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At distinct observed event times $t_1<t_2<\cdots$, let $n_j$ be the risk set size immediately before $t_j$, and let $d_j$ count events at $t_j$. The Kaplan–Meier estimator of the survival function isThe empty product is one. Each factor estimates a conditional survival probability among those still observed and event-free; their product joins the successive steps. This is nonparametric statistics: no specified shape for the survival distribution is imposed. Use independent censoring and the appropriate risk sets to interpret the result as an estimate for the intended population.

The Kaplan–Meier estimator is a right-continuous function with downward jumps at event times. Right censoring changes later denominators without causing a jump. A tick on the curve marks right censoring, rather than a failure. For a tied group of $d_j$ events, use the one factor $(n_j-d_j)/n_j$; subjects right-censored at that recorded time remain in $n_j$ for this factor. The fraction $d_j/n_j$ estimates a discrete hazard, not a continuous hazard function in inverse-time units. With no censoring, the factors telescope to the fraction of the original cohort whose observed survival times exceed $t$, the complement of its empirical distribution function.

Edward L. Kaplan and Paul Meier published ^[ref]. Their paper addresses lifetime estimation when observation ends before failure and explicitly calls attention to the assumption of independence between lifetime and potential observation loss. The date 1958 is the original publication year; the publisher's later online-posting date is not the discovery date.

In his ^[ref], Kaplan traces Meier's interest to cancer-duration data in 1952 and his own to the lifetimes of vacuum tubes in undersea telephone-cable equipment a year later. The journal encouraged them to combine their manuscripts. The shared mathematical question was how to estimate a survival function when some lifetimes are still running or disappear from observation.

Use the usual two mutually independent samples, and rank their $N=n+m$ pooled observations increasingly. Continuity gives no ties with probability one. Let $R_i$ be the pooled rank of $X_i$ and define the Wilcoxon rank sum test statisticThe last identity counts the ranks contributed by the other sample after subtracting the within-$X$ ranks.

When $F_X\ge F_Y$, the $X$ observations are stochastically smaller, so reject for small $W_X$, equivalently small $U_X$. For instance,The inequality follows from $F_X\ge F_Y$ and $\int F_{Y}d{F}_Y=1/2$ for continuous $F_Y$. A common-uniform quantile coupling also makes this direction transparent: $F_X^{-1}(u)\le F_Y^{-1}(u)$, and $U_X$ is nondecreasing in each $X_i$.

Under the common-distribution null hypothesis, the pooled observations are exchangeable. Every selection of $n$ ranks for the $X$ labels has probability ${\left(\genfrac{}{}{0ex}{}{N}{n}\right)}^{-1}$. Thus the exact null law isChoose a lower-tail critical value with probability at most the desired size, optionally randomizing at its boundary to attain the size exactly. This law is independent of the unknown common distribution. Common increasing transformations leave the pooled ranks of observations unchanged, so the statistic is a function of the maximal invariant from part (i). Although a rank sum is not itself maximal, it is an invariant test statistic. These facts justify the test within nonparametric statistics.

Sampling ranks without replacement givesso ${\mathbb{E}}_0{U}_X=nm/2$ with the same variance. For $n,m\to \infty$ with $n/N\to \rho \in (0,1)$, the normal approximation for a rank sum statesEquivalently replace $W_X-n(N+1)/2$ by $U_X-nm/2$. The large-sample one-sided test rejects when this standardized statistic is below the lower $\alpha$ normal distribution quantile. This is the two-sample rank-sum test, rather than the paired Wilcoxon signed-rank test.

Let a group $G$ act on the sample space. A statistic $T$ is a maximal invariant when it is invariant and separates group orbits:Together these conditions say that $T(x)=T(y)$ exactly when $x$ and $y$ belong to the same orbit.

The basic result is that invariant tests factor through a maximal invariant. Specifically, a test function $\phi (x)\in [0,1]$ is invariant exactly when it can be written $b(T(x))$. To prove the nontrivial direction, define $b(t)=\phi (x)$ for any representative with $T(x)=t$. Two representatives are in the same orbit by maximality, and invariance gives them the same test value, so $b$ is well-defined. Conversely $b(T(gx))=b(T(x))$ proves invariance. Measurability is taken on the quotient statistic space; for the finite rank spaces used below it is automatic.

If a null family is transitive under the group action, the null law of $T$ is parameter-free. Indeed, if $P_{{\theta }^{\prime }}$ is the law of $gX$ when $X\sim P_{\theta }$, then for every measurable set $B$,Thus invariant procedures use exactly the information in $T$, and its common null law calibrates tests without estimating the transformed nuisance distribution. This explains the role of maximal invariants in nonparametric statistics.

For simultaneous strictly increasing bijections of the real line, the labeled ranks of observations are a maximal invariant on samples without ties. Increasing maps preserve every comparison. Conversely, two such samples with the same ranks can be matched by a piecewise linear increasing bijection through their corresponding ordered observations, extended with positive-slope tails. This proves ranks as a maximal invariant under increasing transformations. Under independent sampling from any common continuous distribution, every labeled ordering is equally likely by exchangeability. Hence rank tests have distribution-free null calibration even without a global transitivity assertion for the entire class of continuous distributions.

A smoothing bandwidth $h>0$ determines the scale over which a regression kernel averages data, or a kernel for density estimation spreads each observation. It has the same units as the observations. In kernel density estimation, changing $h$ changes the visible detail while preserving the total integral one. For a Gaussian density kernel, each bump has standard deviation $h$ and its individual contribution has peak height $1/(nh\sqrt{2\pi })$.

If the smoothing bandwidth is too small relative to the spacings in a statistical sample, the kernel density estimator shows narrow spikes tied to individual observations: apparent peaks may be sampling accidents. If it is too large relative to features of the unknown probability density function, it blurs distinct groups together and spreads fitted probability far from the observations. An intermediate smoothing bandwidth can reveal broad structure, but visual smoothness alone cannot establish accuracy.

The first three panels use exactly $(-1.8,-1.4,-1.1,-0.7,0.7,1.1,1.4,1.8)$ with smoothing bandwidths $0.08$, $0.35$ and $1.2$, respectively. Every curve has area one; the two groups in the observed statistical sample become less distinct as $h$ increases. No true probability density function is specified in this illustration, so the middle panel is illustrative, not a certified optimal fit. The fourth panel uses $(0.1,0.4,0.9)$ and $h=0.5$ to illustrate boundary bias of a kernel density estimator.

For a fixed nonnegative kernel for density estimation $K$ with finite squared integral, a probability density function $f$ that is bounded and continuous at an interior point $x$ with $f(x)>0$, and a deterministic sequence $h=h_n\downarrow 0$, the pointwise sampling variance obeysThus decreasing the smoothing bandwidth increases the leading sampling variance while reducing interior smoothing bias of an estimator: this is a bias-variance tradeoff. The conditions $h_n\to 0$ and $n{h}_n\to \infty$ express two goals, increasingly fine resolution and enough observations contributing locally. Practical methods such as plug-in bandwidth selection use the data to choose $h$; pointwise optimal kernel bandwidth depends on local features of the unknown probability density function. The smoothing bandwidth remains a tuning choice in nonparametric statistics, and knowing the support of a probability distribution can require more than a width adjustment near a boundary.

A bound on ${\sup }_x\mathbb{P}(|Z_n(x)|>t)$ does not give the same bound on $\mathbb{P}({\sup }_x|Z_n(x)|>t)$. Even independent standard normal random variables each have uniformly bounded tails while their Gaussian maximum over $N$ indices grows at scale $\sqrt{\log N}$. This distinction matters in nonparametric statistics when a shrinking bandwidth permits many spatial windows.