i_2\).

Simultaneous testing for normality

We are testing the following hypotheses:

\(H_0\): For every \(j=1,2,\dots,86\):

\[\varepsilon_{j1},\dots,\varepsilon_{jn} \sim \mathcal{N}(\mu_j,\sigma^2_j)\]

\(H_1\): Otherwise, i.e., for some \(j\), \(\varepsilon_{ji}\) is not normally-distributed.

The following sections describe proposed methods to test these hypotheses.

Approach 1. Standardize-Combine-Assess

This approach relies on the assumption that all 86 branches share the same shape and differ only in mean and variance.

  1. For the \(j\)-th branch calculate:

\[\hat{\mu}_j = \frac{1}{n}\sum_{i=1}^{n}\varepsilon_{ji},\quad \hat{\sigma}^2_j = \frac{1}{n-1}\sum_{i=1}^n(\varepsilon_{ji} - \hat{\mu}_j)^2.\] 2. Obtain standardized values:

\[z_{ji} = \frac{\varepsilon_{ji}-\hat{\mu}_j}{\hat{\sigma}_j}.\]

  1. Collate all standardized values into one big vector \(\mathbf{z}\in \mathbb{R}^{86\times15}\).

  2. Use the Shapiro-Wilk/Shapiro-Francia/Anderson-Darling test to assess the normality of \(\mathbf{z}\).

Approach 2. Meta-analytic multiple testing

  1. Compute a normality test (Shapiro-Wilk) p-value for each sample \(j\): \(p_j\).

  2. Use Fisher’s method to obtain a test statistic for the combined p-values:

\[\chi^2 = -2\sum_{j=1}^{86}\log(p_j).\] This test statistic is chi-squared distributed with \(2\times 86\) degrees of freedom under \(H_0\). A large test statistic implies that the combined p-value is small.

Note: This method inherently tests “are any non-normal?” If all p-values are modestly non-significant, the combined test likely won’t reject. However, if any single sample is severely non-normal, it may pull down the combined p-value.

Simultaneous testing for a specified parametric family

In this section we formulate testing frameworks for a specific parametric family \(\mathcal{F}_\theta\) for each class of IRI.

Approach 1. Pooled approach

Assumption \(\theta_1 = \theta_2 = \dots = \theta_M\), i.e., the parameter of the IRI distribution is equal for all \(M\) branches we can employ the following approach:

  1. Use Maximum Likelihood Estimation on the entire sample vector \(\mathbf{\varepsilon}\) to obtain the estimator \(\hat{\theta}\).

  2. Use a goodness of fit test (Pearson’s chi-squared for discrete and Kolmogorov-Smirnov) to assess whether the distribution is appropriate. Use a visualization like a QQ-plot or a histogram with an overlay of the density.

Approach 2. Meta-analytic multiple testing

Same framework as in the normal case with the appropriate estimation scheme for \(\theta\).

Selecting distribution family for IRI from specified alternatives

Some IRI types require choosing a parametric family. We consider the following alternatives:

  • Normal
  • Laplace
  • Log-normal
  • Beta

  1. For each candidate model obtain an estimate \(\hat{\theta}\) of the parameter \(\theta\) via Maximum Likelihood based on \(\mathbf{\varepsilon}\).

  2. Compute the log-likelihood for each fitted distribution: \[\ell(\hat{\theta}) = \sum_{t=1}^{M\times n} \ln\left(f(x_t\mid\hat{\theta})\right)\]

  3. Calculate the AIC (Akaike Information Criterion): \[\text{AIC} = -2\ell(\hat{\theta}) + 2p\] where \(p\) is the number of estimated parameters.

  4. Select the distribution that yields the lowest AIC (implying a better fit to the data).

  5. Validate the fit using a goodness-of-fit test and a visual assessment.