Why Zeros, Ties, and Boundaries Need Special Attention
The ordinary nonparametric bootstrap resamples observed units, with replacement, from the empirical distribution. When the data are continuous and the statistic behaves smoothly, the resulting bootstrap distribution can provide a useful approximation to sampling uncertainty. But empirical insurance data often contain many zeros, repeated recorded amounts, or estimates that reach a permitted boundary. In these cases, the bootstrap distribution may be discrete, with probability concentrated at a few values. That structure matters when interpreting standard errors and percentile intervals.
A point mass is positive probability at one exact value. A sample containing many zero claims, for example, gives the empirical distribution a point mass at zero. Resamples can then contain no positive claims at all. Ties also make bootstrap statistics repeat: several distinct resamples may yield the same sample median or estimated parameter. If a parameter estimate is constrained to be nonnegative, resamples may repeatedly return the boundary value zero.
The word conditional is important: after observing the data, the empirical distribution is fixed, and the bootstrap describes variation under repeated resampling from that distribution. It is not the unknown sampling distribution itself. The bootstrap distribution can be especially limited when the observed sample has little representation of a feature that matters, such as rare positive claims.
Atoms and Percentile Endpoints
For a discrete bootstrap statistic, define its cumulative distribution function by \(F^*(t)=P(T^*\leq t\mid x_1,\ldots,x_n)\). In this tutorial, the \(p\)-quantile is the smallest value \(t\) for which \(F^*(t)\geq p\). Thus, a percentile interval with nominal endpoints at probabilities 0.025 and 0.975 uses \(q_{0.025}^*\) and \(q_{0.975}^*\). Because the distribution can jump over a probability level, the interval can have endpoints that are far apart or can include a large amount of probability at an endpoint. Software that interpolates between ordered replicates may report different numerical endpoints; the quantile convention should therefore be stated.
This result follows directly from independent resampling. Each draw has probability \(r\) of selecting a zero, and all \(n\) draws must do so, giving \(r\times r\times\cdots\times r=r^n\). More generally, the number of draws from any selected subset with empirical probability \(r\) has a binomial distribution with parameters \(n\) and \(r\).
Worked Example: A Mean With a Large Zero Mass
A hypothetical insurer observes losses of \(0,0,\) and \(2\) thousand dollars for three independent, comparable exposure units. Use a nonparametric bootstrap of size three to find the exact distribution of the bootstrap sample mean and its 2.5th and 97.5th percentiles. Treat the recorded values as the complete data; no interest or other model assumption is involved.
Method. Each bootstrap draw is zero with probability \(2/3\) and 2 with probability \(1/3\). Let \(K\) be the number of 2s in a resample. Then \(K\sim\operatorname{Binomial}(3,1/3)\), and the bootstrap mean is \(\bar X^*=2K/3\), in thousands of dollars.
For \(K=0,1,2,3\), the probabilities are respectively \(8/27,12/27,6/27,1/27\). The corresponding means are \(0,2/3,4/3,2\) thousand dollars. The cumulative probabilities at these ordered values are \(8/27=0.2963\), \(20/27=0.7407\), \(26/27=0.9630\), and \(1\).
Since the cumulative probability at zero already exceeds 0.025, \(q_{0.025}^*=0\). The cumulative probability through \(4/3\) is \(26/27=0.9630\), below 0.975, while through 2 it is 1. Therefore \(q_{0.975}^*=2\) thousand dollars. The percentile interval is \([0,2]\) thousand dollars.
Check. The probabilities sum to \((8+12+6+1)/27=1\), and the interval endpoints are values in the support of the bootstrap distribution. The substantial zero probability reflects the two observed zero losses; it is not evidence that the true mean loss is zero.
Ties Can Make a Statistic Coarse
Repeated observations do not merely create repeated bootstrap data sets. They can make the statistic itself take only a few possible values. This is particularly visible for medians, quantiles, and other rank-based statistics. The percentile interval then depends on the jumps of the bootstrap cumulative distribution, rather than varying smoothly with each replicate.
The next example uses the usual sample median: for an even sample size, it is the average of the two middle ordered observations. That convention must be specified because alternate conventions for an even-sample median can give different bootstrap values.
Worked Example: Tied Values and a Bootstrap Median
A hypothetical portfolio analyst records four claim amounts, in thousands of dollars: \(1,1,3,3\). Resample four observations with replacement and calculate the exact bootstrap distribution of the median and its 2.5th and 97.5th percentiles. Use the even-sample median convention just stated.
Method. Let \(K\) be the number of 1s in a bootstrap sample. Since each draw is 1 with probability \(1/2\), \(K\sim\operatorname{Binomial}(4,1/2)\). If \(K=0\) or \(K=1\), the two middle values are both 3, so the median is 3. If \(K=2\), the middle values are 1 and 3, so the median is 2. If \(K=3\) or \(K=4\), the two middle values are both 1, so the median is 1.
Thus the bootstrap median is 1 with probability \((4+1)/16=5/16\), is 2 with probability \(6/16\), and is 3 with probability \((1+4)/16=5/16\). In increasing order, its cumulative probabilities are \(5/16=0.3125\), \(11/16=0.6875\), and 1. Both requested percentile levels fall in the jump at 1 for the lower endpoint and the jump at 3 for the upper endpoint: \(q_{0.025}^*=1\) and \(q_{0.975}^*=3\).
Check. The three probabilities sum to 1, and the bootstrap distribution is symmetric around the observed median of 2. The endpoints are not estimates of an underlying range of possible claim amounts; they summarize this particular conditional resampling distribution.
Boundary Estimates and Degenerate Replicates
A boundary estimate occurs when an estimate reaches the edge of its allowed parameter space. For example, a probability parameter \(p\) must lie in \([0,1]\), and a nonnegative severity parameter cannot be negative. Bootstrap behavior can be uninformative when the observed estimate is at a boundary: resampling cannot create an outcome that was absent from the empirical data.
For Bernoulli data, the maximum likelihood estimate of the event probability \(p\) is the observed proportion of events. If all observed outcomes are zero, the estimate is \(\hat p=0\). Every ordinary nonparametric bootstrap resample then also consists entirely of zeros, so every replicate estimate is zero. This is an exact statement about the empirical bootstrap, not a claim that the population probability is zero.
Worked Example: All Observations at a Parameter Boundary
Suppose a hypothetical administrator records whether each of five independent service contracts has a particular costly event. The outcomes are \(0,0,0,0,0\), where 1 denotes an event. Model each outcome as Bernoulli with event probability \(p\in[0,1]\), and estimate \(p\) by maximum likelihood. Find the nonparametric bootstrap distribution of the estimate and interpret a percentile interval.
Original estimate. The likelihood is proportional to \((1-p)^5\), which is maximized over \(0\leq p\leq1\) at \(p=0\). Equivalently, \(\hat p=0/5=0\).
Bootstrap calculation. The empirical distribution places probability 1 on zero. Thus every size-five bootstrap sample has zero events, and \(\hat p^*=0/5=0\) in every replicate. In particular, \(P(\hat p^*=0\mid\text{data})=1\), and the 2.5th and 97.5th bootstrap percentiles are both zero.
Interpretation and check. The interval \([0,0]\) describes no variation under resampling from these five observed zeros. It should not be read as proof that the population event probability is known to be zero: the empirical distribution contains no event to resample. The calculation is internally consistent—every replicate is the same—but that degeneracy is a warning that the ordinary nonparametric bootstrap cannot represent uncertainty about unobserved event outcomes.
Worked Example: A Boundary Atom With One Observed Event
Now suppose the five-contract setting is replaced by four independent Bernoulli observations, \(1,0,0,0\). Find the exact bootstrap distribution of the estimated event probability and its 2.5th and 97.5th percentiles. The parameter space remains \([0,1]\), and the bootstrap resample size is four.
Method. The observed estimate is \(\hat p=1/4\). If \(K\) is the number of events in a bootstrap sample, then \(K\sim\operatorname{Binomial}(4,1/4)\) and \(\hat p^*=K/4\). The probabilities for \(K=0,1,2,3,4\), using denominator 256, are \(81,108,54,12,1\), respectively.
The cumulative probabilities through \(K=0,1,2,3,4\) are \(81/256=0.3164\), \(189/256=0.7383\), \(243/256=0.9492\), \(255/256=0.9961\), and 1. Therefore the 2.5th percentile is \(0\), since the mass at zero alone exceeds 0.025. The cumulative probability through \(K=2\) is below 0.975, while through \(K=3\) it is above 0.975. Thus the 97.5th percentile is \(3/4=0.75\), and the percentile interval is \([0,0.75]\).
Check. The probabilities sum to \(256/256=1\). The zero estimate occurs with probability \(81/256\), even though the original estimate is positive. One observed event makes an all-zero resample possible, but not rare; the resulting percentile interval is highly discrete.
What These Features Mean for Practice
A point mass, a tie, or a boundary does not automatically invalidate every bootstrap analysis. It does mean that the shape and support of the bootstrap distribution deserve direct inspection before reporting an interval. The percentile endpoints may be determined by one large atom, and adding more bootstrap replicates only estimates that conditional distribution more accurately; it does not add information absent from the original data.
- Check the data support. Record the proportion of zeros and the number of distinct observed values. For an all-zero sample, verify whether the statistic must be zero in every resample.
- Inspect replicate frequencies. Tabulate how often the statistic equals each value, especially at a boundary. A histogram can conceal a large atom; exact frequencies are often more informative.
- State the quantile convention. Discrete cumulative probabilities jump. Give the definition used for percentile endpoints and be aware that software interpolation can change the reported values.
- Separate resampling precision from inferential adequacy. A large number of bootstrap replicates reduces Monte Carlo noise in the estimated bootstrap distribution, but cannot restore unobserved positive outcomes or justify a degenerate interval.
- Check whether the data-generating model matches the feature. If zeros are a real structural part of the process, a model or resampling design should preserve them. If the sample has no rare events, a nonparametric bootstrap alone cannot reveal their unseen frequency.
A common exam trap is to interpret the bootstrap percentile interval as automatically informative because its endpoints have familiar probability labels. For the all-zero Bernoulli sample above, the calculation \(q_{0.025}^*=q_{0.975}^*=0\) is correct, but concluding that uncertainty is zero is not. The interval is degenerate because the empirical distribution puts all its mass at zero. The calculation answers what happens under empirical resampling; it does not establish that empirical resampling captures all relevant sampling uncertainty.
Check Your Understanding
For each problem, use resampling with replacement from the stated empirical data and the inverse-CDF quantile convention defined above.
- A sample of three hypothetical losses, in thousands of dollars, is \(0,0,3\). Resample three observations and let \(K\) be the number of positive losses. Find \(P(\bar X^*=0)\) and the 95th percentile of \(\bar X^*\). (Answer: \(P(\bar X^*=0)=8/27=0.2963\); \(q_{0.95}^*=2\) thousand dollars.)
- The observed values are \(2,2,6,6\). Resample four values and use the average-of-middle-two median. What is the probability that the bootstrap median is 4? (Answer: \(6/16=0.3750\).)
- Five Bernoulli observations contain two events and three non-events. What is the probability that a size-five bootstrap replicate has estimated event probability zero? (Answer: \((3/5)^5=243/3125=0.0778\).)
- Four observed losses, in thousands of dollars, are \(0,0,0,4\). Resample four observations and use the bootstrap mean. Find the probability that it is zero and its 97.5th percentile. (Answer: \(P(\bar X^*=0)=(3/4)^4=81/256=0.3164\); \(q_{0.975}^*=3\) thousand dollars.)
- For observed values \(1,1,2\), resample three values and use the sample maximum as the statistic. Find the probability that the bootstrap maximum is 1 and the 90th percentile of the bootstrap maximum. (Answer: \(P(\max^*=1)=(2/3)^3=8/27=0.2963\); \(q_{0.90}^*=2\).)