# Methods and formulas for Warranty Prediction

## Summary of current warranty claims

After reformatting the data with Pre-Process Warranty Data, the data are interval censored, grouped in intervals of the form (t0, t1), (t1, t2),...,(tk-1, tk) such that each interval (ti-1, ti) contains ni failures (if ti is finite) or ni suspensions (if ti is infinite), i = 1, 2,..., k.

Total number of units = the total number of units shipped up to the present time

Observed number of failures = the number of shipped units that failed during the warranty period

If you do not specify a length of warranty (L), the expected number of failures (ENF) is given by:
where IC = 1 if condition C is met, IC = 0 otherwise.
If you specify a length of warranty (L), the expected number of failures is given by:

Number of units at risk for future time periods = the total number of right censored units under warranty

### Notation

TermDescription
R(t)the reliability function

## Predicted future failures

Calculations for the expected number of future failures are based only on "suspended units" (right-censored units). Units that have already failed have no impact on future failures.

The predicted number of failures (PNF) for an additional period of time Δ is given by:
If you specify production quantities d1, d2,...,dr for future time periods 1, 2,...,r, the PNF for any future time period Δ is given by:
where q = min{r, int(Δ)} and int(Δ) is the integer part of Δ.
If you specify a warranty limit L, then only units still within the warranty period contribute to PNF, which is given by:
where IC = 1 if condition C is met, and IC = 0 otherwise
If you specify a warranty limit L and production quantities d1, d2,...,dr for future time periods 1, 2,...,r, then the PNF for any future time period Δ is given by:
where q = min{r, int(Δ)} and int(Δ) is the integer part of (Δ).

### Notation

TermDescription
tithe suspension times
nithe number of units suspended at time ti, i = 1, 2,...,m
m the number of distinct suspension times

## Confidence intervals for the expected number of failures

An approximate 100(1-α)% confidence interval for the predicted number of failures (x) is given by:
An approximate one-sided 100(1-α)% lower confidence bound is given by:
An approximate one-sided 100(1-α)% upper confidence bound is given by:

These confidence intervals and bounds are based on the assumption that failures occur according to an approximate Poisson process with a constant rate.

### Notation

TermDescription
s calculated predicted number of failures (the statistic)
xtrue predicted number of failures (the parameter)
the 100(1-α)th percentile of the chi-square distribution with f degrees of freedom
αthe level of significance (alpha)

## Graphs

Predicted Number of Failures Plot
The predicted number of failures are plotted against future time periods. The range of the x-axis is the range of future time periods. If you do not specify future time periods (that is, if the PREDICT subcommand is not given), the range of the x-axis is (0, 5].
Predicted Cost of Failures Plot
If you specify an average cost per failure (that is, if you use the COST subcommand), the predicted cost of failures are plotted against future time periods. The range of the x-axis is the range of future time periods. If you do not specify future time periods (that is, if the PREDICT subcommand is not given), the range of the x-axis is (0, 5].
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